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This article is cited in 5 scientific papers (total in 5 papers) Elements of hyperbolic theory on an infinite-dimensional torus S. D. Glyzina, A. Yu. Kolesov a Demidov Yaroslavl' State University
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     IntroductionThe history of the development of hyperbolic theory and its main achievements were thoroughly described in the surveys [1] and [2] and the monographs [3]–[12] (of course, this list is not full). As regards infinite-dimensional hyperbolic dynamical systems, many authors have repeatedly advanced to their investigation (for instance, see [13]–[17]), and our paper is part of this trend. This is a continuation of our series of papers [18]–[22], and it is devoted to the foundation of hyperbolic theory on an infinite-dimensional torus. Our motivation for developing such a theory was its possible applications to dynamical systems with infinite-dimensional phase space. As for the infinite-dimensional torus, it is the most natural model example of an infinite-dimensional manifold. This paper consists of three sections. In the first we give the definitions of an abstract integer lattice $\mathbb{Z}^{\infty}$ and an infinite-dimensional torus $\mathbb{T}^{\infty}$. In particular, we show that this torus is an analytic Banach manifold, closed and non-compact. In addition, we introduce such basic concepts as a tangent space, a differential, a diffeomorphism, and we describe a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ which we consider in what follows. We prove a theorem on the structure of maps in this class. In the second section we present a criterion of hyperbolicity for the class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$. First of all we give the definition of hyperbolicity and establish a number of auxiliary results on hyperbolic diffeomorphisms (the boundedness away from zero of the angle between the stable and unstable subspaces, the uniform boundedness of the relevant projections, and so on). Next we describe the hyperbolicity criterion itself and, using it, prove a result that for diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ the property of hyperbolicity is $C^1$-rough. We also analyze a non-trivial example of a hyperbolic diffeomorphism in $\operatorname{Diff}(\mathbb{T}^{\infty})$. The third section, which is technically the most cumbersome one, contains the proofs of two cornerstone results of hyperbolic theory: the Hadamard–Perron theorem on the existence of stable and unstable local manifolds and the theorem on the existence of a stable and an unstable invariant foliation. Both results turn out to hold for any hyperbolic diffeomorphism in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$ which we introduce. 1. Main constructions1.1. The definition of an infinite-dimensional torusSince we present elements of hyperbolic theory on an infinite-dimensional torus $\mathbb{T}^{\infty}$, first, following [18]–[22], we give the definition of $\mathbb{T}^{\infty}$ itself. To do this we fix an infinite-dimensional real Banach space $E$ with norm $\|\,{\cdot}\,\|$ and start by presenting the definition of an infinite-dimensional integer lattice $\mathbb{Z}^{\infty}$. Definition 1.1. By the infinite-dimensional integer lattice (or simply the integer lattice) we mean a non-empty subset $\mathbb{Z}^{\infty}$ of $ E$ that satisfies the following axioms. 1) Linearity: for any $l_1,l_2\in\mathbb{Z}^{\infty}$ and $k_1,k_2\in\mathbb{Z}$ we have $k_1l_1+k_2l_2\in\mathbb{Z}^{\infty}$. 2) Discreteness:
$$
\begin{equation}
\mu_0\overset{\rm def}{=}\inf_{l_1, l_2\in\mathbb{Z}^{\infty},\ l_1\ne l_2} \|l_1-l_2\|>0.
\end{equation}
\tag{1.1}
$$
3) The closure of the linear span of the vectors in $\mathbb{Z}^{\infty}$ is the original space $E$ (this property can naturally be called maximality). In constructing particular examples of integer lattices the following notion of a kernel is useful. Namely, a kernel $\Omega$ of a lattice $\mathbb{Z}^{\infty}$ is a non-empty subset of $\mathbb{Z}^{\infty}$ such that each vector $l\in\mathbb{Z}^{\infty}$ is a finite linear combination of vectors in $\Omega$ with integer coefficients. Clearly, the so-called maximal kernel $\Omega=\mathbb{Z}^{\infty}$ always exists. Now we say that a kernel $\Omega$ is minimal if it consists of linearly independent vectors. A characteristic feature of a minimal kernel $\Omega$ is as follows: for any subset $\Omega_0\subset\Omega$ that is also a kernel we must have $\Omega_0=\Omega$. In general the question of the existence of a minimal kernel is open; however, in some particular cases such a kernel can be presented explicitly. One example of an integer lattice in the space $\ell_{p}$, $p\geqslant 1$, of vectors
$$
\begin{equation}
\begin{gathered} \, \varphi=\operatorname{colon}(\varphi_{(1)},\varphi_{(2)},\dots, \varphi_{(k)},\dots),\qquad \varphi_{(k)}\in \mathbb{R},\quad k\geqslant 1, \\ \|\varphi\|\overset{\rm def}{=} \biggl(\,\sum_{k=1}^{\infty}|\varphi_{(k)}|^{p}\biggr)^{1/p}<\infty, \end{gathered}
\end{equation}
\tag{1.2}
$$
is the set
$$
\begin{equation}
\mathbb{Z}^{\infty}=\bigl\{l=\operatorname{colon}(l_{(1)},l_{(2)},\dots, l_{(k)},\dots) \in\ell_{p}\colon l_{(k)}\in\mathbb{Z},\ k\geqslant 1\bigr\}.
\end{equation}
\tag{1.3}
$$
As the series $\sum^{\infty}_{k=1}|l_{(k)}|^p$ is convergent (see (1.2)), each vector $l\in\mathbb{Z}^{\infty}$ has only a finite number of non-zero components. As regards a kernel $\Omega$, here we can take it to be the set $\{e_k, k\in\mathbb{N}\}$ (where $e_k$ is the vector with $k$th component equal to one and all the others equal to zero). Now note that these vectors are linearly independent, and therefore this kernel is minimal. For the space $\ell_{\infty}$ of vectors
$$
\begin{equation}
\begin{gathered} \, \varphi=\operatorname{colon}(\varphi_{(1)},\varphi_{(2)},\dots, \varphi_{(k)},\dots),\qquad \varphi_{(k)}\in \mathbb{R},\quad k\geqslant 1, \\ \|\varphi\|\overset{\rm def}{=} \sup_{k\geqslant 1}|\varphi_{(k)}|<\infty, \end{gathered}
\end{equation}
\tag{1.4}
$$
an integer lattice similar to (1.3) has the form
$$
\begin{equation}
\mathbb{Z}^{\infty}=\{l=\operatorname{colon}(l_{(1)},l_{(2)},\dots, l_{(k)},\dots)\in \ell_{\infty}\colon l_{(k)}\in\mathbb{Z},\ k\geqslant 1\}.
\end{equation}
\tag{1.5}
$$
Here a kernel is the set $\Omega=\operatorname{Bin}(\ell_{\infty})$ of so-called binary vectors $l\in\ell_{\infty}$ with components $l_{(k)}$, $k\geqslant 1$, taking the values $0$ and $1$ independently of one another. In fact, consider a vector $l\in\mathbb{Z}^{\infty}$ and note that, as the sequence $l_{(k)}$ in (1.5) is bounded, there exists a finite tuple of pairwise distinct integers $m_1,m_2,\dots,m_s$ such that
$$
\begin{equation*}
l_{(k)}\in\{m_1,m_2,\dots,m_s\}\quad \forall\,k\geqslant 1.
\end{equation*}
\notag
$$
This means that $l=m_1e_{m_1}+m_2e_{m_2}+\cdots+m_se_{m_s}$, where the binary vectors $e_{m_j}$, $1\leqslant j\leqslant s$, are defined by
$$
\begin{equation*}
e_{m_j}=\operatorname{colon}(e_{m_j}^1,e_{m_j}^2,\dots, e_{m_j}^k,\dots),\qquad e_{m_j}^k=\begin{cases} 0&\text{for}\ l_{(k)}\ne m_j, \\ 1&\text{for}\ l_{(k)}=m_j. \end{cases}
\end{equation*}
\notag
$$
Also note that the set $\operatorname{Bin}(\ell_{\infty})$ is certainly not a minimal kernel because, for example, $\operatorname{Bin}(\ell_{\infty})\setminus\{l_0\}$, where $l_0=\operatorname{colon}(1,1,\dots,1,\dots)$, is also a kernel. The question of whether or not a minimal kernel exists in this case is open, just as in general. We can also define an analogue of an integer lattice (1.5) in the Lebesgue space $L_{\infty}(0,1)$ of classes of measurable functions $x(t)$ such that the norm
$$
\begin{equation*}
\|x\|= \operatorname{ess\,sup} _{0\leqslant t\leqslant 1}|x(t)|
\end{equation*}
\notag
$$
is finite. Namely, now the set
$$
\begin{equation*}
\mathbb{Z}^{\infty}=\{x(t)\in L_{\infty}(0,1)\colon x(t)\in \mathbb{Z} \text{ for almost all } t\in [0,1]\}
\end{equation*}
\notag
$$
is an integer lattice. Another natural example of an integer lattice is constructed as follows. Let $E$ be the infinite-dimensional real Hilbert space and let $\{e_{\alpha}\in E\colon \alpha\in\Sigma\}$ be a complete orthonormal system in it (where the index set $\Sigma$ is known to be infinite). Then it is easy to verify that this system is a minimal kernel for the corresponding integer lattice $\mathbb{Z}^{\infty}$. Elements of this lattice are arbitrary finite linear combinations of vectors $e_{\alpha}$ with integer coefficients. We turn to the definition of the torus $\mathbb{T}^{\infty}$. In this connection, throughout what follows we assume that we have fixed an integer lattice $\mathbb{Z}^{\infty}$ in $E$. Using it we introduce an equivalence relation on $E$ as follows. We say that two vectors $x, y\in E$ are equivalent if there exists $l\in\mathbb{Z}^{\infty}$ such that $x-y=2\pi l$. Definition 1.2. By an infinite-dimensional torus $\mathbb{T}^{\infty}$ we mean the set of equivalence classes generated by the above relation. In other words we have $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}=\operatorname{pr}(E)$ where $\operatorname{pr}\colon E\to\mathbb{T}^{\infty}$ is the so-called natural projection. It acts by the formula where $\varphi$ is an arbitrary element of $E$ and $\{\varphi\}$ is an equivalence class in $\mathbb{T}^{\infty}$ containing $\varphi$. For brevity, in what follows we use the same letter $\varphi$ for a vector in $E$ and the corresponding class $\{\varphi\}\in\mathbb{T}^{\infty}$ (the particular object under consideration will always be clear from the context).We define a metric on $\mathbb{T}^{\infty}$ by
$$
\begin{equation}
\forall\,\varphi_1,\varphi_2\in\mathbb{T}^{\infty}\qquad \rho(\varphi_1,\varphi_2)= \inf_{l\in\mathbb{Z}^{\infty}}\|\operatorname{pr}^{-1}(\varphi_1)- \operatorname{pr}^{-1}(\varphi_2)+2\pi l\|,
\end{equation}
\tag{1.7}
$$
where we recall that $\|\,{\cdot}\,\|$ is the norm in $E$ and $\operatorname{pr}^{-1}(\varphi_1),\operatorname{pr}^{-1}(\varphi_2)\in E$ are arbitrary preimages of the points $\varphi_1,\varphi_2\in\mathbb{T}^{\infty}$. Since such preimages are defined up to addition of vectors of the form $2\pi l$, $l\in\mathbb{Z}^{\infty}$, the metric (1.7) is independent of the particular choice of them. Note also that by the discreteness (1.1) and formula (1.7) the map (1.6) is a local isometry, that is,
$$
\begin{equation}
\rho(\operatorname{pr}(\varphi_1),\operatorname{pr}(\varphi_2))= \|\varphi_1-\varphi_2\|\qquad \forall\,\varphi_1,\varphi_2\in E\colon\ \|\varphi_1-\varphi_2\|\leqslant\varepsilon_0,
\end{equation}
\tag{1.8}
$$
where $\varepsilon_0=\operatorname{const}\in(0,\pi\mu_0)$. So the metric space $(\mathbb{T}^{\infty},\rho)$ is complete. Below we need the concept of a fundamental set of the torus $\mathbb{T}^{\infty}$. This is a set $\mathscr{U}\subset E$ such that $\operatorname{pr}(\mathscr{U})=\mathbb{T}^{\infty}$ and the map $\operatorname{pr}\colon\mathscr{U}\to\mathbb{T}^{\infty}$ is bijective. We stress that such a set exists by the axiom of choice applied to the family of non-empty disjoint sets $\operatorname{pr}^{-1}(\varphi)$, $\varphi\in\mathbb{T}^{\infty}$ (here $\operatorname{pr}^{-1}(\varphi)$ denotes the full preimage of the element $\varphi\in\mathbb{T}^{\infty}$). In the case (1.2), (1.3) the domain
$$
\begin{equation}
\mathscr{U}=\{\varphi=\operatorname{colon}(\varphi_{(1)},\varphi_{(2)}, \dots,\varphi_{(k)},\dots)\in\ell_{p}\colon -\pi\leqslant\varphi_{(k)}<\pi,\ k\geqslant 1\}
\end{equation}
\tag{1.9}
$$
is fundamental, while in the case (1.4), (1.5)
$$
\begin{equation}
\mathscr{U}=\{\varphi=\operatorname{colon}(\varphi_{(1)},\varphi_{(2)}, \dots,\varphi_{(k)},\dots)\in\ell_{\infty}\colon -\pi\leqslant\varphi_{(k)}<\pi,\ k\geqslant 1\}
\end{equation}
\tag{1.10}
$$
is a fundamental domain. It is easy to see that the set (1.10) is bounded, but (1.9) does not have this property. We must note that usually authors mean an ‘infinite-dimensional torus’ to be the product of a countable system of circles with Tychonoff topology (for instance, see [16] or [23]–[25]). In this case an introduction of an appropriate metric makes of $\mathbb{T}^{\infty}$ a compact metric space. However, this definition does not suit our aims because such a torus is not a manifold. As hyperbolic theory is usually developed for smooth manifolds, the definition of an infinite-dimensional torus must be adapted accordingly. In our case the torus $\mathbb{T}^{\infty}$ has the required properties. Namely, the following theorem holds. Theorem 1.1. The infinite-dimensional torus $\mathbb{T}^{\infty}$ is an analytic Banach manifold. It is always closed and non-compact. In the case when $\mathbb{T}^{\infty}$ has a bounded fundamental set $\mathscr{U}$, this manifold is bounded (that is, the corresponding metric space $(\mathbb{T}^{\infty},\rho)$ is bounded). Proof. To define a differentiable structure on $\mathbb{T}^{\infty}$ we fix some fundamental set $\mathscr{U}$ for the torus, and for each $\varphi_0\in\mathbb{T}^{\infty}$ we let $u_0\in \mathscr{U}$ denote the corresponding preimage $\operatorname{pr}^{-1}(\varphi_0)$ (recall that the latter is well defined). Now consider the map
$$
\begin{equation}
h_{\varphi_0}(\varphi)=\operatorname{pr}^{-1}_{u_0}(\varphi),\qquad \varphi\in O(\varphi_0,r_0)\overset{\rm def}{=} \{\varphi\in\mathbb{T}^{\infty}\colon \rho(\varphi_0,\ \varphi)<r_0\},
\end{equation}
\tag{1.11}
$$
where $\operatorname{pr}^{-1}_{u_0}(\varphi)$ is a continuous branch of the corresponding multivalued map $\operatorname{pr}^{-1}(\varphi)$ such that $\operatorname{pr}^{-1}_{u_0}(\varphi_0)=u_0$, $r_0=\operatorname{const}\in (0,\varepsilon_0/6)$, and $\varepsilon_0$ is the constant in (1.8). We stress that by the choice of $r_0$ and the local isometry property (1.8) the homeomorphism (1.11) is well defined and takes the ball $O(\varphi_0,r_0)\subset \mathbb{T}^{\infty}$ to the analogous ball $O(u_0,r_0)=\{u\in E\colon \|u-u_0\|<r_0\}$ in $E$.
Using the local homeomorphisms (1.11) we can define the atlas of charts
$$
\begin{equation}
\{(O(\varphi_0,r_0),h_{\varphi_0}),\ \varphi_0\in\mathbb{T}^{\infty}\}
\end{equation}
\tag{1.12}
$$
on $\mathbb{T}^{\infty}$ in a natural way. Now we show that any two local maps in (1.12) are analytically compatible.
In fact, if for some $\varphi_1,\varphi_2\in\mathbb{T}^{\infty}$ we have
$$
\begin{equation}
O(\varphi_1, r_0)\cap O(\varphi_2,r_0)\ne\varnothing,
\end{equation}
\tag{1.13}
$$
then for the corresponding homeomorphisms $h_{\varphi_j}(\varphi)=\operatorname{pr}^{-1}_{u_j}(\varphi)$, where $u_j=\operatorname{pr}^{-1}(\varphi_j)$, $u_j\in\mathscr{U}$, $j=1,2$, we have the relation
$$
\begin{equation}
\forall\,\varphi\in O(\varphi_2,r_0)\quad h_{\varphi_2}(\varphi)=h_{\varphi_1}(\varphi)+2\pi l_0,\quad l_0=\operatorname{const}\in\mathbb{Z}^{\infty}.
\end{equation}
\tag{1.14}
$$
This is because, as $r_0$ is small, the map $h_{\varphi_1}(\varphi)$ extends to the ball $O(\varphi_1,3r_0)\supset O(\varphi_1,r_0)\cup O(\varphi_2,r_0)$ by continuity. On the other hand, since the $h_{\varphi_j}(\varphi)$, $j=1,2$, are distinct continuous branches of the same map $\operatorname{pr}^{-1}(\varphi)$, and the lattice $\mathbb{Z}^{\infty}$ is discrete, these branches differ one from the other by a constant $2\pi l_0$, $l_0\in \mathbb{Z}^{\infty}$.
Now consider the transition map (1.13)
$$
\begin{equation}
h_{\varphi_2}\circ h^{-1}_{\varphi_1}\colon h_{\varphi_1}(O(\varphi_1,r_0) \cap O(\varphi_2,r_0))\to h_{\varphi_2}(O(\varphi_1,r_0)\cap O(\varphi_2,r_0))
\end{equation}
\tag{1.15}
$$
corresponding to the case (1.13) and note that
$$
\begin{equation}
\forall\,u\in h_{\varphi_1}(O(\varphi_1,r_0)\cap O(\varphi_2,r_0))\quad h^{-1}_{\varphi_1}(u)=\operatorname{pr}(u).
\end{equation}
\tag{1.16}
$$
Combining (1.14) and (1.16) we conclude that
$$
\begin{equation*}
h_{\varphi_2}(h^{-1}_{\varphi_1}(u))=h_{\varphi_2}(\operatorname{pr}(u))= h_{\varphi_1}(\operatorname{pr}(u))+2\pi l_0=u+2\pi l_0.
\end{equation*}
\notag
$$
Thus, the map (1.15) can be expressed as $u\mapsto u+2\pi l_0$, so it is analytic in the local variable $u$.
We have thus shown that $\mathbb{T}^{\infty}$ is an analytic Banach manifold. The fact that it is closed (that is, has no boundary) now follows because any ball $O(\varphi_0,r_0)\subset\mathbb{T}^{\infty}$ is homeomorphic to some $O(u_0,r_0)\subset E$. As balls in $E$ are obviously non-compact, this also shows that $\mathbb{T}^{\infty}$ is non-compact. Finally, if $\mathbb{T}^{\infty}$ has a bounded fundamental set $\mathscr{U}$, then the metric space $(\mathbb{T}^{\infty},\rho)$ is also bounded. This is a direct consequence of (1.7), where we can always choose $\operatorname{pr}^{-1}(\varphi_1)$ and $\operatorname{pr}^{-1}(\varphi_2)$ in a bounded set $\mathscr{U}$. $\Box$ To complete our discussion of $\mathbb{T}^{\infty}$ we add that, since we can define the differentiable structure (1.12) on it, we can also carry over such concepts as tangent space, differential, diffeomorphism, and so on to this manifold. For example, the tangent space $T_{\varphi}\mathbb{T}^{\infty}$ to $\mathbb{T}^{\infty}$ at a point $\varphi\in\mathbb{T}^{\infty}$ is defined as follows. Consider various continuous curves $\varkappa(t)\in\mathbb{T}^{\infty}$, $t\in (-a,a)$, $a>0$, such that $\varkappa(0)=\varphi$. Assume that the function $h_{\varphi_0}(\varkappa(t))$ is differentiable at $t=0$ in some local chart $(O(\varphi_0,r_0),h_{\varphi_0})\colon \varphi\in O(\varphi_0,r_0)$ (and therefore in each such chart). Next we say that two curves $\varkappa_1(t)$ and $\varkappa_2(t)$ with these properties are equivalent if
$$
\begin{equation*}
\frac{d}{dt}h_{\varphi_0}(\varkappa_1(t))\bigg|_{t=0}= \frac{d}{dt}h_{\varphi_0}(\varkappa_2(t))\bigg|_{t=0}.
\end{equation*}
\notag
$$
As concerns the space $T_{\varphi}\mathbb{T}^{\infty}$, it is the set of equivalence classes $\{\varkappa\}$ obtained (see [4], § П 3). It is a characteristic feature of our case that there exists a natural isomorphism $\Gamma_{\varphi}\colon T_{\varphi}\mathbb{T}^{\infty}\to E$ between the classes $\{\varkappa\}$ and the vectors in $E$, which is defined by
$$
\begin{equation*}
\begin{aligned} \, \Gamma_{\varphi}\colon \{\varkappa\}&\mapsto e_{\{\varkappa\}} \overset{\rm def}{=}\frac{d}{dt}h_{\varphi_0}(\varkappa(t))\big|_{t=0}, \\ \Gamma^{-1}_{\varphi}\colon e\in E&\mapsto \{\operatorname{pr}[\operatorname{pr}^{-1}(\varphi)+t e]\}\in T_{\varphi}\mathbb{T}^{\infty}. \end{aligned}
\end{equation*}
\notag
$$
This definition is consistent in the following sense: the vector $e_{\{\varkappa\}}\in E$ is independent of the choice of a particular curve $\varkappa(t)$ in the corresponding class $\{\varkappa\}$ (because such curves are equivalent) and of the local homeomorphism $h_{\varphi_0}$ (because the transition from one homeomorphism to another is described by (1.14)). Thus each class $\{\varkappa\}\in T_{\varphi}\mathbb{T}^{\infty}$ can be identified with its canonical representative $\varkappa(t)=\operatorname{pr}[\operatorname{pr}^{-1}(\varphi)+t e]$, $e\in E$, and therefore with $e$. Thus, abusing the language slightly we can say that at any point $\varphi\in\mathbb{T}^{\infty}$ the tangent space $T_{\varphi}\mathbb{T}^{\infty}$ to $\mathbb{T}^{\infty}$ coincides with $E$. Since the norm $\|\,{\cdot}\,\|$ is defined in $E$, $\mathbb{T}^{\infty}$ is also a Finsler manifold (see [4], § П 4) with Finsler metric (1.7). Recall that by contrast to Riemann manifolds, at each point $x$ in a Finsler manifold $M$, in the tangent space $T_xM$ we have a norm, rather than an inner product, and this norm depends continuously on $x$ in a certain sense. 1.2. The description of the class $\operatorname{Diff}(\mathbb{T}^{\infty})$First we give a number of auxiliary definitions. We start with so-called local liftings, which can be introduced for any continuous map $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$. Fix a point $\varphi_0\in\mathbb{ T}^{\infty}$, set $\varphi_1=G(\varphi_0)$, and consider arbitrary preimages $v_0=\operatorname{pr}^{-1}(\varphi_0)$ and $v_1=\operatorname{pr}^{-1}(\varphi_1)$ of these points. By a local lifting of $G$ we mean a map of the form
$$
\begin{equation}
\overline{G}_{v_0}(v)= \operatorname{pr}^{-1}_{v_1}[G(\operatorname{pr}(v))],\qquad \overline{G}_{v_0}(v_0)=v_1.
\end{equation}
\tag{1.17}
$$
As in (1.11), here we let $\operatorname{pr}^{-1}_{v_1}(\varphi)$ denote a continuous branch of the multivalued map $\operatorname{pr}^{-1}(\varphi)$ distinguished by the equality $\operatorname{pr}^{-1}_{v_1}(\varphi_1)=v_1$ and defined on the ball $O(\varphi_1, r_0)\subset\mathbb{ T}^{\infty}$ (the constant $r_0>0$ is the same as in (1.11)). As concerns the variable $v$ in (1.17), it ranges in the ball where $\delta_0\in(0,r_0)$ satisfies
$$
\begin{equation}
\rho(G(\operatorname{pr}(v)),\varphi_1)<r_0\quad \forall\,v\in O(v_0,\delta_0).
\end{equation}
\tag{1.18}
$$
We also add that, since the map $G(\operatorname{pr}(v))$ is continuous and $G(\operatorname{pr}(v_0))=\varphi_1$, such $\delta_0$ certainly exists. Now, we say that a continuous map $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ is continuously differentiable if each local lifting (1.17) of it is differentiable and the corresponding Fréchet derivative $D(\overline{G}_{v_0}(v))$ depends on $v\in O(v_0,\delta_0)$ continuously in the uniform operator topology (that is, in the norm of the Banach space $L(E;E)$ of linear bounded operators). Given a continuously differentiable map $G$, the local representations (1.17) enable us to introduce its differential $DG(\varphi)$, $\varphi\in\mathbb{T}^{\infty}$. By definition this is a linear bounded operator from $E$ to $E$ given by
$$
\begin{equation}
DG(\varphi)|_{\varphi=\operatorname{pr}(v)}=D(\overline{G}_{v_0}(v))\quad \forall\,v\in O(v_0,\delta_0).
\end{equation}
\tag{1.19}
$$
We must note that (1.19) is a well-defined formula in the following sense: we cannot obtain different values of $DG(\varphi)$ for the same point $\varphi\in\mathbb{T}^{\infty}$ by choosing different local representation. In fact, since any two local branches of $\operatorname{pr}^{-1}(\varphi)$ in the intersection of their domains of definition (or more precisely, on each connected component of this intersection) differ by an additive constant of the form $2\pi l$, $l\in\mathbb{Z}^{\infty}$, the same holds for any two local liftings $\overline{G}_{\widetilde{v}_0}(v)$ and $\overline{G}_{\widetilde{\widetilde{v}}_0}(v)$. Hence on their common domain of definition we have $D(\overline{G}_{\widetilde{v}_0}(v)) =D(\overline{G}_{\widetilde{\widetilde{v}}_0}(v))$. Now let $G$ be a homeomorphism of $\mathbb{T}^{\infty}$ such that both $G$ and $G^{-1}$ are continuously differentiable. We call such maps diffeomorphisms (and say that $G$ is diffeomorphic). Writing out for $G^{-1}$ formulae similar to (1.17) and (1.19), it is easy to show that for each $\varphi\in\mathbb{T}^{\infty}$ the operator $DG(\varphi)$ is invertible and
$$
\begin{equation}
D(G^{-1}(\varphi))=[DG(\theta)]^{-1}\big|_{\theta=G^{-1}(\varphi)}\quad \forall\,\varphi\in\mathbb{T}^{\infty}.
\end{equation}
\tag{1.20}
$$
Now availing of the auxiliary concepts introduced above, we can introduce the class $\operatorname{Diff}(\mathbb{T}^{\infty})$ that is of interest to us. Definition 1.3. The class $\operatorname{Diff}(\mathbb{T}^{\infty})$ consists of auxiliary diffeomorphisms $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ satisfying the condition of boundedness and uniform continuity: Recall that the induced operator norm of a linear operator $A\colon E\to E$ is the quantity $\|A\|_{E\to E}=\sup_{x\in E, \|x\|=1}\|Ax\|$ (provided that it is finite). We also use such norm in what follows. Namely, below we let $\|\,{\cdot}\,\|_{V_1\to V_2}$, where the $V_j$, $j=1,2$, are closed subsets of $E$, denote the corresponding induced operator norms. Here we assume that the norms in the $V_j$, $j=1,2$, are inherited from $E$ (unless otherwise stated) or explicitly specified. Note that in view of (1.20) diffeomorphisms $G$ and $G^{-1}$ belong or do not belong simultaneously to the class $\operatorname{Diff}(\mathbb{T}^{\infty})$, so that there is some symmetry. However, in what follows we will need more details of the structure of maps in $\operatorname{Diff}(\mathbb{T}^{\infty})$. For the corresponding result we introduce some additional objects. Let $L(\mathbb{Z}^{\infty})$ denote the class of bounded linear operators $\Lambda$ from $E$ to $E$ such that each $\Lambda\in L(\mathbb{Z}^{\infty})$ is an invertible operator and $\Lambda\mathbb{Z}^{\infty}=\mathbb{Z}^{\infty}$. Next we let $B^1_{\rm per}(E)$ denote the set of vector-valued functions $g(\varphi)\in E$, $\varphi\in E$, with the following properties. Assume that, first, all functions $g(\varphi)$ in $B^1_{\rm per}(E)$ and their Fréchet derivatives $g'(\varphi)$ are continuous in $\varphi\in E$; second, they are $2\pi$-periodic, bounded, and uniformly continuous:
$$
\begin{equation}
\begin{gathered} \, g(\varphi+2\pi l)\equiv g(\varphi)\quad \forall\,l\in\mathbb{Z}^{\infty},\quad \sup_{\varphi\in E}\|g'(\varphi)\|_{E\to E}<\infty, \\ \lim_{\varepsilon\to 0}\ \sup_{\substack{\varphi_1,\varphi_2\in E: \\ \|\varphi_1-\varphi_2\|<\varepsilon}} \|g'(\varphi_1)-g'(\varphi_2)\|_{E\to E}=0. \end{gathered}
\end{equation}
\tag{1.22}
$$
With any pair $\lambda\in L(\mathbb{Z}^{\infty})$, $g(\varphi)\in B^1_{\rm per}(E)$ we associate two maps $\overline{G}\colon E\to E$ and $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ by the formulae
$$
\begin{equation}
\overline{G}\colon\varphi \mapsto \overline{G}(\varphi) \overset{\rm def}{=}\Lambda\varphi+g(\varphi),
\end{equation}
\tag{1.23}
$$
$$
\begin{equation}
G\colon\varphi \mapsto \overline{G}(\varphi) \,(\operatorname{mod}\,2\pi) = \Lambda\varphi+g(\varphi) \,(\operatorname{mod}\,2\pi) .
\end{equation}
\tag{1.24}
$$
Here the element $\overline{G}(\varphi) \,(\operatorname{mod}\,2\pi) $ of $\mathbb{T}^{\infty}$ is defined by
$$
\begin{equation}
\forall\,\varphi\in\mathbb{T}^{\infty}\quad \overline{G}(\varphi) \,(\operatorname{mod}\,2\pi) =\Lambda\varphi+g(\varphi) \,(\operatorname{mod}\,2\pi) = \operatorname{pr}[\overline{G}(\operatorname{pr}^{-1}(\varphi))],
\end{equation}
\tag{1.25}
$$
where $\operatorname{pr}$ is the projection (1.6) and $\operatorname{pr}^{-1}(\varphi)\in E$ is an arbitrary preimage of $\varphi\in\mathbb{T}^{\infty}$. The difference between two such preimages has the form $2\pi l$, $l\in\mathbb{Z}^{\infty}$. Furthermore, $\Lambda\mathbb{Z}^{\infty}=\mathbb{Z}^{\infty}$ and $g(\varphi)$ is a $2\pi$-periodic function (see (1.22)), so formula (1.25) is independent of the particular choice of $\operatorname{pr}^{-1}(\varphi)$. Below we call $\overline{G}$ a global lifting of the map $G$, and we call $G$ the pushdown of $\overline{G}$ to the torus $\mathbb{T}^{\infty}$. By (1.19) and (1.23)–(1.25), both maps are continuously differentiable and their differentials are connected by
$$
\begin{equation}
DG(\operatorname{pr}(\varphi))=D\overline{G}(\varphi)= \Lambda+g'(\varphi)\quad \forall\,\varphi\in E.
\end{equation}
\tag{1.26}
$$
We have the following result on the structure of diffeomorphisms in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$ under consideration. Theorem 1.2. Each diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ has a representation (1.24), (1.25) for some $\Lambda\in L(\mathbb{Z}^{\infty})$ and $g(\varphi)\in B^1_{\rm per}(E)$, and the corresponding global lifting (1.23) is a diffeomorphism of $E$ onto itself such that Proof. The first claim is verified as follows. First we show that each local lifting (1.17) of $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ can be extended to a global lifting $\overline{G}(v)$ defined on the whole of $E$. Then we verify that, first, $\overline{G}(v)$ is a diffeomorphism from $E$ to $E$ which has a representation
$$
\begin{equation}
\overline{G}(v)=\Lambda v+g(v),\qquad \Lambda\in L(\mathbb{Z}^{\infty}),\quad g\in B^1_{\rm per}(E);
\end{equation}
\tag{1.28}
$$
and, second, the operators $\Lambda$ and $g$ in (1.28) satisfy the inequality (1.27).
Fix an arbitrary $v_0\in E$ and consider the corresponding local lifting (1.17) defined on the ball $O(v_0,\delta_0)$. First we show that if (1.21) holds, then the radius $\delta_0>0$ of this ball can be taken independent of $v_0$. In fact, for any $v\in O(v_0,\delta_0)$, by (1.19) and (1.21) we have
$$
\begin{equation*}
\begin{aligned} \, \rho(G(\operatorname{pr}(v)),\varphi_1)&\leqslant \|\overline{G}_{v_0}(v)-v_1\|=\|\overline{G}_{v_0}(v)- \overline{G}_{v_0}(v_0)\| \\ &\leqslant\sup_{v\in O(v_0,\delta_0)}\|D(\overline{G}_{v_0}(v))\|_{E\to E} \cdot\|v-v_0\|\leqslant N\delta_0, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
N=\sup_{\varphi\in \mathbb{T}^{\infty}}\|DG(\varphi)\|_{E\to E}<\infty.
\end{equation}
\tag{1.29}
$$
From this, in turn, we conclude that condition (1.18) on $\delta_0$ is certainly satisfied for
$$
\begin{equation}
\delta_0=\operatorname{const}\in \biggl(0,\min\biggl(r_0,\frac{r_0}{N}\biggr)\biggr).
\end{equation}
\tag{1.30}
$$
Taking this choice of $\delta_0$ into account, we fix an arbitrary $\overline{v}_0\in O(v_0,\delta_0)$, set $\overline{v}_1=\overline{G}_{v_0}(\overline{v}_0)$ and consider the vector function
$$
\begin{equation}
\overline{G}_{\overline{v}_0}(v)=\operatorname{pr}^{-1}_{\overline{v}_1} [G(\operatorname{pr}(v))],\qquad v\in O(\overline{v}_0,\delta_0),
\end{equation}
\tag{1.31}
$$
which is similar to (1.17). Here $\operatorname{pr}^{-1}_{\overline{v}_1}(\varphi)$ is the continuous branch of $\operatorname{pr}^{-1}(\varphi)$ such that
$$
\begin{equation*}
\operatorname{pr}^{-1}_{\overline{v}_1}(\overline{\varphi}_1)=\overline{v}_1,
\end{equation*}
\notag
$$
where $\overline{\varphi}_1=G(\operatorname{pr}(\overline{v}_0)) \in\mathbb{T}^{\infty}$.
As mentioned above, on the intersection of their domains of definition two distinct local liftings differ by an additive constant of the form $2\pi l$, $l\in\mathbb{Z}^{\infty}$. In our case we have $\overline{G}_{\overline{v}_0}(\overline{v}_0)=\overline{v}_1= \overline{G}_{v_0}(\overline{v}_0)$, so
$$
\begin{equation}
\overline{G}_{\overline{v}_0}(v)=\overline{G}_{v_0}(v)\quad \forall\,v\in O(v_0,\delta_0)\cap O(\overline{v}_0,\delta_0).
\end{equation}
\tag{1.32}
$$
Thus we can define the map (1.17) on the set
$$
\begin{equation}
\bigcup_{\overline{v}_0\in O(v_0,\delta_0)}O(\overline{v}_0,\delta_0)= O(v_0,2\delta_0)
\end{equation}
\tag{1.33}
$$
by the formula
$$
\begin{equation}
\overline{G}_{v_0}(v)=\{\overline{G}_{\overline{v}_0}(v)\text{ for } v\in O(\overline{v}_0,\delta_0)\}.
\end{equation}
\tag{1.34}
$$
We observe that this is a consistent definition in view of (1.32).
At the next step we consider the map (1.31) for $\overline{v}_0\in O(v_0,2\delta_0)$ and extend the original map (1.17) to the set
$$
\begin{equation*}
\bigcup_{\overline{v}_0\in O(v_0,2\delta_0)}O(\overline{v}_0,\delta_0)= O(v_0,3\delta_0),
\end{equation*}
\notag
$$
which is similar to (1.33) by formula (1.34). Since $\delta_0$ is universal (see (1.30)), it is clear that this process continues indefinitely. As a result, we obtain a global lifting $\overline{G}(v)$, $v\in E$. Just as any local lift, the vector function $\overline{G}(v)$ is continuously Fréchet differentiable, and in view of (1.19) we have an equality
$$
\begin{equation}
D\overline{G}(v)=DG(\varphi)\big|_{\varphi=\operatorname{pr}(v)}\quad \forall\,v\in E,
\end{equation}
\tag{1.35}
$$
which is similar to (1.26). Hence from (1.21) we obtain the following properties:
$$
\begin{equation}
\begin{gathered} \, \sup_{v\in E}\|D\overline{G}(v)\|_{E\to E}= \sup_{\varphi\in \mathbb{T}^{\infty}}\|DG(\varphi)\|_{E\to E}<\infty, \\ \sup_{v\in E}\|(D\overline{G}(v))^{-1}\|_{E\to E}= \sup_{\varphi\in \mathbb{T}^{\infty}}\|(DG(\varphi))^{-1}\|_{E\to E}<\infty, \\ \lim_{\varepsilon\to 0}\, \sup_{\substack{v_1,v_2\in E: \\ \|v_1-v_2\|<\varepsilon}} \|D\overline{G}(v_1)-D\overline{G}(v_2)\|_{E\to E}=0. \end{gathered}
\end{equation}
\tag{1.36}
$$
As concerns the diffeomorphism $G$, local representations of the form (1.17) imply the following global representation for it:
$$
\begin{equation}
G(\varphi)= \operatorname{pr}[\overline{G}(v)]\big|_{v=\operatorname{pr}^{-1}(\varphi)} \quad \forall\,\varphi\in\mathbb{T}^{\infty}.
\end{equation}
\tag{1.37}
$$
From (1.37) we obtain with necessity
$$
\begin{equation*}
\operatorname{pr}[\overline{G}(v+2\pi l)]= \operatorname{pr}[\overline{G}(v)]\quad \forall\,l\in\mathbb{Z}^{\infty},\quad \forall\,v\in E,
\end{equation*}
\notag
$$
so there exists $\overline{l}\in\mathbb{Z}^{\infty}$ such that
$$
\begin{equation*}
\overline{G}(v+2\pi l)=\overline{G}(v)+2\pi\overline{l}.
\end{equation*}
\notag
$$
We also add that, as $\overline{G}(v)$ is continuous and the integer lattice $\mathbb{Z}^{\infty}$ is discrete, $\overline{l}$ is independent of $v\in E$. Thus the operator
$$
\begin{equation}
\Lambda l\overset{\rm def}{=}\frac{1}{2\pi}\,(\overline{G}(v+2\pi l)- \overline{G}(v))\in\mathbb{Z}^{\infty}\quad \forall\,l\in\mathbb{Z}^{\infty}
\end{equation}
\tag{1.38}
$$
is well defined on $\mathbb{Z}^{\infty}$.
We look at some properties of the operator (1.38). It follows from the obvious equalities
$$
\begin{equation*}
\begin{aligned} \, \Lambda(l_1+l_2)&=\frac{1}{2\pi}\bigl(\overline{G}(v+2\pi (l_1+l_2))- \overline{G}(v)\bigr) \\ &=\frac{1}{2\pi}\bigl(\overline{G}((v+2\pi l_2)+2\pi l_1)- \overline{G}(v+2\pi l_2)\bigr)+ \frac{1}{2\pi}(\overline{G}(v+2\pi l_2)-\overline{G}(v)) \\ &=\Lambda l_1+\Lambda l_2 \quad \forall\,l_1,l_2\in\mathbb{Z}^{\infty}, \\ \Lambda(-l)&=\frac{1}{2\pi}\,(\overline{G}(v-2\pi l)-\overline{G}(v)) \\ &=-\frac{1}{2\pi}\bigl(\overline{G}((v-2\pi l)+2\pi l)- \overline{G}(v-2\pi l)\bigr)=-\Lambda l\quad \forall\,l\in\mathbb{Z}^{\infty} \end{aligned}
\end{equation*}
\notag
$$
that
$$
\begin{equation}
\Lambda(k_1l_1+k_2l_2)=k_1\Lambda l_1+k_2\Lambda l_2\quad \forall\,l_1,l_2\in\mathbb{Z}^{\infty},\quad \forall\,k_1,k_2\in\mathbb{Z}.
\end{equation}
\tag{1.39}
$$
Moreover, by (1.36) we have
$$
\begin{equation}
\|\Lambda l\|\leqslant N\|l\|\quad \forall\,l\in\mathbb{Z}^{\infty},
\end{equation}
\tag{1.40}
$$
where $N$ is the constant in (1.29).
Properties (1.39) and (1.40) just established enable us to extend the operator (1.38) from $\mathbb{Z}^{\infty}$ to the whole of $E$ while keeping it linear and bounded. Namely, first, for each finite linear combination
$$
\begin{equation}
v=\alpha_1 l_1+\alpha_2 l_2+\cdots+\alpha_k l_k,\qquad l_j\in\mathbb{Z}^{\infty},\quad \alpha_j\in\mathbb{R},\quad j=1,\dots,k,
\end{equation}
\tag{1.41}
$$
we set
$$
\begin{equation}
\Lambda v\overset{\rm def}{=}\sum_{j=1}^k\alpha_j\Lambda l_j.
\end{equation}
\tag{1.42}
$$
Then we verify that in the case (1.41), (1.42) we still have an estimate of the form (1.40), that is,
Assume first that all coefficients in (1.41) are rational. We can also assume without loss of generality that $\alpha_j=m_j/m$, where $m_j\in\mathbb{Z}$, $m\in\mathbb{N}$. Then from (1.40) and (1.42) we obtain
$$
\begin{equation}
\begin{aligned} \, \biggl\|\Lambda\biggl(\,\sum_{j=1}^k\frac{m_j}{m}\,l_j\biggr)\biggr\|&= \frac{1}{m}\biggl\|\Lambda\biggl(\,\sum_{j=1}^km_jl_j\biggr)\biggr\| \nonumber \\ &\leqslant\frac{N}{m}\biggl\|\,\sum_{j=1}^km_jl_j\biggr\|=N \biggl\|\,\sum_{j=1}^k\frac{m_j}{m}\,l_j\biggr\|. \end{aligned}
\end{equation}
\tag{1.44}
$$
Now fix some element (1.41) and select sequences of rational numbers $\alpha_j^{(n)}$, $j=1,\dots,k$, such that $\alpha_j^{(n)}\to\alpha_j$ as $n\to+\infty$. In this case, by (1.42) and (1.44) we have
$$
\begin{equation}
\biggl\|\,\sum_{j=1}^k\alpha_j^{(n)}\Lambda l_j\biggr\|\leqslant N\biggl\|\,\sum_{j=1}^k\alpha_j^{(n)}l_j\biggr\|.
\end{equation}
\tag{1.45}
$$
Now taking the limit as $n\to+\infty$ in (1.45), we obtain the required estimate (1.43). It remains to add that, since linear combinations of the form (1.41) are dense in $E$ (see the maximality property of the lattice $\mathbb{Z}^{\infty}$), the operator $\Lambda$ extends uniquely to $E$ so that inequality (1.43) is preserved.
We have thus constructed a bounded linear operator $\Lambda\colon E\to E$ with the property $\Lambda\mathbb{Z}^{\infty}\subset\mathbb{Z}^{\infty}$. Furthermore, in view of (1.38) the vector function $g(v)=\overline{G}(v)-\Lambda v$ is $2\pi$-periodic in $v$, Fréchet continuously differentiable, and satisfies
$$
\begin{equation*}
\begin{gathered} \, \sup_{v\in E}\|g'(v)\|_{E\to E}\leqslant \sup_{v\in E}\|D\overline{G}(v)\|_{E\to E}+ \|\Lambda\|_{E\to E}=N+\|\Lambda\|_{E\to E}\leqslant 2N, \\ \|g'(v_1)-g'(v_2)\|_{E\to E}=\|D\overline{G}(v_1)- D\overline{G}(v_2)\|_{E\to E}\to 0\quad\text{as } \|v_1-v_2\|\to 0, \end{gathered}
\end{equation*}
\notag
$$
where $N$ is the constant in (1.29). Hence we have the required inclusion $g(v)\in B^1_{\rm per}(E)$.
The boundedness property (1.27), required in Theorem 1.2, also holds (it follows from (1.35) and the second inequality in (1.36)). Thus, to complete the proof of the first result of the theorem, it remains to verify that $\Lambda$ is bounded and show that $\Lambda^{-1}\mathbb{Z}^{\infty}\subset\mathbb{Z}^{\infty}$ and that $\overline{G}(v)$ is a diffeomorphism. Note that all the above arguments also hold for the inverse map $G^{-1}$, because it also belongs to $\operatorname{Diff}(\mathbb{T}^{\infty})$. In particular, on the ball $O(v_1,\delta_0)\subset E$ we have a well-defined local lifting
$$
\begin{equation}
\overline{H}_{v_1}(v)= \operatorname{pr}^{-1}_{v_0}[G^{-1}(\operatorname{pr}(v))],\qquad \overline{H}_{v_1}(v_1)=v_0.
\end{equation}
\tag{1.46}
$$
Here the vectors $v_0$ and $v_1$ are the same as in (1.17) and $\operatorname{pr}^{-1}_{v_0}(\varphi)$ is the continuous branch of $\operatorname{pr}^{-1}(\varphi)$ such that $\operatorname{pr}^{-1}_{v_0}(\varphi_0)=v_0$. As concerns $\delta_0$, we can choose it in a uniform way from (1.30), where in place of (1.29) we have
$$
\begin{equation*}
N=\sup_{\varphi\in \mathbb{T}^{\infty}} \|(DG(\varphi))^{-1}\|_{E\to E}<\infty.
\end{equation*}
\notag
$$
Now extending the local lifting (1.46) to the whole of $E$ in accordance with the above rules we obtain a continuously Fréchet differentiable vector function $\overline{H}(v)$ whose differential is bounded and uniformly continuous similarly to (1.36):
$$
\begin{equation}
\begin{gathered} \, \sup_{v\in E}\|D\overline{H}(v)\|_{E\to E}= \sup_{\varphi\in \mathbb{T}^{\infty}}\|(DG(\varphi))^{-1}\|_{E\to E}<\infty, \\ \sup_{v\in E}\|(D\overline{H}(v))^{-1}\|_{E\to E}= \sup_{\varphi\in \mathbb{T}^{\infty}}\|DG(\varphi)\|_{E\to E}<\infty, \\ \lim_{\varepsilon\to 0}\,\sup_{\substack{ v_1,v_2\in E: \\ \|v_1-v_2\|<\varepsilon}}\|D\overline{H}(v_1)- D\overline{H}(v_2)\|_{E\to E}=0. \end{gathered}
\end{equation}
\tag{1.47}
$$
In addition, we have by construction a representation similar to (1.37):
$$
\begin{equation}
G^{-1}(\varphi)= \operatorname{pr}[\overline{H}(v)]\big|_{v=\operatorname{pr}^{-1}(\varphi)} \quad \forall\,\varphi\in\mathbb{T}^{\infty}.
\end{equation}
\tag{1.48}
$$
The arguments that follow repeat the above part of proof related to the operator (1.38). Namely, on the basis of (1.47) and (1.48) we verify that where $h(v)\in B^1_{\rm per}(E)$ and $C\colon E\to E$ is a bounded linear operator such that $C\mathbb{Z}^{\infty}\subset\mathbb{Z}^{\infty}$. Next, from the representations (1.37) and (1.48) and the obvious relations $G(G^{-1}(\varphi))=\varphi$ and $G^{-1}(G(\varphi))=\varphi$, for the corresponding liftings $\overline{G}(v)$ and $\overline{H}(v)$ we obtain
$$
\begin{equation}
\overline{G}(\overline{H}(v))=v+2\pi l_1\quad\text{and}\quad \overline{H}(\overline{G}(v))=v+2\pi l_2\quad \forall\,v\in E.
\end{equation}
\tag{1.50}
$$
We also add that, as the functions $\overline{G}(\overline{H}(v))-v$ and $\overline{H}(\overline{G}(v))-v$ are continuous and the lattice $\mathbb{Z}^{\infty}$ is discrete, the elements $l_1,l_2\in \mathbb{Z}^{\infty}$ in (1.50) are independent of the choice of $v\in E$. Furthermore, we show that, in fact, $l_1=l_2=0$.
To do this, first we turn to the first equality in (1.50) and set $v=v_1$ in it. Then from the obvious properties $\overline{H}(v_1)=v_0$ and $\overline{G}(v_0)=v_1$ we obtain $\overline{G}(\overline{H}(v_1))=v_1$, so that $l_1=0$. In a similar way, setting $v=v_0$ in the second equality in (1.50), we conclude that $l_2=0$. Thus, we have shown that
$$
\begin{equation}
\overline{G}(\overline{H}(v))=v,\quad \overline{H}(\overline{G}(v))=v\quad \forall\,v\in E.
\end{equation}
\tag{1.51}
$$
By the way, these formulae show that $\overline{G}(v)$ is a diffeomorphism of $E$ onto itself. Now we replace the argument $v$ in (1.51) by $v+2\pi l$, $l\in\mathbb{Z}^{\infty}$, and using equality (1.49) in combination with the similar representation $\overline{G}(v)=\Lambda v+g(v)$, after simple manipulations we conclude that
$$
\begin{equation}
\Lambda Cl=l,\quad C\Lambda l=l\quad \forall\,l\in\mathbb{Z}^{\infty}.
\end{equation}
\tag{1.52}
$$
On the other hand it has been shown above that any bounded linear operator on the lattice $\mathbb{Z}^{\infty}$ extends to $E$ uniquely (as a bounded linear operator); so it follows automatically from (1.52) that $\Lambda C=I$, $C\Lambda=I$, where, as usual, $I$ is the identity operator in $E$. Thus the operator $\Lambda$ under consideration is invertible and $\Lambda^{-1}\mathbb{Z}^{\infty}=C\mathbb{Z}^{\infty}\subset \mathbb{Z}^{\infty}$. The first part of Theorem 1.2 is proved.
To prove the converse result claimed by Theorem 1.2 it is sufficient to verify that if for some $\Lambda\in L(\mathbb{Z}^{\infty})$ and $g\in B^1_{\rm per}(E)$ the map $\overline{G}$ (see (1.23)) is a diffeomorphism, then its pushdown $G$ to the torus $\mathbb{T}^{\infty}$ is too. So let $\overline{G}$ be a diffeomorphism. Then by (1.26), as we assume that the differential $D\overline{G}(\varphi)$ is invertible, $G$ is a local diffeomorphism. Furthermore, we conclude from the obvious relations $\operatorname{pr}^{-1}(\mathbb{T}^{\infty})=E$, $\overline{G}(E)=E$, $\operatorname{pr}(E)=\mathbb{T}^{\infty}$ and (1.37) that $G(\mathbb{T}^{\infty})=\mathbb{T}^{\infty}$. Thus, to verify that $G$ is a diffeomorphic map, it remains to prove that it is injective. Assuming on the contrary that there exist points $\varphi_1,\varphi_2\in\mathbb{T}^{\infty}$, $\varphi_1\ne\varphi_2$, such that $G(\varphi_1)=G(\varphi_2)$, by (1.37) we have
$$
\begin{equation*}
\operatorname{pr}[\overline{G}(\operatorname{pr}^{-1}(\varphi_1))]= \operatorname{pr}[\overline{G}(\operatorname{pr}^{-1}(\varphi_2))]
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation}
\overline{G}(\operatorname{pr}^{-1}(\varphi_1))= \overline{G}(\operatorname{pr}^{-1}(\varphi_2))+2\pi l
\end{equation}
\tag{1.53}
$$
for some $l\in\mathbb{Z}^{\infty}$. Now, to analyze (1.53) we use the obvious property
$$
\begin{equation*}
\overline{G}(\varphi+2\pi l)\equiv 2\pi\Lambda l+\overline{G}(\varphi)\quad \forall\,l\in\mathbb{Z}^{\infty}.
\end{equation*}
\notag
$$
Taking this into account, as $\overline{G}$ is one-to-one, we conclude that
$$
\begin{equation*}
\overline{G}(\operatorname{pr}^{-1}(\varphi_1))= \overline{G}(\operatorname{pr}^{-1}(\varphi_2)+2\pi\Lambda^{-1}l)\quad\text{and}\quad \operatorname{pr}^{-1}(\varphi_1)= \operatorname{pr}^{-1}(\varphi_2)+2\pi\Lambda^{-1}l.
\end{equation*}
\notag
$$
Recall however that $\Lambda^{-1}l\in\mathbb{Z}^{\infty}$; hence $\varphi_1,\varphi_2\in\mathbb{T}^{\infty}$ must coincide. This contradiction proves that $G$ is injective as required. By the above this completes the verification of Theorem 1.2. $\Box$ 1.3. Some sufficient conditions of a diffeomorphismIn constructing explicit examples of diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ we encounter the following problem. Consider a map $G$ of $\mathbb{T}^{\infty}$ to itself of the following form:
$$
\begin{equation}
G\colon \varphi\mapsto \Lambda\varphi+g(\varphi) \,(\operatorname{mod}\,2\pi) ,\qquad \Lambda\in L(\mathbb{Z}^{\infty}),\quad g(\varphi)\in B^1_{\rm per}(E),
\end{equation}
\tag{1.54}
$$
where we assume that for each $\varphi\in E$ the linear operator $\Lambda+g'(\varphi)\colon E\to E$ is invertible. The question is: how can we verify that this map is a diffeomorphism? In other words, can we find sufficient conditions ensuring that the map is diffeomorphic? An answer is given by the following statement. Theorem 1.3. Assume that the vector-valued function $g(\varphi)$ in (1.54) is bounded: and the map is completely continuous in $E$. Then the operator (1.54) is a diffeomorphism of $\mathbb{T}^{\infty}$. Proof. As shown in the proof of Theorem 1.2, if a lifting $\overline{G}$ (see (1.23)) of an operator $G$ is a diffeomorphism of $E$ to $E$, then the map (1.54) itself is too. Thus we only need to verify that $\overline{G}$ is a diffeomorphism.
First we show that $\overline{G}$ is surjective. To do this we fix an arbitrary $z\in E$ and consider the equation $\overline{G}(\varphi)=z$ with respect to $\varphi\in E$. Next we make the substitution $\varphi=\Lambda^{-1}z+h$ in it and arrive at the equation with respect to $h\in E$.On the basis of the assumed boundedness (1.55) and the complete continuity of (1.56) we conclude that, with respect to $h$, the right-hand side of (1.57) is a completely continuous operator on $E$, which transforms the ball with centre at zero and radius
$$
\begin{equation*}
r=\|\Lambda^{-1}\|_{E\to E}\cdot\sup_{\varphi\in E}\|g(\varphi)\|
\end{equation*}
\notag
$$
into itself. Thus, by the Schauder principle the above equation has at least one solution $h\in E$, so that $\overline{G}(E)=E$.
We can add that, apart from being surjective, $\overline{G}$ is a local diffeomorphism (because we assume that the differential $D\overline{G}(\varphi)$ is invertible for all $\varphi\in E$). By results due to Banach and Masur (see [26] and [27]), if $\overline{G}$ is also proper (that is, the preimage $\overline{G}^{\,-1}(Y)$ of any compact set $Y$ is compact), then it is a global diffeomorphism of $E$ onto itself. Hence we must verify that $\overline{G}^{\,-1}(Y)$ is compact for each compact set $Y\subset E$. Fix some compact set $Y\subset E$ and an infinite sequence of points $x_n\in\overline{G}^{\,-1}(Y)$, $n\geqslant 1$. We can assume without loss of generality that the corresponding sequence $y_n=\Lambda x_n+g(x_n)\in Y$ tends to some $y_*\in Y$ as $n\to +\infty$. Bearing this in mind, from the relation and property (1.55) we conclude that the sequence $x_n$ is bounded. Since the operator (1.56) is compact, from $g(x_n)$ we can extract a convergent subsequence. We assume without loss of generality that $g(x_n)\to z_*\in E$ as $n\to+\infty$. Now, on the basis of (1.58) and the continuity of $g(\varphi)$ we obtain in succession
$$
\begin{equation*}
\begin{gathered} \, x_n\to x_*=\Lambda^{-1}y_*-\Lambda^{-1}z_*,\quad g(x_n)\to g(x_*)=z_*,\qquad n\to+\infty, \\ \Lambda x_*+g(x_*)=y_*. \end{gathered}
\end{equation*}
\notag
$$
Thus we have shown that $x_*\in \overline{G}^{\,-1}(Y)$, which proves that the set $\overline{G}^{\,-1}(Y)$ is compact.
We see that $\overline{G}$ is a diffeomorphism of $E$ onto itself. Then, as mentioned above, the original operator (1.54) is a diffeomorphism of the torus $\mathbb{T}^{\infty}$. $\Box$ 2. A criterion for hyperbolicity2.1. The simplest properties of hyperbolic diffeomorphismsFirst we define hyperbolicity for an arbitrary diffeomorphism $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$. To do this, apart from the differential $DG(\varphi)$ we require the linear operators $D(G^n(\varphi))$ and $D(G^{-n}(\varphi))$, $n\in\mathbb{N}$, defined by
$$
\begin{equation}
\begin{aligned} \, D(G^n(\varphi))&=DG(\varphi_{n-1})\circ DG(\varphi_{n-2}) \circ\cdots\circ DG(\varphi_{0}), \\ D(G^{-n}(\varphi))&=[DG(\varphi_{-n})]^{-1}\circ [DG(\varphi_{-(n-1)})]^{-1}\circ\cdots\circ[DG(\varphi_{-1})]^{-1}, \end{aligned}
\end{equation}
\tag{2.1}
$$
where $\varphi_j=G^j(\varphi)$, $j\in\mathbb{Z}$. Definition 2.1. We say that a diffeomorphism $G\colon \mathbb{T}^{\infty}\to \mathbb{T}^{\infty}$ is hyperbolic, or call it an Anosov diffeomorphism, if for each $\varphi\in\mathbb{T}^{\infty}$ the space $T_{\varphi}\mathbb{T}^{\infty}=E$ has the representation as a direct sum of non-trivial closed linear subspaces $E_\varphi^{\rm u}$ and $E_\varphi^{\rm s}$ so that the following conditions are satisfied: (a) for each $\varphi\in \mathbb{T}^{\infty}$ we have $DG(\varphi)E_{\varphi}^{\rm u}=E_{G(\varphi)}^{\rm u}$ and $DG(\varphi)E_{\varphi}^{\rm s}=E_{G(\varphi)}^{\rm s}$ (this is called $DG$-invariance); (b) there exist constants $\mu_1,\mu_2\in (0,1)$ and $c_1,c_2>0$ such that
$$
\begin{equation}
\|D(G^{-n}(\varphi))\xi\| \leqslant c_1\mu_1^n\|\xi\| \qquad \forall\,\varphi \in \mathbb{T}^{\infty}, \quad \forall\,\xi \in E_\varphi^{\rm u}, \quad \forall\,n \in \mathbb{N},
\end{equation}
\tag{2.3}
$$
$$
\begin{equation}
\|D(G^n(\varphi))\xi\| \leqslant c_2\mu_2^n\|\xi\| \qquad \forall\,\varphi \in \mathbb{T}^{\infty}, \quad \forall\,\xi \in E_\varphi^{\rm s}, \quad \forall\,n \in\mathbb{N};
\end{equation}
\tag{2.4}
$$
(c) the projections
$$
\begin{equation*}
P_{\varphi}\xi=\xi_1,\quad Q_{\varphi}\xi=\xi_2\quad \forall\,\xi=\xi_1+\xi_2\in E,\quad \xi_1\in E_\varphi^{\rm u},\quad \xi_2\in E_\varphi^{\rm s},
\end{equation*}
\notag
$$
associated with the decomposition (2.2) are uniformly continuous with respect to $\varphi\in \mathbb{T}^{\infty}$ in the uniform operator topology, that is, This is a natural generalization of a definition due to Anosov (see [3], § 1) to the infinite-dimensional torus $\mathbb{T}^{\infty}$. On the other hand we must recall that if we replace $\mathbb{T}^{\infty}$ by a finite-dimensional torus $\mathbb{T}^{m}$, $m\geqslant 2$, then the projections $P_{\varphi}$ and $Q_{\varphi}$ are necessarily uniformly continuous because of the $DG$-invariance of the subspaces $E_\varphi^{\rm u}$ and $E_\varphi^{\rm s}$ and conditions (2.3) and (2.4) (for instance, see [3] and [6]). However, in the infinite-dimensional case the question of this property is open. For this reason we must postulate such continuity, which is one of the basic properties of a hyperbolic structure in Definition 2.1 which we have stated. Below we consider only diffeomorphisms $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$. The reasons why we have chosen just the class $\operatorname{Diff}(\mathbb{T}^{\infty})$ are as follows: for hyperbolic diffeomorphisms $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ we still have some elementary results from hyperbolic theory: the angle between the subspaces $E_\varphi^{\rm u}$ and $E_\varphi^{\rm s}$ is separated from zero, the projections $P_\varphi$ and $Q_\varphi$ are uniformly bounded, and so on. But if we drop conditions (1.21), then we cannot establish such results. We start by discussing conditions for the hyperbolicity of the simplest representatives of $\operatorname{Diff}(\mathbb{T}^{\infty})$, namely, linear diffeomorphisms
$$
\begin{equation}
\varphi\mapsto \Lambda\varphi \,(\operatorname{mod}\,2\pi) ,
\end{equation}
\tag{2.5}
$$
where $\Lambda\in L(\mathbb{Z}^{\infty})$. It is easy to see that they can be formulated as follows in terms of the spectrum $\sigma(\Lambda)$ of the operator $\Lambda$:
$$
\begin{equation}
\begin{gathered} \, \sigma(\Lambda)=\sigma_1\cup\sigma_2,\qquad \sigma_j\ne\varnothing,\quad j=1,2,\qquad \sigma_1\subset\{\lambda\in\mathbb{C}\colon |\lambda|>1\}, \\ \sigma_2\subset\{\lambda\in\mathbb{C}\colon|\lambda|<1,\ \lambda\ne 0\}. \end{gathered}
\end{equation}
\tag{2.6}
$$
In fact, it follows from the above conditions that where the sum is direct and the closed linear subspaces $E_1$ and $E_2$ have the following properties: $\Lambda E_j=E_j$, $j=1,2$, and the spectra of the restrictions $\Lambda_j=\Lambda\big|_{E_j}$, $j=1,2$, coincide with the spectral sets $\sigma_j$, $j=1,2$, in (2.6). It is also clear that the decomposition (2.7) satisfies all conditions in Definition 2.1, and therefore a map (2.5) satisfying (2.6) is hyperbolic. We call such maps linear hyperbolic automorphisms of the torus.Now we turn to describing some of the simplest properties of hyperbolic diffeomorphisms $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$. To do this we need a definition of hyperbolicity for the diffeomorphism $\overline{G}$ (see (1.23)), a global lifting of $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$. Definition 2.2. The diffeomorphism $\overline{G}$ being hyperbolic means that the following conditions are satisfied. 1) For each $\varphi\in E$ the space $E$ has a representation as a direct sum
$$
\begin{equation}
E=\overline{E}_\varphi^{\rm \,u}\oplus \overline{E}_\varphi^{\rm \,s}
\end{equation}
\tag{2.8}
$$
of non-trivial closed linear subspaces $\overline{E}_\varphi^{\rm \,u}$ and $\overline{E}_\varphi^{\rm \,s}$ which satisfy the conditions of invariance
$$
\begin{equation}
D\overline{G}(\varphi)\overline{E}_{\varphi}^{\rm \,u}= \overline{E}_{\overline{G}(\varphi)}^{\rm \,u}\quad\text{and}\quad D\overline{G}(\varphi)\overline{E}_{\varphi}^{\rm \,s}= \overline{E}_{\overline{G}(\varphi)}^{\rm \,s}\quad \forall\,\varphi\in E,
\end{equation}
\tag{2.9}
$$
where we recall that $D\overline{G}(\varphi)=\Lambda+g'(\varphi)$. 2) There exist constants $\mu_1,\mu_2\in (0,1)$ and $c_1,c_2>0$ such that
$$
\begin{equation}
\|D(\overline{G}^{\,-n}(\varphi))\xi\| \leqslant c_1\mu_1^n\|\xi\| \quad \forall\,\varphi \in E, \quad \forall\,\xi \in \overline{E}_\varphi^{\rm \,u}, \quad \forall\,n \in\mathbb{N},
\end{equation}
\tag{2.10}
$$
$$
\begin{equation}
\text{and} \qquad \|D(\overline{G}^{\,n}(\varphi))\xi\| \leqslant c_2\mu_2^n\|\xi\| \quad \forall\,\varphi \in E, \quad \forall\,\xi \in \overline{E}_\varphi^{\rm \,s}, \quad \forall\,n \in\mathbb{N}.
\end{equation}
\tag{2.11}
$$
Here $D(\overline{G}^{\,n}(\varphi))$ and $D(\overline{G}^{\,-n}(\varphi))$ are operators defined by equalities similar to (2.1), with the differential $DG(\varphi)$ replaced by $D\overline{G}(\varphi)$ and iterates of $G$ replaced by the analogous iterates $\varphi_j=\overline{G}^{\,j}(\varphi)$, $j\in\mathbb{Z}$. 3) The projections $\overline{P}_{\varphi}$, $\overline{Q}_{\varphi}$, $\overline{P}_{\varphi}E=\overline{E}_\varphi^{\rm \,u}$ and $\overline{Q}_{\varphi}E=\overline{E}_\varphi^{\rm \,s}$, which are related to the decomposition (2.8), are uniformly continuous with respect to $\varphi\in E$ in the uniform operator topology. On the basis of Definitions 2.1 and 2.2 above and Theorem 1.2 we can establish the following result. Lemma 2.1. A diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ and a global lifting $\overline{G}$ of it are simultaneously hyperbolic or not. Proof. First we assume that $\overline{G}$ is hyperbolic and show that the original map $G$ is too.
Note that when $\overline{G}$ is hyperbolic, apart from the three standard properties involved in Definition 2.2, we also have periodicity of the following form:
$$
\begin{equation}
\overline{E}_{\varphi+2\pi l}^{\rm \,u}\equiv \overline{E}_\varphi^{\rm \,u},\quad \overline{E}_{\varphi+2\pi l}^{\rm \,s}\equiv \overline{E}_\varphi^{\rm \,s},\quad \overline{P}_{\varphi+2\pi l}\equiv\overline{P}_\varphi,\quad \overline{Q}_{\varphi+2\pi l}\equiv\overline{Q}_\varphi
\end{equation}
\tag{2.12}
$$
for each $l\in\mathbb{Z}^{\infty}$. To verify it, we look at the representation
$$
\begin{equation}
\begin{gathered} \, \overline{G}(\varphi)=\Lambda\varphi+g(\varphi),\quad \overline{G}^{\,-1}(\varphi)=\Lambda^{-1}\varphi+h(\varphi), \\ \Lambda\in L(\mathbb{Z}^{\infty}),\qquad g,h\in B^1_{\rm per}(E), \end{gathered}
\end{equation}
\tag{2.13}
$$
which holds by Theorem 1.2. Next, on the basis of (2.13) and mathematical induction we conclude sequentially that
$$
\begin{equation}
\begin{gathered} \, \overline{G}^{\,m}(\varphi+2\pi l)\equiv 2\pi\Lambda^ml+ \overline{G}^{\,m}(\varphi),\quad D(\overline{G}^{\,m}(\varphi+2\pi l))\equiv D(\overline{G}^{\,m}(\varphi)) \\ \forall\,l\in \mathbb{Z}^{\infty},\quad \forall\,m\in\mathbb{Z}. \end{gathered}
\end{equation}
\tag{2.14}
$$
It turn, this shows that for each fixed $l\in \mathbb{Z}^{\infty}$ the decomposition $E=\overline{E}_{\varphi+2\pi l}^{\rm \,u}\oplus \overline{E}_{\varphi+2\pi l}^{\rm \,s}$ also forms a hyperbolic structure.
In fact, by (2.9) and (2.14) the subspaces $\overline{E}_{\varphi+2\pi l}^{\rm \,u}$ and $\overline{E}_{\varphi+2\pi l}^{\rm \,s}$ possess the required properties of invariance
$$
\begin{equation*}
D\overline{G}(\varphi)\overline{E}_{\varphi+2\pi l}^{\rm \,u}= \overline{E}_{\overline{G}(\varphi+2\pi l)}^{\rm \,u},\quad D\overline{G}(\varphi)\overline{E}_{\varphi+2\pi l}^{\rm \,s}= \overline{E}_{\overline{G}(\varphi+2\pi l)}^{\rm \,s}\quad \forall\,\varphi\in E
\end{equation*}
\notag
$$
and estimates (2.10) and (2.11) hold (with $\xi\in\overline{E}_{\varphi}^{\rm \,u}$ and $\xi\in\overline{E}_{\varphi}^{\rm \,s}$ replaced by $\xi\in\overline{E}_{\varphi+2\pi l}^{\rm \,u}$ and $\xi\in\overline{E}_{\varphi+2\pi l}^{\rm \,s}$, respectively, but with the same constants $c_j$ and $\mu_j$, $j=1,2$). However, the hyperbolic structure for $\overline{G}$ is unique, and therefore identities (2.12), which we need, are automatically satisfied.
As we also use similar arguments below, we consider the verification of identities (2.12) in slightly greater detail. Assuming the converse, consider, for example, the case when $\overline{E}_{\varphi}^{\rm \,u}\ne\overline{E}_{\varphi+2\pi l}^{\rm \,u}$ for some $\varphi\in E$, $l\in\mathbb{Z}^{\infty}$ and there exists a vector $\xi_0\ne 0$ such that $\xi_0\in\overline{E}_{\varphi}^{\rm \,u}$ and $\xi_0\not\in\overline{E}_{\varphi+2\pi l}^{\rm \,u}$. Then it is obvious that
$$
\begin{equation*}
\|D(\overline{G}^{\,-n}(\varphi))\xi_0\|\leqslant c_1\mu_1^n\|\xi_0\|.
\end{equation*}
\notag
$$
On the other hand, setting $\xi_0=\xi_0^1+\xi_0^2$, $\xi_0^1\in\overline{E}_{\varphi+2\pi l}^{\rm \,u}$ and $\xi_0^2\in\overline{E}_{\varphi+2\pi l}^{\rm \,s}$ and using the inequalities analogous to (2.10) and (2.11) for the subspaces $\overline{E}_{\varphi+2\pi l}^{\rm \,u}$ and $\overline{E}_{\varphi+2\pi l}^{\rm \,s}$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \|D(\overline{G}^{\,-n}(\varphi))\xi_0\|&= \|D(\overline{G}^{\,-n}(\varphi))(\xi_0^1+\xi_0^2)\| \geqslant\|D(\overline{G}^{\,-n}(\varphi))\xi_0^2\|- \|D(\overline{G}^{\,-n}(\varphi))\xi_0^1\| \\ &\geqslant\frac{1}{c_2}\biggl(\frac{1}{\mu_2}\biggr)^n\|\xi_0^2\|- c_1\mu_1^n\|\xi_0^1\|. \end{aligned}
\end{equation*}
\notag
$$
It remains to add that, since $\xi_0^2\ne 0$ (because $\xi_0\notin\overline{E}_{\varphi+2\pi l}^{\rm \,u}$), letting $n\to+\infty$ we obtain contradictory estimates. The other cases are considered in a similar way.
Returning to $G$, for each $\varphi\in\mathbb{T}^{\infty}$ we set
$$
\begin{equation}
E_\varphi^{\rm \,u}= \overline{E}_{\operatorname{pr}^{-1}(\varphi)}^{\rm \,u},\quad E_\varphi^{\rm \,s}= \overline{E}_{\operatorname{pr}^{-1}(\varphi)}^{\rm \,s},\quad P_{\varphi}=\overline{P}_{\operatorname{pr}^{-1}(\varphi)},\quad Q_{\varphi}=\overline{Q}_{\operatorname{pr}^{-1}(\varphi)}.
\end{equation}
\tag{2.15}
$$
We stress that, because of periodicity (2.12), formulae (2.15) are independent of the particular choice of the preimage $\operatorname{pr}^{-1}(\varphi)\in E$. Moreover, they define a required hyperbolic structure for $G$.
In fact, the $DG$-invariance of $E_\varphi^{\rm \,u}$ and $E_\varphi^{\rm \,s}$ and properties (2.2)–(2.4) are consequences of (1.26) and conditions (2.8)–(2.11). It only remains to add that the projections $P_{\varphi}$ and $Q_{\varphi}$ in (2.15) possess the required uniform continuity in $\varphi\in\mathbb{T}^{\infty}$. Thus the diffeomorphism $G$ satisfies the conditions in Definition 2.1. The converse part of the lemma is rather simple. In fact, if $G$ is hyperbolic, then we have (2.2)–(2.4). Hence we can set
$$
\begin{equation}
\overline{E}_\varphi^{\rm \,u}=E_{\operatorname{pr}(\varphi)}^{\rm \,u},\quad \overline{E}_\varphi^{\rm \,s}=E_{\operatorname{pr}(\varphi)}^{\rm \,s},\quad \overline{P}_\varphi=P_{\operatorname{pr}(\varphi)},\quad \overline{Q}_\varphi=Q_{\operatorname{pr}(\varphi)}\quad \forall\,\varphi\in E.
\end{equation}
\tag{2.16}
$$
Relying on (1.26) again, it is easy to see that formulae (2.16) define a hyperbolic structure for $\overline{G}$. $\Box$ To state the next result, fix an arbitrary diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ and some positive integer $n_0$. Then the following holds. Proof. If the original maps $G$ is hyperbolic, then $G^{n_0}$ is obviously too: the hyperbolic structure for $G^{n_0}$ also has the form (2.2), and estimates similar to (2.3) and (2.4) hold with the constants $c_1$, $\mu_1^{n_0}$ and $c_2$, $\mu_2^{n_0}$, respectively.
Now let $G^{n_0}$ be hyperbolic. Also let $\overline{G}$ be a global lifting of $G$. Then it is obvious that the diffeomorphism $f=\overline{G}^{n_0}$ is a global lifting from $G^{n_0}$. Hence it follows from Lemma 2.1 that verifying Lemma 2.2 reduces to verifying that $\overline{G}$ is hyperbolic when $f$ is. Thus, assume that there exists a decomposition (2.8) satisfying all the requirements in Definition 2.2 with respect to $f$. In particular, this means that
$$
\begin{equation}
Df(\varphi)\overline{E}^{\rm\,u}_\varphi= \overline{E}^{\rm\,u}_{f(\varphi)},\quad Df(\varphi)\overline{E}^{\rm\,s}_\varphi= \overline{E}^{\rm\,s}_{f(\varphi)}\quad \forall\,\varphi\in E,
\end{equation}
\tag{2.17}
$$
and
$$
\begin{equation}
\|D(f^{\,-n}(\varphi))\xi\| \leqslant c_1\mu_1^n\|\xi\| \quad \forall\,\varphi \in E, \quad \forall\,\xi \in \overline{E}_\varphi^{\rm \,u}, \quad \forall\,n \in\mathbb{N},
\end{equation}
\tag{2.18}
$$
$$
\begin{equation}
\|D(f^{\,n}(\varphi))\xi\| \leqslant c_2\mu_2^n\|\xi\| \quad \forall\,\varphi \in E, \quad \forall\,\xi \in \overline{E}_\varphi^{\rm \,s}, \quad \forall\,n \in\mathbb{N},
\end{equation}
\tag{2.19}
$$
where $c_1,c_2>0$ and $\mu_1,\mu_2\in (0,1)$. Now it is an immediate verification that, apart from (2.8), we have the decomposition $E=\widetilde{E}_\varphi^{\rm \,u}\oplus\widetilde{E}_\varphi^{\rm \,s}$, where the subspaces
$$
\begin{equation}
\widetilde{E}_\varphi^{\rm \,u}=D\overline{G}(\theta) \overline{E}_\theta^{\rm \,u}\big|_{\theta=\overline{G}^{\,-1}(\varphi)}\quad\text{and}\quad \widetilde{E}_\varphi^{\rm \,s}=D\overline{G}(\theta) \overline{E}_\theta^{\rm \,s}\big|_{\theta=\overline{G}^{\,-1}(\varphi)}
\end{equation}
\tag{2.20}
$$
are $Df$-invariant, just as the original subspaces $\overline{E}_\varphi^{\rm \,u}$ and $\overline{E}_\varphi^{\rm \,s}$.
In fact, taking the explicit formula for the differential $Df(\varphi)=D(\overline{G}^{\,n_0}(\varphi))$ (which is obtained from (2.1) by replacing $G$ by $\overline{G}$ and $n$ by $n_0$) into account, we can write conditions (2.17) for $Df$-invariance in the form
$$
\begin{equation}
\begin{aligned} \, D\overline{G}(\varphi_{n_0-1})\circ D\overline{G}(\varphi_{n_0-2})\circ\cdots \circ D\overline{G}(\varphi_{0})\overline{E}_{\varphi_0}^{\rm \, u}&= \overline{E}_{\varphi_{n_0}}^{\rm \,u}, \\ D\overline{G}(\varphi_{n_0-1})\circ D\overline{G}(\varphi_{n_0-2})\circ\cdots \circ D\overline{G}(\varphi_{0})\overline{E}_{\varphi_0}^{\rm\, s}&= \overline{E}_{\varphi_{n_0}}^{\rm \,s}, \end{aligned}
\end{equation}
\tag{2.21}
$$
where $\varphi_j=\overline{G}^{\,j}(\varphi)$, $j\in \mathbb{Z}$. Now applying $D\overline{G}(\varphi_{n_0})$ to the resulting relations we conclude that
$$
\begin{equation*}
\begin{aligned} \, D(\overline{G}^{\,n_0}(\varphi_1))(D\overline{G}(\varphi_{0})\overline{E} _{\varphi_0}^{\rm \,u})&= D\overline{G}(\varphi_{n_0})\overline{E}_{\varphi_{n_0}}^{\rm \,u} \\ \text{and} \qquad D(\overline{G}^{\,n_0}(\varphi_1))(D\overline{G}(\varphi_{0})\overline{E} _{\varphi_0}^{\rm \,s})&= D\overline{G}(\varphi_{n_0})\overline{E}_{\varphi_{n_0}}^{\rm \,s}. \end{aligned}
\end{equation*}
\notag
$$
It remains to add that these formulae imply automatically that the subspaces (2.20) are $Df$-invariant.
Note also that, as an easy verification shows, for the above subspaces (2.20) we have
$$
\begin{equation*}
\begin{aligned} \, D(f^{-n}(\varphi))\widetilde{E}_\varphi^{\rm \,u}&= D\overline{G}(\varphi_{-nn_0-1})\circ D(f^{-n}(\varphi_{-1}))\overline{E}_{\varphi_{-1}}^{\rm \,u} \\ \text{and} \qquad D(f^{n}(\varphi))\widetilde{E}_\varphi^{\rm \,s}&= D\overline{G}(\varphi_{nn_0-1})\circ D(f^{n}(\varphi_{-1}))\overline{E}_{\varphi_{-1}}^{\rm \,s}, \end{aligned}
\end{equation*}
\notag
$$
where, as in (2.21), we have $\varphi_j=\overline{G}^{\,j}(\varphi)$, $j\in \mathbb{Z}$. Hence, as $D\overline{G}$ is bounded (see (1.36)), we obtain estimates of the form (2.18) and (2.19) for the subspaces (2.20).
To summarize, it follows from our constructions that the subspaces (2.20) define another hyperbolic structure for $f$ (in addition to (2.8)). However, we know already that there can be only one hyperbolic structure for $f$ (see a similar place in the proof of (2.12)). Thus, $\widetilde{E}_\varphi^{\rm \,u}=\overline{E}_\varphi^{\rm \,u}$ and $\widetilde{E}_\varphi^{\rm \,s}=\overline{E}_\varphi^{\rm \,s}$. It remains to add that, in combination with (2.20), these equalities yield (2.9). In other words, we have shown that, apart from the properties (2.17), the subspaces $\overline{E}_\varphi^{\rm \,u}$ and $\overline{E}_\varphi^{\rm \,s}$ are also $D\overline{G}$-invariant. Estimates of the form (2.10) and (2.11) also hold for them. More precisely, we can deduce these from (2.18), (2.19), and the representations
$$
\begin{equation*}
\begin{gathered} \, D(\overline{G}^{\,n}(\varphi))= D(f^{\,k}(\theta))\big|_{\theta=\overline{G}^{\,r}(\varphi)}\circ D(\overline{G}^{\,r}(\varphi)),\quad \\ D(\overline{G}^{\,-n}(\varphi))= D(f^{\,-k}(\theta))\big|_{\theta=\overline{G}^{\,-r}(\varphi)}\circ D(\overline{G}^{\,-r}(\varphi)), \\ n=kn_0+r,\quad r\in\{0,1,\dots,n_0-1\}, \end{gathered}
\end{equation*}
\notag
$$
in view of the boundedness
$$
\begin{equation*}
\sup_{\varphi\in E}\|D(\overline{G}^{\,r}(\varphi))\|_{E\to E}+ \sup_{\varphi\in E}\|D(\overline{G}^{\,-r}(\varphi))\|_{E\to E}<\infty,\qquad 0\leqslant r\leqslant n_0-1,
\end{equation*}
\notag
$$
which follows from (1.36). Thus we have shown that the diffeomorphism $\overline{G}$ is hyperbolic, which completes the proof of Lemma 2.2. $\Box$ A further well-known result in hyperbolic theory is connected with the angle between the subspaces $E_\varphi^{\rm \,u}$ and $E_\varphi^{\rm \,s}$ in (2.2). This angle is given by
$$
\begin{equation}
\angle(E_\varphi^{\rm \,u}, E_\varphi^{\rm \,s})= \inf\{\|v-w\|\colon v\in E_\varphi^{\rm \,u},\ w\in E_\varphi^{\rm \,s},\ \|v\|=\|w\|=1\}.
\end{equation}
\tag{2.22}
$$
We have the following result (see [6], Lemma 7.3). Proof. Fix some vectors $v\in E_\varphi^{\rm \,u}$ and $w\in E_\varphi^{\rm \,s}$, $\|v\|=\|w\|=1$, and an integer $k\geqslant 0$. Set $a(k)=D(G^{\,k}(\varphi))(v-w)$ and observe that where by (1.21)
For a lower estimate for $\|a(k)\|$ we consider (2.4) and the inequality
$$
\begin{equation*}
\|D(G^{\,n}(\varphi))\xi\|\geqslant \frac{1}{c_1}\biggl(\frac{1}{\mu_1}\biggr)^n\|\xi\|\quad \forall\,\varphi\in \mathbb{T}^{\infty},\quad \forall\,\xi\in E_\varphi^{\rm \,u},\quad \forall\,n\in\mathbb{N},
\end{equation*}
\notag
$$
which is equivalent to (2.3). On this basis we obtain
$$
\begin{equation*}
\|a(k)\|\geqslant \|D(G^{\,k}(\varphi))v\|-\|D(G^{\,k})(\varphi)w\|\geqslant \frac{1}{c_1}\biggl(\frac{1}{\mu_1}\biggr)^k-c_2\mu_2^k\to+\infty
\end{equation*}
\notag
$$
as $k\to+\infty$. Hence there exists a positive integer $k_0$ independent of $v$ and $w$ such that $\|a(k_0)\|\geqslant 1$. Then it follows from (2.24) (for $k=k_0$) that
$$
\begin{equation*}
\|v-w\|\geqslant N^{-k_0}\quad \forall\,v\in E_\varphi^{\rm \,u},\ w\in E_\varphi^{\rm \,s}\colon \|v\|=\|w\|=1.
\end{equation*}
\notag
$$
Thus, in view of (2.22) we can take $N^{-k_0}$ as $\alpha$ in (2.23). $\Box$ We complete the discussion of the simplest properties of hyperbolic diffeomorphisms by analysing the question of the uniform boundedness of the projections $P_{\varphi}$ and $Q_{\varphi}$ in Definition 2.1. Note that for a finite-dimensional torus these projections are uniformly bounded just because they are continuous. However, in the case of $\mathbb{T}^{\infty}$ this property must be verified separately. Lemma 2.4. For any hyperbolic diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ the following boundedness relations hold: Proof. It follows from the obvious equality $P_{\varphi}+Q_{\varphi}=I$, where $I$ is the identity operator in $E$, that it is sufficient to establish the first property in (2.26).
Assume on the contrary that there exists a sequence of points $\varphi_n\in \mathbb{T}^{\infty}$ such that $\|P_{\varphi_n}\|_{E\to E}\to+\infty$ as $n\to+\infty$. Then by the principle of location of singularities (see [28], Ch. VII, § 1) there exists $\xi\in E$ such that We will assume without loss of generality that Then it follows from the relation that we must haveNow we show that
$$
\begin{equation}
\lim_{n\to+\infty}\frac{\|P_{\varphi_n}\xi\|}{\|Q_{\varphi_n}\xi\|}=1.
\end{equation}
\tag{2.30}
$$
To do this we look at (2.28); this formula shows that
$$
\begin{equation*}
\begin{aligned} \, \|Q_{\varphi_n}\xi\|-\|\xi\|&\leqslant \|P_{\varphi_n}\xi\|\leqslant \|Q_{\varphi_n}\xi\|+\|\xi\|, \\ \text{so} \qquad 1-\frac{\|\xi\|}{\|Q_{\varphi_n}\xi\|}&\leqslant \frac{\|P_{\varphi_n}\xi\|}{\|Q_{\varphi_n}\xi\|}\leqslant 1+\frac{\|\xi\|}{\|Q_{\varphi_n}\xi\|}\,. \end{aligned}
\end{equation*}
\notag
$$
Hence we will obtain the required equality (2.30) automatically once we will have taken (2.29) into account.
Combining the limit relations (2.27), (2.29), and (2.30), which we have established we conclude that
$$
\begin{equation*}
\begin{aligned} \, \biggl\|\frac{P_{\varphi_n}\xi}{\|P_{\varphi_n}\xi\|}+ \frac{Q_{\varphi_n}\xi}{\|Q_{\varphi_n}\xi\|}\biggr\|&= \frac{1}{\|P_{\varphi_n}\xi\|}\biggl\| P_{\varphi_n}\xi+ \frac{\|P_{\varphi_n}\xi\|}{\|Q_{\varphi_n}\xi\|}\,Q_{\varphi_n}\xi\biggr\| \\ &=\frac{1}{\|P_{\varphi_n}\xi\|} \biggl\|\xi+\biggl(\frac{\|P_{\varphi_n}\xi\|}{\|Q_{\varphi_n}\xi\|}-1\biggr) \,Q_{\varphi_n}\xi\biggr\| \\ &\leqslant\frac{\|\xi\|}{\|P_{\varphi_n}\xi\|}+ \biggl|1-\frac{\|Q_{\varphi_n}\xi\|}{\|P_{\varphi_n}\xi\|}\biggr|\to 0,\qquad n\to+\infty, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\angle(E_{\varphi_n}^{\rm u}, E_{\varphi_n}^{\rm s})\leqslant \biggl\|\frac{P_{\varphi_n}\xi}{\|P_{\varphi_n}\xi\|}+ \frac{Q_{\varphi_n}\xi}{\|Q_{\varphi_n}\xi\|}\biggr\|\to 0,\qquad n\to+\infty.
\end{equation*}
\notag
$$
The last relation contradicts (2.23). Thus our assumption fails, and in fact we have (2.26). $\Box$ 2.2. A list of results on hyperbolicity criteriaNow we turn to describing a criterion of hyperbolicity for diffeomorphisms in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$. As in the well-known cone criterion (see [5], Theorem 10.1.5), we postulate the existence of a certain decomposition of $E$ into a direct sum of closed linear subspaces. Condition 2.1. For each $\varphi\in E$ To state the next two conditions we fix an arbitrary diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ and a positive integer $n_0$. Set where $\overline{G}$ is a global lifting of the map $G$ (see (1.23)). Then, drawing upon (2.31) and (2.32), consider the linear operators
$$
\begin{equation}
\Lambda_{j,1}(\varphi) =P(f(\varphi))Df(\varphi)\colon E_j(\varphi)\to E_1(f(\varphi)), \qquad j =1,2,
\end{equation}
\tag{2.35}
$$
$$
\begin{equation}
\text{and} \qquad \Lambda_{j,2}(\varphi) =Q(f(\varphi))Df(\varphi)\colon E_j(\varphi)\to E_2(f(\varphi)), \qquad j =1,2,
\end{equation}
\tag{2.36}
$$
where $f$ is the map in (2.34). Condition 2.2. For each $\varphi\in E$ the operator $\Lambda_{1,1}(\varphi)$ in (2.35) is invertible and the following boundedness condition is satisfied: To state the last condition we need the constants
$$
\begin{equation}
\begin{aligned} \, &\alpha_1=\sup_{\varphi\in E} \|\Lambda_{1,1}^{-1}(\varphi)\|_{E_1(f(\varphi))\to E_1(\varphi)}, \\ &\alpha_2=\sup_{\varphi\in E} \|\Lambda_{2,2}(\varphi)\|_{E_2(\varphi)\to E_2(f(\varphi))}, \end{aligned}
\end{equation}
\tag{2.38}
$$
$$
\begin{equation}
\begin{aligned} \, &\beta_1=\sup_{\varphi\in E} \|\Lambda_{1,2}(\varphi)\|_{E_1(\varphi)\to E_2(f(\varphi))}, \\ &\beta_2=\sup_{\varphi\in E} \|\Lambda_{1, 1}^{-1}(\varphi)\Lambda_{2,1}(\varphi)\|_{E_2(\varphi) \to E_1(\varphi)}, \end{aligned}
\end{equation}
\tag{2.39}
$$
$$
\begin{equation}
\begin{aligned} \, &\gamma_1=\sup_{\varphi\in E}\|\Lambda_{1, 2}(\varphi) \Lambda_{1, 1}^{-1}(\varphi)\|_{E_1(f(\varphi))\to E_2(f(\varphi))}, \\ &\gamma_2=\sup_{\varphi\in E} \|\Lambda_{2,1}(\varphi)\|_{E_2(\varphi)\to E_1(f(\varphi))}, \end{aligned}
\end{equation}
\tag{2.40}
$$
which are certainly finite because of (1.36), (2.33), and (2.37). Using the techniques developed in [19] and [29] we can establish the following two results, combining which we obtain the hyperbolicity criterion in question. Theorem 2.1. A diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ satisfying Conditions 2.1–2.3 is hyperbolic. Theorem 2.2. If a diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ is hyperbolic, then there exists a decomposition (2.31) and a positive integer $n_0$ such that Condiitons 2.1–2.3 are satisfied. Another consequence of Theorems 2.1 and 2.2 concerns the $C^1$-roughness of the property of hyperbolicity. To state it, consider a Banach space $C^1_{\rm per}(E)\subset B^1_{\rm per}(E)$ of vector functions $g(\varphi)$ in $B^1_{\rm per}(E)$ that have additionally the property of boundedness (1.55). We define the norm in this space by
$$
\begin{equation}
\|g\|_{C^1_{\rm per}}=\sup_{\varphi\in E}\|g(\varphi)\|+ \sup_{\varphi\in E}\|g'(\varphi)\|_{E\to E}.
\end{equation}
\tag{2.42}
$$
The following result on the $C^1$-roughness of hyperbolic diffeomorphisms is true. Theorem 2.3. Let $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ be a hyperbolic diffeomorphism and let $\overline{G}$ be the corresponding global lifting (1.23). Then there exists a sufficiently small $\varepsilon=\varepsilon(G)> 0$ such that for each vector function $\Delta(\varphi)\in C^1_{\rm per}(E)$, $\|\Delta\|_{C^1_{\rm per}}<\varepsilon$ ($\|\,{\cdot}\,\|_{C^1_{\rm per}}$ is the norm (2.42)) the corresponding perturbed map is also a hyperbolic diffeomorphism in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$. Since we gave detailed proofs of similar results in [22], here we only present the general schemes of such proofs. We start with Theorem 2.1. It follows from Lemmas 2.1 and 2.2 that verifying this theorem reduces to verifying that the map (2.34) is hyperbolic. In this connection we discuss the scheme for finding a hyperbolic structure (2.8) for a diffeomorphism $f$. Note first of all that, although the subspaces $\overline{E}^{\rm\,u}_\varphi$ and $\overline{E}^{\rm\,s}_\varphi$ in (2.8) are generally speaking distinct from $E_1(\varphi)$ and $E_2(\varphi)$ in (2.31), there is a link between these subspaces. Namely, the required subspaces $\overline{E}^{\rm\,u}_\varphi$ and $\overline{E}^{\rm\,s}_\varphi$ can be represented parametrically as
$$
\begin{equation}
\overline{E}^{\rm\,u}_\varphi =\{\xi=u_1+u_2\in E\colon u_2=a(\varphi)u_1,\ u_1\in E_1(\varphi)\}
\end{equation}
\tag{2.44}
$$
$$
\begin{equation}
\text{and} \qquad \overline{E}^{\rm\,s}_\varphi =\{\xi=u_1+u_2\in E\colon u_1=b(\varphi)u_2,\ u_2\in E_2(\varphi)\}.
\end{equation}
\tag{2.45}
$$
Here $u_1\in E_1(\varphi)$ and $u_2\in E_2(\varphi)$ are vector parameters on $\overline{E}^{\rm\,u}_\varphi$ and $\overline{E}^{\rm\,s}_\varphi$, respectively, and the linear operators $a(\varphi)\colon E_1(\varphi)\to E_2(\varphi)$ and $b(\varphi)\colon E_2(\varphi)\to E_1(\varphi)$ to be determined have the following properties. First, they are $2\pi$-periodic in $\varphi$ and uniformly bounded, that is,
$$
\begin{equation*}
\sup_{\varphi\in E}\|a(\varphi)\|_{E_1(\varphi)\to E_2(\varphi)}<\infty\quad\text{and}\quad \sup_{\varphi\in E}\|b(\varphi)\|_{E_2(\varphi)\to E_1(\varphi)}<\infty.
\end{equation*}
\notag
$$
Second, the operators $a(\varphi)P(\varphi)\colon E\to E$ and $b(\varphi)Q(\varphi)\colon E\to E$ are uniformly continuous in $\varphi\in E$ in the uniform operator topology. It was shown in [22] that the $Df$-invariance of the subspaces (2.44) and (2.45) implies certain nonlinear operator equations for $a(\varphi)$ and $b(\varphi)$, and one can apply the contracting mapping principle to these equations (inequalities (2.41) ensure the applicability of this principle in appropriate function spaces). As a result, we show that the required subspaces $\overline{E}^{\rm\,u}_\varphi$ and $\overline{E}^{\rm\,s}_\varphi$ exist. Using their parametric representations (2.44) and (2.45) we can verify estimates of the form (2.18) and (2.19), as well as the decomposition (2.8) (in combination with the uniform continuity of the projections $\overline{P}_{\varphi}$ and $\overline{Q}_{\varphi}$). The proof of Theorem 2.2 is much simpler. By Lemma 2.1, the lifting $\overline{G}$ of a hyperbolic diffeomorphism $G$ is also hyperbolic, so we have relations of the form (2.8)–(2.11) for $\overline{G}$. Next we consider the operator $f(\varphi)=\overline{G}^{\,n_0}(\varphi)$, where the positive integer $n_0$ satisfies
$$
\begin{equation}
c_1\mu_1^{n_0}<1\quad\text{and}\quad c_2\mu_2^{n_0}<1
\end{equation}
\tag{2.46}
$$
(the $c_j$ and $\mu_j$, $j=1,2$, are the constants in (2.10) and (2.11)), and we set
$$
\begin{equation}
E_1(\varphi)=\overline{E}_\varphi^{\rm \,u} \quad\text{and}\quad E_2(\varphi)=\overline{E}_\varphi^{\rm \,s} \quad \forall\,\varphi\in E,
\end{equation}
\tag{2.47}
$$
where $\overline{E}_\varphi^{\rm \,u}$ and $\overline{E}_\varphi^{\rm \,s}$ are the subspaces from (2.8). It is easy to see that the integer $n_0$ and the subspaces $E_1(\varphi)$ and $E_2(\varphi)$ selected in this way satisfy Conditions 2.1–2.3. Condition 2.1 follows in this case from equalities (2.8), (2.12) and (2.16) and Lemma 2.4. To verify that Conditions 2.2 and 2.3 also hold, note that now the operators (2.35) and (2.36) have the form
$$
\begin{equation}
\begin{alignedat}{2} \Lambda_{1,1}(\varphi)&=Df(\varphi)\colon\overline{E}_\varphi^{\rm \,u}\to \overline{E}_{f(\varphi)}^{\rm \,u},&\qquad \Lambda_{2,1}(\varphi)&=0, \\ \Lambda_{2,2}(\varphi)&=Df(\varphi)\colon\overline{E}_\varphi^{\rm \,s}\to \overline{E}_{f(\varphi)}^{\rm \,s},&\qquad \Lambda_{1,2}(\varphi)&=0. \end{alignedat}
\end{equation}
\tag{2.48}
$$
On the other hand, combining (2.48) and the fact that $Df(\varphi)$ is invertible we conclude that now we have $\beta_1=\beta_2=\gamma_1=\gamma_2=0$, and the constants $\alpha_1$ and $\alpha_2$ satisfy
$$
\begin{equation}
\begin{aligned} \, \alpha_1&=\sup_{\varphi\in E} \|(Df(\varphi))^{-1}\|_{\overline{E}_{f(\varphi)}^{\rm\, u}\to \overline{E}_\varphi^{\rm \,u}}\leqslant c_1\mu_1^{n_0}<1, \\ \alpha_2&=\sup_{\varphi\in E} \|Df(\varphi)\|_{\overline{E}_{\varphi}^{\rm \,s}\to \overline{E}_{f(\varphi)}^{\rm \,s}}\leqslant c_2\mu_2^{n_0}<1. \end{aligned}
\end{equation}
\tag{2.49}
$$
Thus, Conditions 2.2 and 2.3 also hold. Theorem 2.2 is proved. In the case of Theorem 2.3 the situation is as follows. It is easy to show (see similar constructions in [22]) that for small $\Delta(\varphi)\in C^1_{\rm per}(E)$ the map (2.43) continues to belong to $\operatorname{Diff}(\mathbb{T}^{\infty})$. Hence to verify that $G_{\Delta}$ is hyperbolic, it is sufficient by Theorem 2.1 to show that for suitable subspaces $E_1(\varphi)$ and $E_2(\varphi)$ and a positive integer $n_0$ the diffeomorphism under consideration satisfies Conditions 2.1–2.3. Since we assume that the original map $G$ is hyperbolic, its global lifting $\overline{G}$ also has this property by Lemma 2.1. So we have (2.8)–(2.11). Now, as in the proof of Theorem 2.2, taking the subspaces (2.47) for $E_1(\varphi)$ and $E_2(\varphi)$ and setting $f_{\Delta}(\varphi)=\overline{G}_{\Delta}^{\,n_0}(\varphi)$, where $n_0\in\mathbb{N}$ satisfies (2.46), we can verify Conditions 2.1–2.3 in this setting. Recall that the fact that Condition 2.1 holds for the subspaces (2.47) has already been established. As concerns Conditions 2.2 and 2.3, we turn to properties (1.36). They show that for each fixed $n\in\mathbb{Z}$
$$
\begin{equation*}
\sup_{\varphi\in E}\|D(\overline{G}_{\Delta}^{\,n}(\varphi))- D(\overline{G}^{\,n}(\varphi))\|_{E\to E}\to 0\quad\text{as}\quad \|\Delta\|_{C^1_{\rm per}}\to 0.
\end{equation*}
\notag
$$
Hence, as the projections $P(\varphi)$ and $Q(\varphi)$ are uniformly bounded and uniformly continuous (see (2.33)), we conclude that, first, Condition 2.2 holds for all sufficiently small $\Delta\in C^1_{\rm per}(E)$; second, as $\|\Delta\|_{C^1_{\rm per}}\to 0$, the quantities $\alpha_j(\Delta)$, $\beta_j(\Delta)$, and $\gamma_j(\Delta)$, $j=1,2$, corresponding to $f_{\Delta}(\varphi)$ and defined by equalities of the form (2.38)–(2.40) have finite limits $\alpha_j^0$, $\beta_j^0$, and $\gamma_j^0$, $j=1,2$, respectively. Taking (2.48) into account, it is easy to observe that $\beta_1^0=\beta_2^0=\gamma_1^0=\gamma_2^0=0$, $\alpha_1^0=\alpha_1$, and $\alpha_2^0=\alpha_2$, where $\alpha_1$ and $\alpha_2$ are the quantities in (2.49). Thus, for small $\Delta\in C^1_{\rm per}(E)$ we certainly have (2.41), and Theorem 2.3 is proved. We complete the discussion of a criterion for hyperbolicity by observing that, by [22], the analogues of Theorems 2.1 and 2.2 also hold in a weaker version of our theory, when in the definition of the class $\operatorname{Diff}(\mathbb{T}^{\infty})$, in Definition 2.1, and in Condition 2.1, in place of the uniform continuity of the differential $DG(\varphi)$ and the projections $P_{\varphi}$, $Q_{\varphi}$, $P(\varphi)$, and $Q(\varphi)$ we assume just ordinary continuity. However, in that case we cannot establish a full-scale analogue of Theorem 2.3: we can only show that, in the weaker version of the theory, hyperbolic diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ are $C^1$-rough under the additional assumption that the projections $P_{\varphi}$ and $Q_{\varphi}$ are uniformly continuous. 2.3. An analysis of an exampleBelow we present a particular example of a hyperbolic diffeomorphism in the class $\operatorname{Diff}(\mathbb{ T}^{\infty})$, which is different from trivial examples of the form (2.5). First we look at the well-known modified ‘Arnold cat’ mapping that was presented in [30]. This map acts on the torus
$$
\begin{equation*}
\mathbb{T}^2=\{\vartheta=\operatorname{colon}(\xi,\eta)\colon 0\leqslant \xi\leqslant 2\pi \,(\operatorname{mod}\,2\pi) ,\ 0\leqslant \eta\leqslant 2\pi \,(\operatorname{mod}\,2\pi) \}.
\end{equation*}
\notag
$$
In the variables $\xi$ and $\eta$ it has the form
$$
\begin{equation}
\xi\mapsto \xi+\eta+\delta\cos \eta \,(\operatorname{mod}\,2\pi) ,\qquad \eta\mapsto \xi+2\eta \,(\operatorname{mod}\,2\pi) ,
\end{equation}
\tag{2.50}
$$
where $\delta=\operatorname{const}>0$. Its hyperbolicity properties were analysed in [31], where it was shown that for each $\delta\in (0,1)$ this map is an Anosov diffeomorphism. The example in question is a generalization of (2.50) to the infinite-dimensional case. To describe it, we look at the Banach space $E$ of vectors
$$
\begin{equation}
\varphi=\operatorname{colon} (\vartheta_1,\vartheta_2,\dots,\vartheta_k,\dots),\qquad \vartheta_k=\operatorname{colon}(\xi_k,\eta_k)\in\mathbb{R}^2,\quad k\geqslant 1,
\end{equation}
\tag{2.51}
$$
with finite norm (we select the norm $\|\,{\cdot}\,\|_{*}$ in $\mathbb{R}^2$ slightly below). Next, on the infinite-dimensional torus $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where
$$
\begin{equation*}
\mathbb{Z}^{\infty}=\{\varphi=\operatorname{colon} (\vartheta_1, \vartheta_2,\dots,\vartheta_k,\dots)\in E\colon \vartheta_k=\operatorname{colon}(\xi_k,\eta_k)\in\mathbb{Z}^2,\ k\geqslant 1\},
\end{equation*}
\notag
$$
we define an operator $G$ by
$$
\begin{equation}
G\colon \varphi\mapsto\overline{G}(\varphi) \,(\operatorname{mod}\,2\pi) ,
\end{equation}
\tag{2.53}
$$
where $\overline{G}(\varphi)=\Lambda\varphi+g(\varphi)$ and
$$
\begin{equation}
\Lambda\varphi=\operatorname{colon}(\Lambda_0\vartheta_1, \Lambda_0\vartheta_2,\dots, \Lambda_0\vartheta_k,\dots),\quad \Lambda_0=\begin{pmatrix} 1&1 \\ 1&2 \end{pmatrix}.
\end{equation}
\tag{2.54}
$$
As concerns the vector function $g(\varphi)$, it acts on vectors (2.51) in $E$ by the formula
$$
\begin{equation}
g(\varphi)=\operatorname{colon}(\overline{\vartheta}_1, \overline{\vartheta}_2,\dots,\overline{\vartheta}_k,\dots),\qquad \overline{\vartheta}_k=\operatorname{colon}(\delta_k\cos(\eta_{k+1}),0),\quad k\geqslant 1,
\end{equation}
\tag{2.55}
$$
where
$$
\begin{equation}
\delta_k=\operatorname{const}>0\quad \forall\,k\geqslant 1,\qquad \lim_{k\to+\infty}\delta_k=0.
\end{equation}
\tag{2.56}
$$
The map (2.53) introduced above is a possible generalization of the infinite- dimensional diffeomorphism (2.50). More precisely, it is a chain of countably many two-dimensional maps of the form (2.50) which are consecutively linked in one direction. We must also note that (2.54)–(2.56) show that $\Lambda\in L(\mathbb{Z}^{\infty})$, $g(\varphi)\in C^1_{\rm per}(E)$, and the corresponding map (1.56) is completely continuous. Thus, by Theorem 1.3, to prove that $G$ is a diffeomorphism it remains to verify that for each $\varphi\in E$ the linear operator $D\overline{G}(\varphi)\colon E\to E$ is invertible. As regards the inclusion $G\in\operatorname{Diff}(\mathbb{ T}^{\infty})$, to verify it we must also show the property of boundedness (1.27). In what follows we show that all these facts hold under certain additional constraints on the constants $\delta_k$, $k\geqslant 1$, in (2.56). In investigating the invertibility of $D\overline{G}(\varphi)=\Lambda+g'(\varphi)$ we use an auxiliary result. Before stating it, we make the following convention: in the statement below we use the same symbol $\|{\,{\cdot}\,}\|$ for an arbitrary norm in $\mathbb{R}^m$, $m\geqslant 1$, and for the corresponding induced norm in the space of $m\times m$ matrices. Lemma 2.5. Given a positive integer $m$ let $A_k$ and $B_k$, $k\geqslant 1$, be sequences of $m\times m$ matrices such that Proof. Fix an arbitrary sequence of vectors of the form (2.59) and consider the corresponding sequence
$$
\begin{equation}
x_s=A_s^{-1}y_s+\sum_{k=1}^{\infty}(-1)^k\biggl(\,\prod_{n=1}^kA_{s+n-1}^{-1} B_{s+n-1}\biggr)A_{s+k}^{-1}y_{s+k},\qquad s\geqslant 1.
\end{equation}
\tag{2.63}
$$
It is easy to show on the basis of (2.58) and (2.63) that, first, this sequence has the property of boundedness required in (2.61) and satisfies (2.60); second, we have the estimate (2.62) for this sequence. So, to justify the theorem it remains to verify that the corresponding homogeneous system has a unique bounded solution $\{x_k,k\geqslant 1\}$, namely, the trivial one.
Let $\{x_k, k\geqslant 1\}$ be some bounded solution of (2.64). Then taking the first $k$ equations in this system we express $x_1$ in terms of $x_{k+1}$ by the formula
$$
\begin{equation}
x_1=(-1)^k\biggl(\,\prod_{n=1}^kA_{n}^{-1}B_{n}\biggr)x_{k+1}.
\end{equation}
\tag{2.65}
$$
Now we note that $\sup_{k\geqslant 1}\|x_k\|<\infty$ and from (2.57) and (2.58) obtain in succession
$$
\begin{equation*}
\begin{gathered} \, \sup_{s\geqslant 1}\|A_s^{-1}\|<\infty,\qquad \sum_{k=1}^{\infty}\|A_{k+1}^{-1}\|\cdot\prod_{n=1}^{k} \bigl(\|A_{n}^{-1}\|\cdot\|B_{n}\|\bigr)<\infty, \\ \lim_{k\to+\infty}\prod_{n=1}^{k}\bigl(\|A_{n}^{-1}\|\cdot\|B_{n}\|\bigr)=0. \end{gathered}
\end{equation*}
\notag
$$
Combining these facts and taking the limit as $k\to+\infty$ in (2.65) we verify that $x_1=0$.
In a similar way, using the block of relations (2.64) with indices $s,s+1,\dots, s+k-1$, we conclude that
$$
\begin{equation}
x_s=(-1)^k\biggl(\,\prod_{n=1}^kA_{s+n-1}^{-1}B_{s+n-1}\biggr)x_{s+k},\quad s\geqslant 1.
\end{equation}
\tag{2.66}
$$
As in the previous case, it is easy to see that for each fixed $s\geqslant 1$ the right-hand side of (2.66) tends to zero as $k\to+\infty$. By implication $x_s=0$ for all $s\in\mathbb{N}$. $\Box$ Before we go over to the invertibility of the operator $D\overline{G}(\varphi)$, we specify the norm $\|\,{\cdot}\,\|_{*}$ in (2.52). Consider the eigenvalues
$$
\begin{equation}
\lambda_1=\frac{3+\sqrt{5}}{2}>1\quad\text{and}\quad \lambda_2=\frac{3-\sqrt{5}}{2}<1
\end{equation}
\tag{2.67}
$$
of the matrix $\Lambda_0$ in (2.54) and the corresponding eigenvectors
$$
\begin{equation}
e_s=\frac{1}{\sqrt{1+(\lambda_s-1)^2}} \operatorname{colon}(1,\lambda_s-1),\qquad s=1,2.
\end{equation}
\tag{2.68}
$$
Now, using the formulae
$$
\begin{equation}
\vartheta_k=t_k e_1+\tau_k e_2,\qquad t_k=(\vartheta_k, e_1)\quad\text{and}\quad \tau_k=(\vartheta_k, e_2),
\end{equation}
\tag{2.69}
$$
where $(\,{\cdot}\,{,}\,{\cdot}\,)$ is the Euclidean inner product, set
$$
\begin{equation}
\|\vartheta_k\|_{*}=\max(c_1^0|t_k|,c_2^0|\tau_k|)\quad \forall\,k\geqslant 1,
\end{equation}
\tag{2.70}
$$
where the constants $c_j^0>0$, $j=1,2$, can for now be arbitrary. As regards the operator $\Lambda+g'(\varphi)$, in the variables $t_k$ and $\tau_k$ in (2.69), $k\geqslant 1$, it takes the form
$$
\begin{equation}
\begin{aligned} \, t_k&\mapsto\lambda_1t_k+(C_k(\varphi)e_1,e_1)t_{k+1} +(C_k(\varphi)e_2, e_1)\tau_{k+1}, \\ \tau_k&\mapsto\lambda_2\tau_k+(C_k(\varphi)e_1,e_2)t_{k+1}+ (C_k(\varphi)e_2,e_2)\tau_{k+1}, \end{aligned}
\end{equation}
\tag{2.71}
$$
where $k\in\mathbb{N}$, and the $2\times 2$ matrices $C_k(\varphi)$ are given by
$$
\begin{equation}
C_k(\varphi)=-\begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix} \delta_k\sin(\eta_{k+1}).
\end{equation}
\tag{2.72}
$$
We fix two arbitrary sequences of numbers $\overline{t}_k$ and $\overline{\tau}_k$, $k\geqslant 1$, and consider the countable system of equations
$$
\begin{equation}
\begin{aligned} \, \lambda_1t_k+(C_k(\varphi)e_1,e_1)t_{k+1}+(C_k(\varphi)e_2,e_1)\tau_{k+1} &=\overline{t}_k, \\ \lambda_2\tau_k+(C_k(\varphi)e_1,e_2)t_{k+1}+(C_k(\varphi)e_2,e_2)\tau_{k+1} &=\overline{\tau}_k. \end{aligned}
\end{equation}
\tag{2.73}
$$
By the representation (2.71), to verify that the operator $D\overline{G}(\varphi)$ is invertible we only need to show that this system has a unique solution $\{t_k,\tau_k,k\geqslant 1\}$, where $t_k$ and $\tau_k$ are bounded sequences of numbers. For a more convenient analysis we write (2.73) in the vector form (2.60) with suitable sequences (2.59) and (2.61). Namely, we set
$$
\begin{equation}
x_k=\operatorname{colon}(t_k,\tau_k),\quad y_k=\operatorname{colon}(\overline{t}_k,\overline{\tau}_k),\quad A_k=\operatorname{diag}\{\lambda_1,\lambda_2\},\qquad k\geqslant 1,
\end{equation}
\tag{2.74}
$$
$$
\begin{equation}
\text{and} \qquad B_k=\begin{pmatrix} (C_k(\varphi)e_1,e_1)&(C_k(\varphi)e_2,e_1) \\ (C_k(\varphi)e_1,e_2)&(C_k(\varphi)e_2,e_2) \end{pmatrix},\qquad k\geqslant 1.
\end{equation}
\tag{2.75}
$$
As a result, we obtain a system of the form (2.60) and wish to apply Lemma 2.5 to this system. To verify conditions (2.57) and (2.58) in the lemma we estimate the norms of the operators $A_k$ and $B_k$ in (2.74) and (2.75). Taking the definition (2.70) of the norm in $\mathbb{R}^2$ into account, on the basis of the explicit formulae (2.67), (2.68), and (2.72) we conclude that
$$
\begin{equation}
\begin{aligned} \, \|A_k^{-1}\|&=\frac{1}{\lambda_2},\quad \nonumber \\ \|B_k\|&=\max\biggl(|(C_k(\varphi)e_1,e_1)|+c_0|(C_k(\varphi)e_2, e_1)|,\nonumber \\ &\qquad\qquad\qquad\qquad\frac{1}{c_0}|(C_k(\varphi)e_1, e_2)|+|(C_k(\varphi)e_2, e_2)|\biggr) \nonumber \\ &\leqslant\delta_k\max\biggl(\frac{1}{\sqrt{5}}+ c_0\frac{\sqrt{5}-1}{2\sqrt{5}}\,, \frac{1}{\sqrt{5}}+\frac{1}{c_0}\,\frac{\sqrt{5}+1}{2\sqrt{5}}\biggr), \end{aligned}
\end{equation}
\tag{2.76}
$$
where $c_0=c_1^0/c_2^0$. Now we specify the free parameter $c_0>0$ so as to minimize the right-hand side of the inequality in (2.76). Then we see that the corresponding minimum is attained for and with this choice of $c_0$, for the sequence of matrices $B_k$ we obtain
$$
\begin{equation}
\|B_k\|\leqslant\frac{2\delta_k}{\sqrt{5}}\,,\qquad k\geqslant 1.
\end{equation}
\tag{2.77}
$$
Throughout what follows we assume that
$$
\begin{equation}
\theta_0\overset{\rm def}{=}\sup_{s\geqslant 1}\frac{1}{\lambda_2} \biggl(1+\sum_{k=1}^{\infty}\biggl(\frac{4}{\sqrt{5}(3-\sqrt{5})}\biggr)^k \prod_{n=1}^k\delta_{s+n-1}\biggr)<\infty.
\end{equation}
\tag{2.78}
$$
Then by (2.76) and (2.77) we are certainly under the assumptions of Lemma 2.5, which ensures that $D\overline{G}(\varphi)$ is invertible as required. Moreover, we conclude from (2.62) and the definition of the norm in $E$ (see (2.52) and (2.70)) that we have
$$
\begin{equation*}
\sup_{\varphi\in E}\|(\Lambda+g'(\varphi))^{-1}\|_{E\to E}\leqslant \theta_0,
\end{equation*}
\notag
$$
where $\theta_0$ is the constant in (2.78). Thus, provided that (2.78) holds, the map (2.53) under consideration is a diffeomorphism in $\operatorname{Diff}(\mathbb{T}^{\infty})$. To find conditions ensuring that the diffeomorphism (2.53) is hyperbolic, we use Theorem 2.1. Since the operator (2.54) is clearly hyperbolic, now we can take $E_1(\varphi)=E_1$ and $E_2(\varphi)=E_2$ as the subspaces in the decomposition (2.31). Here $E_1$ and $E_2$ are the root subspaces of $\Lambda$ which have the form
$$
\begin{equation}
E_1 =\Bigl\{u=\operatorname{colon} (\vartheta_1,\vartheta_2,\dots,\vartheta_k,\dots)\colon \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad\vartheta_k=t_ke_1,\ t_k\in\mathbb{R},\ k\geqslant 1,\ \sup_{k\geqslant 1}|t_k|<\infty\Bigr\}
\end{equation}
\tag{2.79}
$$
$$
\begin{equation}
\text{and} \qquad E_2 =\Bigl\{v=\operatorname{colon} (\vartheta_1,\vartheta_2,\dots,\vartheta_k,\dots)\colon \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad\vartheta_k=\tau_ke_2,\ \tau_k\in\mathbb{R},\ k\geqslant 1,\ \sup_{k\geqslant 1}|\tau_k|<\infty\Bigr\}.
\end{equation}
\tag{2.80}
$$
Clearly, the space $E$ of vectors (2.51) introduced above is the direct sum $E_1\oplus E_2$, and we have
$$
\begin{equation}
\Lambda u=\lambda_1 u\quad \forall\,u\in E_1,\qquad \Lambda v=\lambda_2 v\quad \forall\,v\in E_2.
\end{equation}
\tag{2.81}
$$
As concerns the projections $P$ and $Q$ corresponding to this decomposition, they act by the formulae
$$
\begin{equation}
P\xi=u\quad\text{and}\quad Q\xi=v\qquad \forall\,\xi=\operatorname{colon}(\xi_1,\xi_2,\dots,\xi_k,\dots)\in E,
\end{equation}
\tag{2.82}
$$
where $\xi_k\in\mathbb{R}^2$, $k\geqslant 1$, and the vectors $u\in E_1$ and $v\in E_2$ are given by
$$
\begin{equation}
\begin{alignedat}{4} u&=\operatorname{colon} (\vartheta_1,\vartheta_2,\dots,\vartheta_k,\dots),&\qquad \vartheta_k&=t_ke_1,&\quad t_k&=(\xi_k,e_1),&\quad k&\geqslant 1, \\ v&=\operatorname{colon} (\vartheta_1,\vartheta_2,\dots,\vartheta_k,\dots),&\qquad \vartheta_k&=\tau_ke_2,&\quad \tau_k&=(\xi_k,e_2),&\quad k&\geqslant 1. \end{alignedat}
\end{equation}
\tag{2.83}
$$
Now we turn to the map (2.34) for $n_0=1$ and observe that, for the relevant operators (2.35) and (2.36), from (2.79)–(2.83) we can obtain explicit formulae. To deduce these we identify vectors $u\in E_1$ and $v\in E_2$ with the infinite-dimensional vectors
$$
\begin{equation*}
t=\operatorname{colon}(t_1,t_2,\dots,t_k,\dots)\quad\text{and}\quad \tau=\operatorname{colon}(\tau_1,\tau_2,\dots,\tau_k,\dots),
\end{equation*}
\notag
$$
where the $t_k$ and $\tau_k$ are the variables in (2.79) and (2.80). As a result, we obtain the series of representations
$$
\begin{equation}
\Lambda_{1,1}(\varphi)\colon t\mapsto\overline{t} = \operatorname{colon}\bigl(\lambda_1t_1+(C_1(\varphi)e_1,e_1)t_2,\lambda_1t_2+ (C_2(\varphi)e_1,e_1)t_3,\dots, \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad\lambda_1t_k+(C_k(\varphi)e_1, e_1)t_{k+1},\dots\bigr),
\end{equation}
\tag{2.84}
$$
$$
\begin{equation}
\Lambda_{2,1}(\varphi)\colon\tau\mapsto t =\operatorname{colon} \bigl((C_1(\varphi)e_2,e_1)\tau_2,(C_2(\varphi)e_2,e_1)\tau_3,\dots, \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad(C_k(\varphi)e_2,e_1)\tau_{k+1},\dots\bigr),
\end{equation}
\tag{2.85}
$$
$$
\begin{equation}
\Lambda_{1,2}(\varphi)\colon t\mapsto \tau =\operatorname{colon} \bigl((C_1(\varphi)e_1,e_2)t_2,(C_2(\varphi)e_1,e_2)t_3,\dots, \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad(C_k(\varphi)e_1,e_2)t_{k+1},\dots\bigr),
\end{equation}
\tag{2.86}
$$
$$
\begin{equation}
\Lambda_{2,2}(\varphi)\colon\tau\mapsto\overline{\tau} = \operatorname{colon}\bigl(\lambda_2\tau_1+ (C_1(\varphi)e_2, e_2)\tau_2, \,\,\lambda_2\tau_2+ (C_2(\varphi)e_2, e_2)\tau_3,\dots, \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\qquad\qquad\lambda_2\tau_k+(C_k(\varphi)e_2,e_2)\tau_{k+1},\dots\bigr),
\end{equation}
\tag{2.87}
$$
where we recall that the $C_k(\varphi)$, $k\geqslant 1$, are the matrices in (2.72). To verify that Condition 2.2 holds here we consider the infinite countable system
$$
\begin{equation}
\lambda_1t_k+(C_k(\varphi)e_1,e_1)t_{k+1}=\overline{t}_k,\qquad k\geqslant 1,
\end{equation}
\tag{2.88}
$$
where $\{\overline{t}_k,k\geqslant 1\}$ is an arbitrary bounded sequence of numbers. Next, on the basis of the obvious bounds
$$
\begin{equation*}
|(C_k(\varphi)e_1,e_1)|\leqslant\frac{\delta_k}{\sqrt{5}}\,,\qquad k\geqslant 1,
\end{equation*}
\notag
$$
using Lemma 2.5 we conclude that, under the additional assumption
$$
\begin{equation}
\alpha_1^0\overset{\rm def}{=}\sup_{s\geqslant 1}\frac{1}{\lambda_1} \biggl(1+\sum_{k=1}^{\infty}\biggl(\frac{2}{\sqrt{5}\,(3+\sqrt{5}\,)}\biggr)^k \prod_{n=1}^k\delta_{s+n-1}\biggr)<\infty,
\end{equation}
\tag{2.89}
$$
system (2.88) has a unique bounded solution $\{t_k,k\geqslant 1\}$. Hence we conclude from the representation (2.84) that the operator $\Lambda_{1,1}(\varphi)$ is invertible; moreover,
$$
\begin{equation}
\sup_{\varphi\in E}\|\Lambda_{1,1}^{-1}(\varphi)\|_{E_1\to E_1} \leqslant\alpha_1^0.
\end{equation}
\tag{2.90}
$$
Estimates for the norms of the remaining operators $\Lambda_{1,2}(\varphi)$, $\Lambda_{2,1}(\varphi)$, and $\Lambda_{2, 2}(\varphi)$ go on without complications. In fact, it is easy to show on the basis of (2.85)–(2.87) that
$$
\begin{equation}
\|\Lambda_{1,2}(\varphi)\|_{E_1\to E_2} \leqslant\frac{c_2^0}{c_1^0}\cdot \sup_{\varphi\in E,k\geqslant 1}|(C_k(\varphi)e_1,e_2)|\leqslant \frac{c_2^0}{c_1^0}\,\frac{\sqrt{5}+1}{2\sqrt{5}}\max_{k\geqslant 1}\delta_k,
\end{equation}
\tag{2.91}
$$
$$
\begin{equation}
\|\Lambda_{2,1}(\varphi)\|_{E_2\to E_1} \leqslant\frac{c_1^0}{c_2^0}\cdot \sup_{\varphi\in E,k\geqslant 1}|(C_k(\varphi)e_2,e_1)|\leqslant \frac{c_1^0}{c_2^0}\,\frac{\sqrt{5}-1}{2\sqrt{5}}\max_{k\geqslant 1}\delta_k,
\end{equation}
\tag{2.92}
$$
$$
\begin{equation}
\|\Lambda_{2,2}(\varphi)\|_{E_2\to E_2} \leqslant \sup_{\varphi\in E,k\geqslant 1}(\lambda_2+|(C_k(\varphi)e_2, e_2)|)\leqslant \lambda_2+\frac{1}{\sqrt{5}}\max_{k\geqslant 1}\delta_k.
\end{equation}
\tag{2.93}
$$
Now we turn to Condition 2.3, the last one, and verify that assumptions (2.41) are certainly satisfied for
$$
\begin{equation}
\alpha_1^0<1,\quad \alpha_2^0<1,\quad\text{and}\quad \frac{\alpha_1^0}{5}\Bigl(\,\max_{k\geqslant 1}\delta_k\Bigr)^2< (1-\alpha_1^0)(1-\alpha_2^0),
\end{equation}
\tag{2.94}
$$
where $\alpha_1^0$ is the quantity (2.89) and the constant $\alpha_2^0$ is defined by
$$
\begin{equation*}
\alpha_2^0=\lambda_2+\frac{1}{\sqrt{5}}\max_{k\geqslant 1}\delta_k.
\end{equation*}
\notag
$$
In fact, it follows from (2.38)–(2.40) and the bounds (2.90)–(2.93) that
$$
\begin{equation*}
\begin{gathered} \, \alpha_1\leqslant\alpha_1^0,\quad \alpha_2\leqslant\alpha_2^0,\quad \beta_1\leqslant\frac{c_2^0}{c_1^0}\,\frac{\sqrt{5}+1}{2\sqrt{5}} \max_{k\geqslant 1}\delta_k,\quad \beta_2\leqslant\alpha_1^0\,\frac{c_1^0}{c_2^0}\, \frac{\sqrt{5}-1}{2\sqrt{5}}\,\max_{k\geqslant 1}\delta_k, \\ \gamma_1\leqslant\alpha_1^0\,\frac{c_2^0}{c_1^0}\, \frac{\sqrt{5}+1}{2\sqrt{5}}\,\max_{k\geqslant 1}\delta_k,\quad \gamma_2\leqslant\frac{c_1^0}{c_2^0}\,\frac{\sqrt{5}-1}{2\sqrt{5}}\, \max_{k\geqslant 1}\delta_k. \end{gathered}
\end{equation*}
\notag
$$
This implies in turn that
$$
\begin{equation*}
\min(\beta_1\beta_2,\gamma_1\gamma_2)\leqslant\frac{\alpha_1^0}{5} \Bigl(\,\max_{k\geqslant 1}\delta_k\Bigr)^2<(1-\alpha_1^0)(1-\alpha_2^0) \leqslant (1-\alpha_1)(1-\alpha_2).
\end{equation*}
\notag
$$
Summarizing, we point out that, if (2.78) holds, then the map (2.53) in question is in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$, while under the additional assumptions (2.94) it is hyperbolic by Theorem 2.1. However, these assumptions are difficult to verify, so it can be reasonable to replace them by some stronger, but simpler restrictions. To do this we use the obvious bounds
$$
\begin{equation}
\prod_{n=1}^k\delta_{s+n-1}\leqslant \Bigl(\,\max_{m\geqslant 1}\delta_m\Bigr)^k\quad \forall\,k,s\in\mathbb{N}.
\end{equation}
\tag{2.95}
$$
Taking account of these bounds in the formula for $\theta_0$ (see (2.78)), we conclude that
$$
\begin{equation*}
\theta_0\leqslant\frac{1}{\lambda_2}\biggl(1+\sum_{k=1}^{\infty} \biggl(\frac{4}{\sqrt{5}\,(3-\sqrt{5}\,)}\, \max_{m\geqslant 1}\delta_m\biggr)^k\,\biggr).
\end{equation*}
\notag
$$
Thus, condition (2.78) certainly holds for
$$
\begin{equation}
\max_{m\geqslant 1}\delta_m<\frac{\sqrt{5}\,(3-\sqrt{5}\,)}{4}\,.
\end{equation}
\tag{2.96}
$$
To verify (2.94) in the case when (2.96) is fulfilled we look at (2.89) first. Using (2.95) again, we verify that
$$
\begin{equation}
\alpha_1^0\leqslant\biggl(\frac{3+\sqrt{5}}{2}-\frac{1}{\sqrt{5}}\, \max_{m\geqslant 1}\delta_m\biggr)^{-1}.
\end{equation}
\tag{2.97}
$$
Now, using (2.97), from (2.94) we go over to the stronger conditions
$$
\begin{equation*}
\begin{gathered} \, \biggl(\frac{3+\sqrt{5}}{2}-\frac{1}{\sqrt{5}}\,\max_{m\geqslant 1} \delta_m\biggr)^{-1}<1,\qquad \max_{m\geqslant 1}\delta_m<\frac{\sqrt{5}\,(\sqrt{5}-1)}{2}\,, \\ \frac{1}{5}\Bigl(\max_{m\geqslant 1}\delta_m\Bigr)^2< \biggl(\frac{\sqrt{5}+1}{2}-\frac{1}{\sqrt{5}}\, \max_{m\geqslant 1}\delta_m\biggr) \biggl(\frac{\sqrt{5}-1}{2}-\frac{1}{\sqrt{5}}\, \max_{m\geqslant 1}\delta_m\biggr) \end{gathered}
\end{equation*}
\notag
$$
and note that in the case of (2.96) they are automatically satisfied. That is, (2.96) is the required fairly simple condition that ensures both the inclusion $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ and the hyperbolicity of the diffeomorphism (2.53).
3. Stable and unstable invariant foliations3.1. An infinite-dimensional version of the Hadamard–Perron theoremThe well-known Hadamard–Perron theorem discusses the existence of so-called stable and unstable local manifolds for a hyperbolic diffeomorphism. As noted in [10], such manifolds can be constructed using either Hadamard’s method [32] (also known as the graph transformation method) or Perron’s method [33]. Both methods are based on representing the relevant local manifolds as the graphs of some functions satisfying the so-called equations of invariance. The difference between the two methods lies in approaches to finding solutions of these equations. In Hadamard’s method solutions are fixed points of contracting operators acting on appropriate function spaces. In Perron’s method solutions can be obtained from the implicit function theorem as applied to nonlinear operators acting on certain spaces of sequences. Below we decide upon Hadamard’s method, which is more geometric. This choice has its shortcomings: we must separately prove that the local manifolds are smooth. Before we present our version of the existence theorem for local manifold, we recall the following relevant phrase from [3]: “Every five years or so, if not more often, someone ‘discovers’ the theorem of Hadamard and Perron, proving it either by Hadamard’s method of proof or by Perron’s.” We have to make yet another ‘discovery’ of this type. In our defence we say that in the case of the torus $\mathbb{T}^{\infty}$, which is non-compact, lacking the usual finite-dimensional machinery (such as the exponential map, normally used to write out the equations of invariance), we can use no known result. So we fix an arbitrary hyperbolic diffeomorphism $G$ in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$ under consideration and consider a global lifting $\overline{G}$ of it. By Lemma 2.1, $\overline{G}$ is also a hyperbolic diffeomorphism, so we have relations (2.8)–(2.12). On this basis we start by proving the existence of local unstable and stable manifolds for $\overline{G}$, and then we extend the corresponding results to $G$. We recall that, as in the finite-dimensional case, for fixed $\delta>0$ and each $\varphi_0\in E$, these manifolds for $\overline{G}$ (which we denote by $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$ and $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)$, respectively) are embedded $C^1$-smooth discs, which are diffeomorphic to the discs
$$
\begin{equation*}
\{\varphi=\varphi_0+u\colon u\in\overline{E}^{\rm \,u}_{\varphi_0},\ \|u\|\leqslant \delta\} \quad\text{and}\quad \{\varphi=\varphi_0+v\colon v\in\overline{E}^{\rm \,s}_{\varphi_0},\ \|v\|\leqslant \delta\},
\end{equation*}
\notag
$$
where $\overline{E}^{\rm \,u}_{\varphi_0}$ and $\overline{E}^{\rm \,s}_{\varphi_0}$ are the subspaces in (2.8) tangent to these discs at the points $u=0$ and $v=0$, respectively. Before stating the result on the existence of local manifolds $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$ and $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)$ we introduce further notation. First of all, we define the so-called Lyapunov norms in $\overline{E}^{\rm \,u}_{\varphi_0}$ and $\overline{E}^{\rm \,s}_{\varphi_0}$. To do this we fix constants $\sigma_1\in(\mu_1,1)$ and $\sigma_2\in(\mu_2,1)$, where $\mu_1$ and $\mu_2$ are the constants in (2.10) and (2.11), and set
$$
\begin{equation}
\begin{alignedat}{3} \|\xi\|_{\varphi_0}&=\sup_{n\in \mathbb{Z}, n\geqslant 0}\, \biggl\|\frac{1}{\sigma_1^n}D(\overline{G}^{\,-n}(\varphi_0))\xi\biggr\|&\quad \forall\,\xi&\in\overline{E}^{\rm \,u}_{\varphi_0},&\quad \forall\,\varphi_0&\in E, \\ \|\xi\|_{\varphi_0}&=\sup_{n\in \mathbb{Z}, n\geqslant 0}\, \biggr\|\frac{1}{\sigma_2^n}D(\overline{G}^{\,n}(\varphi_0))\xi\biggr\|&\quad \forall\,\xi&\in\overline{E}^{\rm \,s}_{\varphi_0},&\quad \forall\,\varphi_0&\in E. \end{alignedat}
\end{equation}
\tag{3.1}
$$
We point out the following. First, by the choice of $\sigma_1$ and $\sigma_2$ and the bounds (2.10) and (2.11), the supremuma in (3.1) are certainly finite. Second – which is more important, in the new norms we have the inequalities
$$
\begin{equation}
\begin{alignedat}{2} \|(D\overline{G}(\varphi_0))^{-1}\xi\|_{\varphi_0}&= \!\sup_{n\in \mathbb{Z},n\geqslant 0}\,\biggl\|\frac{1}{\sigma_1^n} D(\overline{G}^{\,-(n+1)}(\varphi_1))\xi\biggr\|\leqslant \sigma_1\|\xi\|_{\varphi_1}&\quad \forall\,\xi&\in\overline{E}^{\rm \,u}_{\varphi_1}, \\ \|D\overline{G}(\varphi_0)\xi\|_{\varphi_1}&= \!\sup_{n\in \mathbb{Z}, n\geqslant 0}\,\biggl\|\frac{1}{\sigma_2^n} D(\overline{G}^{\,(n+1)}(\varphi_0))\xi\biggr\|\leqslant \sigma_2\|\xi\|_{\varphi_0}&\quad \forall\,\xi&\in\overline{E}^{\rm \,s}_{\varphi_0}; \end{alignedat}
\end{equation}
\tag{3.2}
$$
here and below $\varphi_1=\overline{G}(\varphi_0)$. Third, the norms (3.1) are uniformly equivalent to the original norm in $E$. More precisely, we have
$$
\begin{equation}
\|\xi\|\leqslant\|\xi\|_{\varphi_0}\leqslant c\|\xi\|,
\end{equation}
\tag{3.3}
$$
where $c=\max(1,c_1,c_2)$ and the constants $c_1$ and $c_2$ are borrowed from (2.10) and (2.11). For simpler calculations we use the same notation $\|\xi\|_{\varphi_0}$ for the norms in (3.1). There can be no confusion because the particular norm under consideration will always be clear from the context. Moreover, by $\|\,{\cdot}\,\|_{H_1\to H_2}$, where $(H_1,H_2)$ are some combinations of the subspaces $\overline{E}^{\rm \,u}_{\varphi_0}$, $\overline{E}^{\rm \,s}_{\varphi_0}$, $\overline{E}^{\rm \,u}_{\varphi_1}$, and $\overline{E}^{\rm \,s}_{\varphi_1}$, we mean the corresponding operator norms induced by (3.1). We also add that if a linear operator $A\colon H_1\to H_2$ has an extension to an operator $A\colon E\to E$, then by (3.3) we have Completing the discussion of the norms (3.1), note that (3.2) obviously yields the inequalities
$$
\begin{equation}
\|(D\overline{G}(\varphi_0))^{-1}\|_{\overline{E}^{\rm \,u}_{\varphi_1}\to \overline{E}^{\rm \,u}_{\varphi_0}}\leqslant\sigma_1<1 \quad\text{and}\quad \|D\overline{G}(\varphi_0)\|_{\overline{E}^{\rm \,s}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}}\leqslant\sigma_2<1.
\end{equation}
\tag{3.5}
$$
These bounds will be significant for what follows. In fact, we have introduced the special norms (3.1) just to have these bounds. In what follows we also need the special sets
$$
\begin{equation}
U_{\delta} =\bigl\{(u,\varphi_0)\in E\times E\colon\varphi_0\in E,\ u\in\overline{E}^{\rm \,u}_{\varphi_0},\ \|u\|_{\varphi_0}\leqslant \delta\bigr\}
\end{equation}
\tag{3.6}
$$
$$
\begin{equation}
\text{and} \qquad V_{\delta} =\bigl\{(v,\varphi_0)\in E\times E\colon\varphi_0\in E,\ v\in\overline{E}^{\rm \,s}_{\varphi_0},\ \|v\|_{\varphi_0}\leqslant \delta\bigr\},
\end{equation}
\tag{3.7}
$$
where $\delta$ is a fixed positive constant and $\|u\|_{\varphi_0}$ and $\|v\|_{\varphi_0}$ are the norms (3.1). We define metrics in $U_{\delta}$ and $V_{\delta}$ by the equalities
$$
\begin{equation}
\begin{alignedat}{3} \forall\,w_j&=(u_j,\varphi_0^j)\in U_{\delta},&\quad j&=1,2, &\qquad \rho(w_1,w_2)&=\|u_1-u_2\|+\|\varphi_0^1-\varphi_0^2\|, \\ \forall\,w_j&=(v_j,\varphi_0^j)\in V_{\delta}, &\quad j&=1,2, &\qquad \rho(w_1,w_2)&=\|v_1-v_2\|+\|\varphi_0^1-\varphi_0^2\|, \end{alignedat}
\end{equation}
\tag{3.8}
$$
respectively, where as usual, $\|\,{\cdot}\,\|$ is the norm in $E$. Now, we let $C(U_{\delta})$ and $C(V_{\delta})$ denote the sets of vector functions $\Phi(u,\varphi_0)$ and $\Psi(v,\varphi_0)$, respectively. These functions are defined on the sets (3.6) and (3.7), take values in $E$, and are uniformly continuous with respect to their arguments in the metrics (3.8). We also assume that
$$
\begin{equation}
\sup_{(u,\varphi_0)\in U_{\delta}}\|\Phi(u,\varphi_0)\|<\infty,\quad \sup_{(v,\varphi_0)\in V_{\delta}}\|\Psi(v,\varphi_0)\|<\infty,
\end{equation}
\tag{3.9}
$$
and
$$
\begin{equation}
\begin{aligned} \, \Phi(u,\varphi_0+2\pi l)&=\Phi(u,\varphi_0)\qquad \forall\,l\in\mathbb{Z}^{\infty},\quad \forall\,(u,\varphi_0)\in U_{\delta}, \\ \Psi(v,\varphi_0+2\pi l)&=\Psi(v,\varphi_0)\qquad \forall\,l\in\mathbb{Z}^{\infty},\quad \forall\,(v,\varphi_0)\in V_{\delta}. \end{aligned}
\end{equation}
\tag{3.10}
$$
We stress that conditions (3.10), which can be called conditions of $2\pi$-periodicity, make sense. For example, in the case of $(u,\varphi_0+2\pi l)$ the component $u$ must lie in the subspace $\overline{E}^{\rm \,u}_{\varphi_0+2\pi l}$ and satisfy $\|u\|_{\varphi_0+2\pi l}\leqslant \delta$. However, $\overline{E}^{\rm \,u}_{\varphi_0+2\pi l}= \overline{E}^{\rm \,u}_{\varphi_0}$ (see (2.12)) and $\|\xi\|_{\varphi_0+2\pi l}=\|\xi\|_{\varphi_0}$ for each $\xi\in \overline{E}^{\rm \,u}_{\varphi_0}$, so that $(u,\varphi_0+2\pi l)\in U_{\delta}$, and thus the expression $\Phi(u,\varphi_0+2\pi l)$ is meaningful. Similar arguments also hold in the case of $(v,\varphi_0+2\pi l)$. Now we turn to the statement of the Hadamard–Perron theorem itself. For the hyperbolic map $\overline{G}$ its analogue is as follows. Theorem 3.1. There exists a sufficiently small $\delta>0$ such that for each point $\varphi_0\in E$ the diffeomorphism $\overline{G}$ has an unstable and a stable local manifold, with the following properties. 1) $\Phi_*(u,\varphi_0)\in C(U_{\delta})$, $\Psi_*(v,\varphi_0)\in C(V_{\delta})$, and in addition,
$$
\begin{equation}
\Phi_*(0,\varphi_0) \equiv 0, \qquad \Phi_*(u,\varphi_0) \in\overline{E}^{\rm \,s}_{\varphi_0} \quad \forall\,(u,\varphi_0) \in U_{\delta},
\end{equation}
\tag{3.13}
$$
$$
\begin{equation}
\Psi_*(0,\varphi_0) \equiv 0, \qquad \Psi_*(v, \varphi_0) \in\overline{E}^{\rm \,u}_{\varphi_0} \quad \forall\,(v,\varphi_0) \in V_{\delta}.
\end{equation}
\tag{3.14}
$$
2) The Fréchet partial derivatives $\partial\Phi_*(u,\varphi_0)/\partial u$ and $\partial\Psi_*(v,\varphi_0)/\partial v$ exist and are uniformly continuous with respect to $(u,\varphi_0)\in U_{\delta}$ and $(v,\varphi_0)\in V_{\delta}$ in the uniform operator topology, that is, the operators
$$
\begin{equation*}
\frac{\partial\Phi_*}{\partial u}(u,\varphi_0)\overline{P}_{\varphi_0} \colon E\to E,\quad \frac{\partial\Psi_*}{\partial v}(v,\varphi_0)\overline{Q}_{\varphi_0} \colon E\to E,
\end{equation*}
\notag
$$
are uniformly continuous in the norm of $L(E;E)$, where $\overline{P}_{\varphi_0}$ and $\overline{Q}_{\varphi_0}$ are the projections from Definition 2.2. Moreover,
$$
\begin{equation}
\frac{\partial\Phi_*}{\partial u}(0,\varphi_0) \equiv 0, \quad \sup_{(u,\varphi_0)\in U_{\delta}}\biggl\|\frac{\partial\Phi_*}{\partial u} (u,\varphi_0)\overline{P}_{\varphi_0}\biggr\|_{E\to E} <\infty,
\end{equation}
\tag{3.15}
$$
$$
\begin{equation}
\frac{\partial\Psi_*}{\partial v}(0,\varphi_0) \equiv 0, \quad \sup_{(v,\varphi_0)\in V_{\delta}}\biggl\|\frac{\partial\Psi_*}{\partial v} (v,\varphi_0)\overline{Q}_{\varphi_0}\biggr\|_{E\to E} <\infty.
\end{equation}
\tag{3.16}
$$
3) The following invariance properties hold:
$$
\begin{equation}
\overline{G}(\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)) \supset \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{G}(\varphi_0)) \quad\forall\,\varphi_0\in E
\end{equation}
\tag{3.17}
$$
$$
\begin{equation}
\textit{and} \qquad \overline{G}^{\,-1}(\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)) \supset\overline{\mathcal{F}}^{\rm \,s}_{\delta} (\overline{G}^{\,-1}(\varphi_0)) \quad \forall\,\varphi_0\in E.
\end{equation}
\tag{3.18}
$$
4) There exist positive constants $r_1,r_2$ and constants $\nu_1,\nu_2\in(0,1)$, such that for each $n\in\mathbb{N}$ We supplement this theorem by noting that, as follows from the representations (3.11) and (3.12), for any sufficiently close $\overline{\varphi}$, $\overline{\overline{\varphi}}\in E$ the local manifolds $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})$ and $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\overline{\overline{\varphi}})$ intersect in one point. Drawing on this observation it is easy to show the following result. Corollary 3.1.1. There exists a sufficiently small $\delta'>0$ such that for each $\varphi_0\in E$ and any point $\overline{\varphi}$ in the set the condition that $\|\overline{G}^{\,-n}(\overline{\varphi})- \overline{G}^{\,-n}(\varphi_0)\|\to 0$ as $n\to+\infty$ implies the inclusion $\overline{\varphi}\in \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\varphi_0)$, while the condition that $\|\overline{G}^{\,n}(\overline{\varphi})- \overline{G}^{\,n}(\varphi_0)\|\to 0$ as $n\to+\infty$ ensures the inclusion $\overline{\varphi}\in \overline{\mathcal{F}}^{\rm \,s}_{\delta'}(\varphi_0)$. In fact, assume, for instance, that $\|\overline{G}^{\,-n}(\overline{\varphi})- \overline{G}^{\,-n}(\varphi_0)\|\to 0$ as $n\to+\infty$ but at the same time $\overline{\varphi}\notin \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\varphi_0)$. In this case the point $\overline{\overline{\varphi}}=\overline{\mathcal{F}}^{\rm \,u}_{\delta'} (\overline{\varphi})\cap\overline{\mathcal{F}}^{\rm \,s}_{\delta'}(\varphi_0)$ is certainly distinct from $\varphi_0$. Now it is obvious that
$$
\begin{equation*}
\|\overline{G}^{\,-n}(\overline{\overline{\varphi}})- \overline{G}^{\,-n}(\varphi_0)\|\leqslant \|\overline{G}^{\,-n}(\overline{\varphi})- \overline{G}^{\,-n}(\overline{\overline{\varphi}})\|+ \|\overline{G}^{\,-n}(\overline{\varphi})-\overline{G}^{\,-n}(\varphi_0)\|.
\end{equation*}
\notag
$$
It remains to observe that both terms on the right-hand side tend to zero as $n\to +\infty$, while in view of (3.20) and the inclusion $\overline{\overline{\varphi}}\in \overline{\mathcal{F}}^{\rm \,s}_{\delta'}(\varphi_0)$ the left-hand side has the estimate
$$
\begin{equation*}
\|\overline{G}^{\,-n}(\overline{\overline{\varphi}})- \overline{G}^{\,-n}(\varphi_0)\|\geqslant \frac{1}{r_2}\biggl(\frac{1}{\nu_2}\biggr)^n\|\overline{\overline{\varphi}}- \varphi_0\|\to +\infty,\qquad n\to+\infty.
\end{equation*}
\notag
$$
This contradiction shows that, in fact, $\overline{\varphi}\in\overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\varphi_0)$. Using this supplementary result we can produce an invariant description of local manifolds, which is independent of the choice of coordinates. For example, the unstable local manifold that corresponds to $\varphi_0\in E$ is the set of points $\varphi$ in some ball $O(\varphi_0,\delta)\subset E$, $\delta=\operatorname{const}>0$, such that $\|\overline{G}^{\,-n}(\varphi)-\overline{G}^{\,-n}(\varphi_0)\|\to 0$ as $n\to+\infty$. On the other hand, for a stable local manifold we need that $\|\overline{G}^{\,n}(\varphi)-\overline{G}^{\,n}(\varphi_0)\|\to 0$ as $n\to+\infty$. We also add that from the above invariant descriptions of local manifolds we can extract information about the structure of the possible intersections $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\overline{\varphi}})$ and $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm \,s}_{\delta}(\overline{\overline{\varphi}})$, where $\overline{\varphi},\overline{\overline{\varphi}}\in E$. It is easy to see that these manifolds cannot intersect in an arbitrary way, but are governed by a certain rule. More precisely, we have the following. Corollary 3.1.2. Assume that $\operatorname{int}\overline{\mathcal{F}}^{\rm \,u}_{\delta} (\overline{\varphi})\cap\overline{\mathcal{F}}^{\rm \,u}_{\delta} (\overline{\overline{\varphi}})\ne\varnothing$ for some $\overline{\varphi}, \overline{\overline{\varphi}}\in E$ (here $\operatorname{int}\overline{\mathcal{F}}^{\rm \,u}_{\delta} (\overline{\varphi})$ denotes the set of points of the form $\{\varphi=\varphi_0+u+\Phi_*(u,\varphi_0)\colon u\in\overline{E}^{\rm \,u}_{\varphi_0},\ \|u\|_{\varphi_0}<\delta\}$). Then there exist $\delta'\in(0,\delta)$ and $\theta\in\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\overline{\varphi}})$ such that $\overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\theta)\subset \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\overline{\varphi}})$. The analogous result also holds for stable local manifolds. Let us dwell on the question of finding an analogue of Theorem 3.1 for the original hyperbolic diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$. As a matter of fact, this requires no separate justification and is an automatic consequence of Theorem 3.1. And it remains true if we replace $\delta$ by a smaller quantity $\delta'\in(0,\delta)$. So without loss of generality we let $\delta$ be sufficiently small so that
$$
\begin{equation}
\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)\subset O\biggl(\varphi_0,\frac{\varepsilon_0}{2}\biggr)\quad\text{and}\quad \overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)\subset O\biggl(\varphi_0,\frac{\varepsilon_0}{2}\biggr)\quad \forall\,\varphi_0\in E,
\end{equation}
\tag{3.21}
$$
where $\varepsilon_0$ is the quantity in (1.8) and $O(\varphi_0,\varepsilon_0/2)=\{\varphi\in E\colon \|\varphi-\varphi_0\|<\varepsilon_0/2\}$. In this case the relevant local manifolds for $G$ are defined by
$$
\begin{equation}
\mathcal{F}^{\rm \,u}_{\delta}(\varphi_0)= \operatorname{pr}[\overline{\mathcal{F}}^{\rm \,u}_{\delta} (\operatorname{pr}^{-1}(\varphi_0))],\quad \mathcal{F}^{\rm \,s}_{\delta}(\varphi_0)= \operatorname{pr}[\overline{\mathcal{F}}^{\rm \,s}_{\delta}( \operatorname{pr}^{-1}(\varphi_0))]\quad \forall\varphi_0\in\mathbb{T}^{\infty}.
\end{equation}
\tag{3.22}
$$
Relations (3.6), (3.7), and (3.11)–(3.16) demonstrate that the sets (3.22) are $C^1$-small discs embedded in the original manifold $\mathbb{T}^{\infty}$, and we have $T_{\varphi_0}\mathcal{F}^{\rm \,u}_{\delta}(\varphi_0)= E_{\varphi_0}^{\rm \,u}$ and $T_{\varphi_0}\mathcal{F}^{\rm \,s}_{\delta}(\varphi_0)= E_{\varphi_0}^{\rm \,s}$, where $E_{\varphi_0}^{\rm \,u}$ and $E_{\varphi_0}^{\rm \,s}$ are the subspaces in (2.2). Moreover, (3.22) and the inclusions (3.17) and (3.18) imply directly that
$$
\begin{equation}
G(\mathcal{F}^{\rm \,u}_{\delta}(\varphi_0))\supset \mathcal{F}^{\rm \,u}_{\delta}(G(\varphi_0)),\quad G^{-1}(\mathcal{F}^{\rm \,s}_{\delta}(\varphi_0))\supset \mathcal{F}^{\rm \,s}_{\delta}(G^{-1}(\varphi_0))\quad \forall\,\varphi_0\in\mathbb{T}^{\infty},
\end{equation}
\tag{3.23}
$$
that is, the standard invariance properties hold. The inequalities
$$
\begin{equation}
\rho(G^{-n}(\varphi),G^{-n}(\varphi_0)) \leqslant r_1\nu_1^n\rho(\varphi,\varphi_0) \quad \forall\,\varphi_0,\varphi \in\mathbb{T}^{\infty}\colon \varphi\in\mathcal{F}^{\rm \,u}_{\delta}(\varphi_0), \quad \forall\,n \in\mathbb{N},
\end{equation}
\tag{3.24}
$$
$$
\begin{equation}
\text{and} \qquad \rho(G^{n}(\varphi),G^{n}(\varphi_0)) \leqslant r_2\nu_2^n\rho(\varphi,\varphi_0) \quad \forall\,\varphi_0,\varphi \in\mathbb{T}^{\infty}\colon \varphi\in\mathcal{F}^{\rm \,s}_{\delta}(\varphi_0), \quad \forall\,n \in\mathbb{N},
\end{equation}
\tag{3.25}
$$
which are analogues of (3.19) and (3.20), also hold with the same constants $r_1$, $r_2$, $\nu_1$, and $\nu_2$. We can see this as follows. By the above choice of $\delta$ (see (3.21)) and equality (1.8), the map (1.6) is an isometry of the sets $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$ and $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)$, $\varphi_0\in E$. Hence we conclude from (3.23) that for each $n\in\mathbb{N}$ we have
$$
\begin{equation*}
\begin{gathered} \, \rho(G^{-n}(\operatorname{pr}(\varphi)),G^{-n}(\operatorname{pr}(\varphi_0)))= \|\overline{G}^{\,-n}(\varphi)-\overline{G}^{\,-n}(\varphi_0)\|\quad \forall\,\varphi_0,\varphi\in E\colon \varphi\in\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0), \\ \rho(G^{n}(\operatorname{pr}(\varphi)),G^{n}(\operatorname{pr}(\varphi_0)))= \|\overline{G}^{\,n}(\varphi)-\overline{G}^{\,n}(\varphi_0)\|\quad \forall\,\varphi_0,\varphi\in E\colon \varphi\in\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0), \\ \text{and} \qquad \rho(\operatorname{pr}(\varphi),\operatorname{pr}(\varphi_0))= \|\varphi-\varphi_0\|\quad \forall\,\varphi_0,\varphi\in E\colon\varphi\in \overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)\text{ or } \varphi\in\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0). \end{gathered}
\end{equation*}
\notag
$$
It remains to use the known inequalities (3.19) and (3.20) and verify the required estimates (3.24) and (3.25). 3.2. The proof of the existence of Lipschitz local setsAs usual, to verify Theorem 3.1 we must only prove that unstable local manifolds $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$, $\varphi_0\in E$, exist and establish their properties. As concerns the stable local manifolds $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)$, $\varphi_0\in E$, they need not be considered separately, as they are the unstable manifolds of the inverse map $\overline{G}^{\,-1}$. First we show that at each point $\varphi_0\in E$ the diffeomorphism $\overline{G}$ has a manifold (3.11) with properties (3.13), (3.17), and (3.19). However, in place of (3.15) we only show that the corresponding function $\Phi_*(u,\varphi_0)$ is Lipschitz continuous in $u$. As regards its being $C^1$-smooth in $u\in\overline{E}^{\rm \,u}_{\varphi_0}$, we prove this separately. To use Hadamard’s method, we consider the so-called graph space $\mathscr{H}$. To do this we fix an arbitrary sufficiently small $\delta>0$ (we specify the value of this parameter in what follows). Next we let $\mathscr{H}$ consist of vector functions $\Phi(u,\varphi_0)\in C(U_{\delta})$ with properties similar to (3.13):
$$
\begin{equation}
\Phi(0,\varphi_0)\equiv 0,\qquad \Phi(u,\varphi_0)\in\overline{E}^{\rm \,s}_{\varphi_0}\quad \forall\,(u,\varphi_0)\in U_{\delta}.
\end{equation}
\tag{3.26}
$$
We also assume that
$$
\begin{equation}
\|\Phi(u_1,\varphi_0)-\Phi(u_2,\varphi_0)\|_{\varphi_0}\leqslant L\|u_1-u_2\|_{\varphi_0}\quad \forall\,(u_1,\varphi_0),(u_2,\varphi_0)\in U_{\delta},
\end{equation}
\tag{3.27}
$$
where we use the second norm from (3.1) on the left-hand side and the first norm on the right-hand side. As regards the non-negative constant $L$, it is universal in the following sense: it is independent of $u_1$, $u_2$, $\varphi_0$, and the particular function $\Phi\in\mathscr{H}$. We specify this constant in what follows. We define the metric in $\mathscr{H}$ by:
$$
\begin{equation}
\forall\,\Phi_1,\Phi_2\in\mathscr{H}\qquad \rho(\Phi_1,\Phi_2)=\sup_{(u,\varphi_0)\in U_{\delta}}\|\Phi_1(u,\varphi_0)-\Phi_2(u,\varphi_0)\|_{\varphi_0}.
\end{equation}
\tag{3.28}
$$
Here and below we denote various metrics by $\rho$; there can be no confusion because the particular metric under consideration is always clear from the context. It is easy to see that, by (3.9), the supremum in (3.28) is certainly finite. However, here this is also a direct consequence of conditions (3.26) and (3.27), which imply that
$$
\begin{equation}
\|\Phi(u,\varphi_0)\|_{\varphi_0}\leqslant L\delta.
\end{equation}
\tag{3.29}
$$
We must also note that the space $\mathscr{H}$ with the metric (3.28) is complete. Now, according to Hadamard’s method, we must define the so-called graph transformation operator $T$ in $\mathscr{H}$. We assume that it acts by the formula where $\overline{\Phi}(u,\varphi_0)$ is a vector function such that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\overline{G}\bigl(\{\varphi=\varphi_0+u+\Phi(u,\varphi_0)\colon u\in\overline{E}^{\rm \,u}_{\varphi_0},\ \|u\|_{\varphi_0}\leqslant \delta\}\bigr) \\ &\qquad\supset \{\varphi=\varphi_1+u+\overline{\Phi}(u,\varphi_1)\colon u\in\overline{E}^{\rm \,u}_{\varphi_1},\ \|u\|_{\varphi_1}\leqslant \delta\},\qquad \varphi_1=\overline{G}(\varphi_0). \end{aligned}
\end{equation}
\tag{3.31}
$$
Our immediate aim is to show that $T$ is well defined, that is, we must express the function $\overline{\Phi}(u,\varphi_0)$ in (3.31) in terms of $\Phi(u,\varphi_0)\in\mathscr{H}$. We assume for convenience that $\varphi_0=\overline{G}^{\,-1}(\varphi_1)$ and $\varphi_1$ is an independent variable ranging in $E$. Note that the inclusion (3.31) means that for each $\widetilde{u}\in\overline{E}^{\rm \,u}_{\varphi_1}$, $\|\widetilde{u}\|_{\varphi_1}\leqslant \delta$, there exists $u\in\overline{E}^{\rm \,u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant \delta$, such that
$$
\begin{equation}
\overline{G}(\varphi_0+u+\Phi(u,\varphi_0))=\varphi_1+\widetilde{u}+ \overline{\Phi}(\widetilde{u},\varphi_1).
\end{equation}
\tag{3.32}
$$
Now applying the projection $\overline{P}_{\varphi_1}$ (see Definition 2.2) to (3.32) we conclude that the variables $\widetilde{u}\in\overline{E}^{\rm \,u}_{\varphi_1}$ and $u\in\overline{E}^{\rm \,u}_{\varphi_0}$ are related by
$$
\begin{equation}
\widetilde{u}=\overline{P}_{\varphi_1}\bigl(\overline{G}(\varphi_0+u+ \Phi(u,\varphi_0))-\varphi_1\bigr).
\end{equation}
\tag{3.33}
$$
Thus, to show that the operator (3.30) is well defined we must express $u$ in (3.33) in terms of $\widetilde{u}$ and $\varphi_1$. Before we analyse equation (3.33), we look at the auxiliary equation where $\varphi_1\in E$, $u\in\overline{E}^{\rm \,u}_{\varphi_0}$, $v\in\overline{E}^{\rm \,s}_{\varphi_0}$, and the vector function $F$ to in $\overline{E}^{\rm \,u}_{\varphi_1}$ is defined by
$$
\begin{equation}
F(u,v,\varphi_1)= \overline{P}_{\varphi_1}(\overline{G}(\varphi_0+u+v)-\varphi_1).
\end{equation}
\tag{3.35}
$$
Now we write (3.34) in an equivalent form:
$$
\begin{equation}
u=\Omega(u, \widetilde{u}, v, \varphi_1)\overset{\rm def}{=} \biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1} \biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)u- F(u,v,\varphi_1)+\widetilde{u}\biggr)
\end{equation}
\tag{3.36}
$$
and analyse the resulting equation. First we must verify that the operator $\partial F(0,0,\varphi_1)/\partial u$ involved in (3.36) is invertible. We require the formulae
$$
\begin{equation}
\begin{aligned} \, \frac{\partial F}{\partial u}(u,v,\varphi_1)&= \overline{P}_{\varphi_1}D\overline{G}(\varphi_0+u+v)\colon \overline{E}^{\rm \,u}_{\varphi_0}\to\overline{E}^{\rm \,u}_{\varphi_1}, \\ \frac{\partial F}{\partial v}(u,v,\varphi_1)&= \overline{P}_{\varphi_1}D\overline{G}(\varphi_0+u+v)\colon \overline{E}^{\rm \,s}_{\varphi_0}\to\overline{E}^{\rm \,u}_{\varphi_1}. \end{aligned}
\end{equation}
\tag{3.37}
$$
Using the first we obtain
$$
\begin{equation*}
\frac{\partial F}{\partial u}(0,0,\varphi_1)=\overline{P}_{\varphi_1} D\overline{G}(\varphi_0)\big|_{\overline{E}^{\rm \,u}_{\varphi_0}\to \overline{E}^{\rm \,u}_{\varphi_1}}= D\overline{G}(\varphi_0)\big|_{\overline{E}^{\rm \,u}_{\varphi_0}\to \overline{E}^{\rm \,u}_{\varphi_1}}.
\end{equation*}
\notag
$$
Hence it follows from (3.5) that the operator $\partial F(0,0,\varphi_1)/\partial u$ has a continuous inverse; moreover,
$$
\begin{equation}
\sup_{\varphi_1\in E}\biggl\|\biggl(\frac{\partial F}{\partial u} (0,0,\varphi_1)\biggr)^{-1}\biggr\|_{\overline{E}^{\rm \,u}_{\varphi_1}\to \overline{E}^{\rm \,u}_{\varphi_0}}\leqslant \sigma_1<1.
\end{equation}
\tag{3.38}
$$
To state a result on the solvability of (3.36) with respect to $u$ we need the set
$$
\begin{equation*}
W_{\delta}=\bigl\{(\widetilde{u},v,\varphi_1)\colon\varphi_1\in E,\ \widetilde{u}\in\overline{E}^{\rm \,u}_{\varphi_1},\ \|\widetilde{u}\|_{\varphi_1}\leqslant \delta,\ v\in\overline{E}^{\rm \,s}_{\varphi_0},\ \|v\|_{\varphi_0}\leqslant \delta\bigr\},
\end{equation*}
\notag
$$
which is similar to (3.6) and (3.7) and has a metric similar to (3.8):
$$
\begin{equation*}
\begin{gathered} \, \forall\,w_j=(\widetilde{u}_j,v_j,\varphi_1^j)\in W_{\delta},\quad j=1,2, \\ \rho(w_1,w_2)=\|\widetilde{u}_1-\widetilde{u}_2\|+ \|v_1-v_2\|+\|\varphi_1^1-\varphi_1^2\|. \end{gathered}
\end{equation*}
\notag
$$
We have the following. Lemma 3.1. For sufficiently small $\delta$ equation (3.36) uniquely defines a vector function $u=\Sigma(\widetilde{u},v,\varphi_1)\in E$ which is jointly continuous in $(\widetilde{u},v,\varphi_1)\in W_{\delta}$ and satisfies Proof. We will have proved this result once we will have shown that for sufficiently small $\delta$ and any $(\widetilde{u},v,\varphi_1)\in W_{\delta}$ the operator satisifies the conditions of the contracting mapping principle in the ball
First we verify that (3.40) takes the ball (3.41) to itself. To do this we make some auxiliary constructions. Namely, we observe that, in view of (3.35), (3.37), and the obvious relations
$$
\begin{equation*}
\overline{P}_{\varphi_1}D\overline{G}(\varphi_0) \overline{E}^{\rm\, s}_{\varphi_0}= \overline{P}_{\varphi_1}\overline{E}^{\rm\, s}_{\varphi_1}=0
\end{equation*}
\notag
$$
we have the equalities
$$
\begin{equation*}
F(0,0,\varphi_1)\equiv 0,\quad \frac{\partial F}{\partial v}(0,0,\varphi_1)\equiv 0,
\end{equation*}
\notag
$$
which in turn yield the representation
$$
\begin{equation}
\begin{aligned} \, \nonumber F(u,v,\varphi_1)-\frac{\partial F}{\partial u}(0,0,\varphi_1)u&= \int_0^1\biggl[\biggl(\frac{\partial F}{\partial u}(tu,tv,\varphi_1)- \frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)u \\ &\qquad+\biggl(\frac{\partial F}{\partial v}(tu,tv,\varphi_1)- \frac{\partial F}{\partial v}(0,0,\varphi_1)\biggr)v\biggr]\,dt. \end{aligned}
\end{equation}
\tag{3.42}
$$
Formula (3.42) is crucial for the derivation of upper bounds for the norm $\|\Omega\|_{\varphi_0}$. Combining it with (3.37) and taking (3.4) into account we conclude that for all $\varphi_1\in E$, all $u\in\overline{E}^{\rm\, u}_{\varphi_0}$ such that $\|u\|_{\varphi_0}\leqslant \delta$, and all $v\in\overline{E}^{\rm\, s}_{\varphi_0}$ such that $\|v\|_{\varphi_0}\leqslant \delta$ we have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\biggl\|F(u,v,\varphi_1)- \frac{\partial F}{\partial u}(0,0,\varphi_1)u\biggr\|_{\varphi_1} \\ \nonumber &\qquad\leqslant\delta\cdot \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}:\\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \bigl\|\overline{P}_{\varphi_1}(D\overline{G}(\varphi_0+u+v)- D\overline{G}(\varphi_0))\bigr\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_1}} \\ \nonumber &\qquad\qquad+\delta\cdot \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}:\\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \bigl\|\overline{P}_{\varphi_1}(D\overline{G}(\varphi_0+u+v)- D\overline{G}(\varphi_0))\bigr\|_{\overline{E}^{\rm\, s}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_1}} \\ &\qquad \leqslant 2c\cdot\sup_{\varphi_1\in E} \|\overline{P}_{\varphi_1}\|_{E\to E}\cdot \theta_0(\delta)\delta, \end{aligned}
\end{equation}
\tag{3.43}
$$
where $c$ is the constant in (3.3) and $\theta_0(\delta)$ has the form
$$
\begin{equation}
\theta_0(\delta)=\sup_{\varphi_1\in E}\, \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}: \\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \|D\overline{G}(\varphi_0+u+v)-D\overline{G}(\varphi_0)\|_{E\to E}.
\end{equation}
\tag{3.44}
$$
Note also that, as the projections $\overline{P}_{\varphi}$ and $\overline{Q}_{\varphi}$ (see (2.16) and (2.26)) are uniformly bounded and the differential $D\overline{G}$ is uniformly continuous, all the suprema in (3.43) and (3.44) are certainly finite. Moreover, the quantity (3.44) tends to zero as $\delta\to 0$.
Using the above inequalities (3.43) we can fairly easily deal with the question of an estimate for $\|\Omega\|_{\varphi_0}$. In fact, combining these inequalities with (3.38) and taking the explicit formula for $\Omega$ (see (3.36)) into account, we see that
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\Omega(u, \widetilde{u},v,\varphi_1)\|_{\varphi_0}&\leqslant \sigma_1\biggl(\biggl\|F(u,v,\varphi_1)- \frac{\partial F}{\partial u}(0,0,\varphi_1)u\biggr\|_{\varphi_1}+ \|\widetilde{u}\|_{\varphi_1}\biggr) \\ &\leqslant\sigma_1\Bigl(2c\cdot\sup_{\varphi_1\in E} \|\overline{P}_{\varphi_1}\|_{E\to E}\cdot\theta_0(\delta)+1\Bigr)\delta. \end{aligned}
\end{equation}
\tag{3.45}
$$
In what follows we take $\delta$ to be sufficiently small so that
$$
\begin{equation}
2c\sigma_1\cdot\sup_{\varphi_1\in E}\|\overline{P}_{\varphi_1}\|_{E\to E} \cdot \theta_0(\delta)<1-\sigma_1.
\end{equation}
\tag{3.46}
$$
Then, as follows from (3.45), the operator (3.40) takes the ball (3.41) in question into itself.
Verifying that (3.40) is contracting is not a problem either, but we must impose an additional restriction on $\delta$. In fact, using the formula
$$
\begin{equation*}
\Omega(u,\widetilde{u},v,\varphi_1)=u- \biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1} F(u,v,\varphi_1)+ \biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1}\widetilde{u},
\end{equation*}
\notag
$$
which follows from (3.36), we arrive at the inequality
$$
\begin{equation}
\begin{gathered} \, \forall\,u_j\in\overline{E}^{\rm\, u}_{\varphi_0},\quad \|u_j\|_{\varphi_0}\leqslant\delta,\quad j=1, 2,\quad \forall\,(\widetilde{u},v,\varphi_1)\in W_{\delta} \\ \|\Omega(u_1,\widetilde{u},v,\varphi_1)- \Omega(u_2,\widetilde{u},v,\varphi_1)\|_{\varphi_0}\leqslant \theta_1(\delta)\|u_1-u_2\|_{\varphi_0}, \end{gathered}
\end{equation}
\tag{3.47}
$$
where
$$
\begin{equation}
\hspace{-1mm}\theta_1(\delta)=\sup_{\varphi_1\in E}\, \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}:\\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \biggl\|I-\biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1}\, \frac{\partial F}{\partial u} (u,v,\varphi_1)\biggr\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0}},
\end{equation}
\tag{3.48}
$$
and $I$ is the identity operator in $\overline{E}^{\rm\, u}_{\varphi_0}$. Note also that, as in the case of (3.44), the quantity (3.48) is certainly finite and $\theta_1(\delta)\to 0$ as $\delta\to 0$ (for the same reasons as before).
Reducing $\delta$ if necessary, it what follows we assume that, in addition to (3.46), we have Then by (3.47) the operator (3.40) is contracting, so that it has a unique fixed point $u=\Sigma(\widetilde{u},v,\varphi_1)$ in the ball (3.41). We stress that, since the contraction is uniform (the contraction constant (3.48) is independent of $(\widetilde{u},v,\varphi_1)\in W_{\delta}$) and $\Omega(u,\widetilde{u},v,\varphi_1)$ is uniformly continuous in $(u,\widetilde{u},v,\varphi_1)$, the function $\Sigma(\widetilde{u},v,\varphi_1)$ is uniformly continuous with respect to $(\widetilde{u},v,\varphi_1)\in W_{\delta}$. Next, it has the first three properties in (3.39) by construction, and $\Sigma(\widetilde{u},v,\varphi_1)$ is periodic in $\varphi_1$ because the vector function $\Omega$ (see (3.36)) is likewise periodic. $\Box$In what follows we require some further properties of the function $\Sigma(\widetilde{u}, v, \varphi_1)$. To establish these, we verify first that for all $\varphi_1\in E$, $u\in\overline{E}^{\rm\, u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant\delta$, and $v\in\overline{E}^{\rm\, s}_{\varphi_0}$, $\|v\|_{\varphi_0}\leqslant\delta$, the operator $\partial F(u,v,\varphi_1)/\partial u$ is invertible. To see this consider the operator
$$
\begin{equation*}
A(u,v,\varphi_1)= I-\biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1}\, \frac{\partial F}{\partial u}(u,v,\varphi_1)\colon \overline{E}^{\rm\, u}_{\varphi_0}\to\overline{E}^{\rm\, u}_{\varphi_0}
\end{equation*}
\notag
$$
and note that, by (3.48) and (3.49),
$$
\begin{equation*}
\bigl\|(I-A(u,v,\varphi_1))^{-1}\bigr\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0}}\leqslant\frac{1}{1-\theta_1(\delta)}\,.
\end{equation*}
\notag
$$
Hence $\partial F(u,v,\varphi_1)/\partial u$ is invertible, as required, and
$$
\begin{equation}
\biggl(\frac{\partial F}{\partial u}(u, v,\varphi_1)\biggr)^{-1}= \bigl(I-A(u,v,\varphi_1)\bigr)^{-1} \biggl(\frac{\partial F}{\partial u}(0,0,\varphi_1)\biggr)^{-1}.
\end{equation}
\tag{3.50}
$$
Using this invertibility of $\partial F(u,v,\varphi_1)/\partial u$ we can apply the implicit function theorem to equation (3.34) at any fixed point $(u,\widetilde{u},v,\varphi_1)$, where $u=\Sigma(\widetilde{u},v,\varphi_1)$. By this theorem $\Sigma(\widetilde{u},v,\varphi_1)$ is continuously differentiable with respect to $\widetilde{u}$ and $v$ and
$$
\begin{equation}
\begin{aligned} \, \frac{\partial}{\partial\widetilde{u}}\Sigma(\widetilde{u},v,\varphi_1)&= \biggl(\frac{\partial F}{\partial u} (u,v,\varphi_1)\biggr)^{-1}\bigg|_{u=\Sigma(\widetilde{u},v,\varphi_1)}, \\ \frac{\partial}{\partial v}\Sigma(\widetilde{u},v,\varphi_1)&= -\biggl(\frac{\partial F}{\partial u}(u,v,\varphi_1)\biggr)^{-1} \frac{\partial F}{\partial v} (u,v,\varphi_1)\bigg|_{u=\Sigma(\widetilde{u},v,\varphi_1)}. \end{aligned}
\end{equation}
\tag{3.51}
$$
Using formulae (3.51) we obtain some estimates required below. Namely, it is easy to see that for all $(\widetilde{u},v,\varphi_1)\in W_{\delta}$ we have
$$
\begin{equation}
\begin{aligned} \, \biggl\|\frac{\partial}{\partial\widetilde{u}} \Sigma(\widetilde{u},v,\varphi_1)\biggr\|_{\overline{E}^{\rm\, u}_{\varphi_1} \to\overline{E}^{\rm\, u}_{\varphi_0}}&\leqslant q_1(\delta), \\ \biggl\|\frac{\partial}{\partial v} \Sigma(\widetilde{u},v,\varphi_1)\biggr\|_{\overline{E}^{\rm\, s}_{\varphi_0} \to\overline{E}^{\rm\, u}_{\varphi_0}}&\leqslant q_1(\delta)q_2(\delta), \end{aligned}
\end{equation}
\tag{3.52}
$$
where
$$
\begin{equation}
\begin{aligned} \, q_1(\delta)&=\sup_{\varphi_1\in E}\, \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}: \\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \biggl\|\biggl(\frac{\partial F}{\partial u} (u,v,\varphi_1)\biggr)^{-1}\biggr\|_{\overline{E}^{\rm\, u}_{\varphi_1} \to \overline{E}^{\rm\, u}_{\varphi_0}}, \\ q_2(\delta)&=\sup_{\varphi_1\in E}\, \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}: \\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \biggl\|\frac{\partial F}{\partial v} (u,v,\varphi_1)\biggr\|_{\overline{E}^{\rm\, s}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_1}}. \end{aligned}
\end{equation}
\tag{3.53}
$$
Note also that, first of all, in view of (3.37) and (3.50), since the differential $D\overline{G}$ is uniformly continuous, the quantities (3.53) are finite and, as $\delta\to 0$, converge to finite limits
$$
\begin{equation*}
\begin{aligned} \, q_1(0)&=\sup_{\varphi_1\in E}\biggl\|\biggl(\frac{\partial F}{\partial u} (0,0,\varphi_1)\biggr)^{-1}\biggr\|_{\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, u}_{\varphi_0}} \\ \text{and} \qquad q_2(0)&=\sup_{\varphi_1\in E}\biggl\|\frac{\partial F}{\partial v} (0,0,\varphi_1)\biggr\|_{\overline{E}^{\rm\, s}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_1}}. \end{aligned}
\end{equation*}
\notag
$$
Second, $q_2(0)=0$, and we have $q_1(0)\leqslant \sigma_1<1$ by (3.38). Thus, reducing $\delta$ appropriately, we will assume without loss of generality that, in addition to (3.46) and (3.49), we have We can analyse equation (3.33) under consideration on the basis of these auxiliary constructions. To do this, first we write it in the equivalent form
$$
\begin{equation}
u=\Gamma_{\Phi}(u,\widetilde{u},\varphi_1)\overset{\rm def}{=} \Sigma(\widetilde{u},v,\varphi_1)\big|_{v=\Phi(u,\varphi_0)}.
\end{equation}
\tag{3.55}
$$
Note, however, that we can only proceed from (3.33) to (3.55) if
$$
\begin{equation*}
\|v\|_{\varphi_0}=\|\Phi(u,\varphi_0\|_{\varphi_0}\leqslant\delta.
\end{equation*}
\notag
$$
In this connection we assume in what follows that the Lipschitz constant in (3.27) satisfies the condition Then the required condition $\|\Phi(u,\varphi_0\|_{\varphi_0}\leqslant\delta$ is a direct consequence of (3.29), and we can legitimately go over from (3.33) to (3.55). We turn to the question of the solvability of this equation with respect to $u$. The following statement provides an answer. Lemma 3.2. Under certain additional restrictions on the constant $L$ in (3.27) equation (3.55) has a unique solution $u=u_{\Phi}(\widetilde{u}, \varphi_1)\in E$, which is jointly uniformly continuous in the variables $(\widetilde{u},\varphi_1)\in U_{\delta}$ and satisfies Proof. As in the proof of Lemma 3.1, we consider the auxiliary operator and verify that the assumptions of the contracting mapping principle hold for it in the ball (3.41). For any $(\widetilde{u},\varphi_1)\in U_{\delta}$ and $u\in\overline{E}^{\rm\, u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant \delta$, by (3.29), (3.52), and the equality $\Sigma(0,0,\varphi_1)\equiv 0$ (see (3.39)) we obviously have
$$
\begin{equation*}
\begin{aligned} \, \|\Gamma_{\Phi}(u,\widetilde{u},\varphi_1)\|_{\varphi_0}&= \|\Sigma(\widetilde{u},\Phi(u,\varphi_0),\varphi_1)\|_{\varphi_0} \\ &\leqslant\|\Sigma(\widetilde{u},\Phi(u,\varphi_0),\varphi_1)- \Sigma(0,\Phi(u,\varphi_0),\varphi_1)\|_{\varphi_0} \\ &\qquad+\|\Sigma(0,\Phi(u,\varphi_0),\varphi_1)- \Sigma(0,0,\varphi_1)\|_{\varphi_0} \\ &\leqslant q_1(\delta)\delta+q_1(\delta)q_2(\delta) \|\Phi(u,\varphi_0)\|_{\varphi_0}\leqslant q_1(\delta)\delta+q_1(\delta)q_2(\delta)L\delta. \end{aligned}
\end{equation*}
\notag
$$
Thus, if $L$ is subject to the additional condition then the operator (3.58) transforms indeed the ball (3.41) into itself. We can also add that constants $L$ satisfying (3.59) certainly exist because of (3.54).
To verify that the operator (3.58) is contracting, we fix some $u_j\in\overline{E}^{\rm\, u}_{\varphi_0}$, $\|u_j\|_{\varphi_0}\leqslant\delta$, $j=1,2$, and use inequalities (3.27) and (3.52). Then we see that
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\Gamma_{\Phi}(u_1, \widetilde{u}, \varphi_1)- \Gamma_{\Phi}(u_2,\widetilde{u},\varphi_1)\|_{\varphi_0} \\ \nonumber &\qquad=\|\Sigma(\widetilde{u},\Phi(u_1,\varphi_0),\varphi_1)- \Sigma(\widetilde{u},\Phi(u_2,\varphi_0),\varphi_1)\|_{\varphi_0} \\ &\qquad\leqslant q_1(\delta)q_2(\delta)L\|u_1-u_2\|_{\varphi_0}. \end{aligned}
\end{equation}
\tag{3.60}
$$
Note also that $q_1(\delta)q_2(\delta)L<1$ as a direct consequence of (3.59). Hence we conclude from the contracting mapping principle that the operator (3.58) has a unique fixed point $u=u_{\Phi}(\widetilde{u},\varphi_1)$ in the ball (3.41).
It remains ro add that, as the contraction is uniform and $\Gamma_{\Phi}$ is uniformly continuous, the function $u_{\Phi}(\widetilde{u},\varphi_1)$ is uniformly continuous in $(\widetilde{u},\varphi_1)\in U_{\delta}$. As regards the additional properties (3.57), just as in the case of (3.39), the first three of them hold by construction, and the $2\pi$-periodicity in $\varphi_1$ follows from the fact that the vector function $\Gamma_{\Phi}(u,\widetilde{u},\varphi_1)$ is periodic in $\varphi_1$ (see (3.55)). $\Box$ Now we turn back to finding the function $\overline{\Phi}(u,\varphi_0)$ in (3.31). We substitute $u=u_{\Phi}(\widetilde{u},\varphi_1)$ into (3.32) and apply the projection $\overline{Q}_{\varphi_1}$ to the result. Then we conclude that
$$
\begin{equation}
\overline{\Phi}(\widetilde{u},\varphi_1)= R(u,v,\varphi_1)\big|_{v=\Phi(u,\varphi_0),\ u=u_{\Phi}(\widetilde{u},\varphi_1),\ \varphi_0=\overline{G}^{\,-1}(\varphi_1)},
\end{equation}
\tag{3.61}
$$
where
$$
\begin{equation}
R(u,v,\varphi_1)=\overline{Q}_{\varphi_1}(\overline{G}(\varphi_0+u+v)- \varphi_1),\quad u\in\overline{E}^{\rm\, u}_{\varphi_0},\quad v\in\overline{E}^{\rm\, s}_{\varphi_0},\quad \varphi_1\in E.
\end{equation}
\tag{3.62}
$$
Now, as the set of variables $(\widetilde{u},\varphi_1)$ in (3.61) ranges over $U_{\delta}$, we can denote $(\widetilde{u},\varphi_1)$ by $(u,\varphi_0)$ again. As concerns the function $\overline{\Phi}(u,\varphi_0)$, in this case we have
$$
\begin{equation}
\overline{\Phi}(u, \varphi_0)= \overline{\Phi}(\widetilde{u},\varphi_1)\big|_{\widetilde{u}\to u,\, \varphi_1\to \varphi_0},
\end{equation}
\tag{3.63}
$$
where $\overline{\Phi}(\widetilde{u},\varphi_1)$ is the function in (3.61) and $\widetilde{u}\to u$, $\varphi_1\to \varphi_0$ is the redenoting mentioned above. Thus we have shown that the operator $T$ is well defined on the graph space $\mathscr{H}$. Now we establish the following result, which is in the core of Hadamard’s method. Lemma 3.3. For sufficiently small $\delta$ and appropriate choice of the Lipschitz constant $L$ in (3.27) the operator (3.30) takes the space $\mathscr{H}$ to itself and is contracting there. Proof. First, assuming that all the above restrictions on $\delta$ and $L$ are satisfied (see (3.46), (3.49), (3.54), (3.56), and (3.59)) we show that, provided these parameters satisfy certain additional condition, we have the inclusion $T(\mathscr{H})\subset\mathscr{H}$.
We note some obvious points straight away: relations (3.61)–(3.63) and the properties (3.57) of $u_{\Phi}(\widetilde{u},\varphi_1)$, which we have established, imply directly that $\overline{\Phi}(u,\varphi_0)\in C(U_{\delta})$ and the assumptions (3.26) are fulfilled for $\overline{\Phi}(u,\varphi_0)$. Hence we only need to verify that $\overline{\Phi}(u,\varphi_0)$ satisfies the Lipschitz condition (3.27). First we show that the function $u_{\Phi}(\widetilde{u},\varphi_1)$ satisfies the Lipschitz condition with respect to $\widetilde{u}$. To do this, fox some $\widetilde{u}_j\in\overline{E}^{\rm\, u}_{\varphi_1}$, $\|\widetilde{u}_j\|_{\varphi_1}\leqslant\delta$, $j=1,2$, and set $u_j=u_{\Phi}(\widetilde{u}_j,\varphi_1)$, $j=1,2$. Next, from the explicit expression (3.55) for $\Gamma_{\Phi}(u,\widetilde{u},\varphi_1)$ and the bounds (3.52) and (3.60) we deduce sequentially
$$
\begin{equation*}
\begin{aligned} \, \|u_1-u_2\|_{\varphi_0}&=\|\Gamma_{\Phi}(u_1,\widetilde{u}_1,\varphi_1)- \Gamma_{\Phi}(u_2,\widetilde{u}_2,\varphi_1)\|_{\varphi_0} \\ &\leqslant \|\Gamma_{\Phi}(u_1,\widetilde{u}_1,\varphi_1)- \Gamma_{\Phi}(u_2,\widetilde{u}_1,\varphi_1)\|_{\varphi_0} \\ &\qquad+\|\Gamma_{\Phi}(u_2,\widetilde{u}_1,\varphi_1)- \Gamma_{\Phi}(u_2,\widetilde{u}_2,\varphi_1)\|_{\varphi_0} \\ &\leqslant q_1(\delta)q_2(\delta)L\|u_1-u_2\|_{\varphi_0}+ q_1(\delta)\|\widetilde{u}_1-\widetilde{u}_2\|_{\varphi_1}. \end{aligned}
\end{equation*}
\notag
$$
From this and the inequality $q_1(\delta)q_2(\delta)L<1$, which we have already mentioned, we have
$$
\begin{equation}
\|u_{\Phi}(\widetilde{u}_1,\varphi_1)- u_{\Phi}(\widetilde{u}_2,\varphi_1)\|_{\varphi_0}\leqslant \frac{q_1(\delta)}{1-q_1(\delta)q_2(\delta)L}\,\|\widetilde{u}_1- \widetilde{u}_2\|_{\varphi_1}.
\end{equation}
\tag{3.64}
$$
Now we proceed as follows. First we obtain certain bounds for the partial derivatives of the vector function (3.62) with respect to $u\in\overline{E}^{\rm\, u}_{\varphi_0}$ for $\|u\|_{\varphi_0}\leqslant\delta$, and $v\in\overline{E}^{\rm\, s}_{\varphi_0}$ for $\|v\|_{\varphi_0}\leqslant\delta$. Then, combining them with (3.64) we establish the Lipschitz continuity of the function $\overline{\Phi}(\widetilde{u}, \varphi_1)$ with respect to $\widetilde{u}\in\overline{E}^{\rm\, u}_{\varphi_1}$ for $\|\widetilde{u}\|_{\varphi_1}\leqslant\delta$ (see (3.61)). To implement this plan we use the equalities
$$
\begin{equation}
\begin{aligned} \, \frac{\partial R}{\partial u}(u,v,\varphi_1)&=\overline{Q}_{\varphi_1} D\overline{G}(\varphi_0+u+v)\colon\overline{E}^{\rm \,u}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}, \\ \frac{\partial R}{\partial v}(u,v,\varphi_1)&=\overline{Q}_{\varphi_1} D\overline{G}(\varphi_0+u+v)\colon\overline{E}^{\rm \,s}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}, \end{aligned}
\end{equation}
\tag{3.65}
$$
which follow from (3.62). Next we look at the first formula in (3.65) and observe that, since
$$
\begin{equation*}
D\overline{G}(\varphi_0)\overline{E}^{\rm \,u}_{\varphi_0}= \overline{E}^{\rm \,u}_{\varphi_1} \quad\text{and}\quad \overline{Q}_{\varphi_1}D\overline{G}(\varphi_0) \overline{E}^{\rm \,u}_{\varphi_0}=0
\end{equation*}
\notag
$$
we can write it as
$$
\begin{equation*}
\frac{\partial R}{\partial u}(u,v,\varphi_1)=\overline{Q}_{\varphi_1} \bigl(D\overline{G}(\varphi_0+u+v)-D\overline{G}(\varphi_0)\bigr)\colon \overline{E}^{\rm \,u}_{\varphi_0}\to\overline{E}^{\rm \,s}_{\varphi_1}.
\end{equation*}
\notag
$$
Hence we can conclude (see a similar place related to inequality (3.43) above) that for all $\varphi_1\in E$, $u\in\overline{E}^{\rm\, u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant\delta$, and $v\in\overline{E}^{\rm\, s}_{\varphi_0}$, $\|v\|_{\varphi_0}\leqslant\delta$, we have the estimate
$$
\begin{equation}
\biggl\|\frac{\partial R}{\partial u} (u,v,\varphi_1)\biggr\|_{\overline{E}^{\rm \,u}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}}\leqslant q_0(\delta)\overset{\rm def}{=} c\cdot\sup_{\varphi_1\in E}\|\overline{Q}_{\varphi_1}\|_{E\to E} \cdot \theta_0(\delta),
\end{equation}
\tag{3.66}
$$
where we recall that $c$ is the constant in (3.3) and (3.4) and $\theta_0(\delta)$ is the quantity in (3.44).
As concerns the second formula in (3.65), for the same values of $\varphi_1$, $u$, and $v$ we obviously have
$$
\begin{equation}
\begin{aligned} \, \nonumber &\biggl\|\frac{\partial R}{\partial v} (u,v,\varphi_1)\biggr\|_{\overline{E}^{\rm \,s}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}} \\ &\qquad\leqslant q_3(\delta)\overset{\rm def}{=}\sup_{\varphi_1\in E}\, \sup_{\substack{u\in\overline{E}^{\rm\, u}_{\varphi_0},\, v\in\overline{E}^{\rm\, s}_{\varphi_0}: \\ \|u\|_{\varphi_0}\leqslant \delta,\,\|v\|_{\varphi_0}\leqslant \delta}} \bigl\|\overline{Q}_{\varphi_1}D\overline{G} (\varphi_0+u+v)\bigr\|_{\overline{E}^{\rm \,s}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}}. \end{aligned}
\end{equation}
\tag{3.67}
$$
We must also note that, since
$$
\begin{equation*}
\overline{Q}_{\varphi_1}D\overline{G} (\varphi_0)\big|_{\overline{E}^{\rm \,s}_{\varphi_0}\to \overline{E}^{\rm \,s}_{\varphi_1}}= D\overline{G}(\varphi_0)\big|_{\overline{E}^{\rm \,s}_{\varphi_0} \to\overline{E}^{\rm \,s}_{\varphi_1}},
\end{equation*}
\notag
$$
by (3.5) we have
$$
\begin{equation*}
q_3(\delta)\to q_3(0)=\sup_{\varphi_1\in E} \|D\overline{G}(\varphi_0)\|_{\overline{E}^{\rm \,s}_{\varphi_0} \to\overline{E}^{\rm \,s}_{\varphi_1}}\leqslant\sigma_2<1
\end{equation*}
\notag
$$
as $\delta\to 0$. Bearing this in mind, throughout what follows we assume that $\delta$ is sufficiently small so that
Now we verify that $\overline{\Phi}(\widetilde{u},\varphi_1)$ satisfies the Lipschitz condition with respect to $\widetilde{u}$. To do this we fix some $\widetilde{u}_j\in\overline{E}^{\rm \,u}_{\varphi_1}$, $\|\widetilde{u}_j\|_{\varphi_1}\leqslant \delta$, $j=1,2$, set $u_j=u_{\Phi}(\widetilde{u}_j,\varphi_1)$ and $v_j=\Phi(u_j,\varphi_0)$, $j=1,2$, and use (3.61) in combination with (3.64), (3.66), and (3.67). Then we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\overline{\Phi}(\widetilde{u}_1,\varphi_1)- \overline{\Phi}(\widetilde{u}_2,\varphi_1)\|_{\varphi_1}&= \|R(u_1,v_1,\varphi_1)-R(u_2,v_2,\varphi_1)\|_{\varphi_1} \\ \nonumber &\leqslant q_0(\delta)\|u_1-u_2\|_{\varphi_0}+q_3(\delta) \|v_1-v_2\|_{\varphi_0} \\ \nonumber &\leqslant (q_0(\delta)+q_3(\delta)L)\|u_1-u_2\|_{\varphi_0} \\ &\leqslant \frac{q_1(\delta)(q_0(\delta)+q_3(\delta)L)}{1-q_1(\delta)q_2(\delta)L}\, \|\widetilde{u}_1-\widetilde{u}_2\|_{\varphi_1}. \end{aligned}
\end{equation}
\tag{3.69}
$$
It follows from the obvious relation between $\overline{\Phi}(\widetilde{u},\varphi_1)$ and $\overline{\Phi}(u,\varphi_0)$ (see (3.63)) that these functions satisfy the Lipschitz conditions with respect to $\widetilde{u} $ and $u$, respectively, with the same constant. Hence we conclude from (3.69) that if we take $L$ to be the root of the equation
$$
\begin{equation}
\frac{q_1(\delta)(q_0(\delta)+q_3(\delta)L)}{1-q_1(\delta)q_2(\delta)L}=L
\end{equation}
\tag{3.70}
$$
that satisfies (3.56) and (3.59), then $\overline{\Phi}(u,\varphi_0)$ satisfies the Lipschitz condition (3.27). Thus, the problem of verifying the inclusion $T(\mathscr{H})\subset\mathscr{H}$ reduces to finding such a root.
It is easy to see that (3.70) is equivalent to the quadratic equation
$$
\begin{equation}
P(L)\overset{\rm def}{=}q_1(\delta)q_2(\delta)L^2+ (q_1(\delta)q_3(\delta)-1)L+q_1(\delta)q_0(\delta)=0.
\end{equation}
\tag{3.71}
$$
Now, if the assumptions
$$
\begin{equation}
q_1(\delta)q_3(\delta)<1 \quad\text{and}\quad (1-q_1(\delta)q_3(\delta))^2>4q_1^2(\delta)q_2(\delta)q_0(\delta)
\end{equation}
\tag{3.72}
$$
are fulfilled, then (3.71) has two distinct roots on the half-axis $L\geqslant 0$. We take the smallest as $L$ in (3.27), that is, we set
$$
\begin{equation}
L=L_*(\delta)\overset{\rm def}{=}\frac{2q_1(\delta)q_0(\delta)} {1-q_1(\delta)q_3(\delta)+\sqrt{(1-q_1(\delta)q_3(\delta))^2- 4q_1^2(\delta)q_2(\delta)q_0(\delta)}}\,.
\end{equation}
\tag{3.73}
$$
Then (3.59) holds automatically (as a consequence of the inequality $P'(L)\big|_{L=L_*(\delta)}< 0$). Since the relations imply that $L_*(\delta)\to 0$ as $\delta\to 0$, for small $\delta$ we also have
To legitimise such a choice of $L$ it remains to verify inequalities (3.72). Note that the first of them is a consequence of conditions (3.54) and (3.68) imposed before. As concerns the second, it view of (3.74) it certainly holds for $\delta=0$. Hence it also holds for small $\delta>0$. Throughout what follows we assume that $\delta$ is taken sufficiently small for this to hold. We also assume that (3.75) also holds for this choice of $\delta$. Now we show that for $L=L_*(\delta)$ (see (3.73)) and suitably small $\delta$ the operator $T$ in $\mathscr{H}$ is contracting. To do this, first we identify the dependence of $\Gamma_{\Phi}(u,\widetilde{u},\varphi_1)$ and $u_{\Phi}(\widetilde{u},\varphi_1)$ on $\Phi\in\mathscr{H}$. On the basis of the explicit form of $\Gamma_{\Phi}$ (see (3.55)), taking (3.52) into account we conclude that for any $\Phi_1,\Phi_2\in\mathscr{H}$
$$
\begin{equation}
\begin{aligned} \, \nonumber &\|\Gamma_{\Phi_1}(u,\widetilde{u},\varphi_1)- \Gamma_{\Phi_2}(u,\widetilde{u},\varphi_1)\|_{\varphi_0} \\ \nonumber &\qquad=\|\Sigma(\widetilde{u},\Phi_1(u,\varphi_0),\varphi_1)- \Sigma(\widetilde{u},\Phi_2(u,\varphi_0),\varphi_1)\|_{\varphi_0} \\ &\qquad\leqslant q_1(\delta)q_2(\delta)\|\Phi_1(u,\varphi_0)- \Phi_2(u, \varphi_0)\|_{\varphi_0}\leqslant q_1(\delta)q_2(\delta)\rho(\Phi_1,\Phi_2), \end{aligned}
\end{equation}
\tag{3.76}
$$
where we recall that $\rho(\Phi_1,\Phi_2)$ is the metric in (3.28). In a similar way it follows from the chains of inequalities (3.60) and (3.76), which we have established, that
$$
\begin{equation*}
\begin{aligned} \, &\|u_{\Phi_1}(\widetilde{u}, \varphi_1)- u_{\Phi_2}(\widetilde{u},\varphi_1)\|_{\varphi_0} \\ &\qquad=\|\Gamma_{\Phi_1}(u_{\Phi_1}(\widetilde{u},\varphi_1), \widetilde{u},\varphi_1)-\Gamma_{\Phi_2}(u_{\Phi_2}(\widetilde{u},\varphi_1), \widetilde{u},\varphi_1)\|_{\varphi_0} \\ &\qquad\leqslant q_1(\delta)q_2(\delta)\rho(\Phi_1,\Phi_2)+ q_1(\delta)q_2(\delta)L_*(\delta)\|u_{\Phi_1}(\widetilde{u},\varphi_1)- u_{\Phi_2}(\widetilde{u},\varphi_1)\|_{\varphi_0}, \end{aligned}
\end{equation*}
\notag
$$
where $L_*(\delta)$ is the Lipschitz constant (3.73). This in turn yields
$$
\begin{equation}
\|u_{\Phi_1}(\widetilde{u},\varphi_1)- u_{\Phi_2}(\widetilde{u},\varphi_1)\|_{\varphi_0}\leqslant \frac{q_1(\delta)q_2(\delta)}{1-q_1(\delta)q_2(\delta)L_*(\delta)}\, \rho(\Phi_1,\Phi_2).
\end{equation}
\tag{3.77}
$$
Returning to the problem of whether (3.30) is a contracting operator, we consider the functions $\overline{\Phi}_1(\widetilde{u},\varphi_1)$ and $\overline{\Phi}_2(\widetilde{u},\varphi_1)$ defined by (3.61) for $\Phi=\Phi_1$ and $\Phi=\Phi_2$, respectively. Next we set $u_j=u_{\Phi_j}(\widetilde{u},\varphi_1)$ and $v_j=\Phi_j(u_j, \varphi_0)$, $j=1,2$. It is easy to see from (3.66) and (3.67) that
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\overline{\Phi}_1(\widetilde{u},\varphi_1)- \overline{\Phi}_2(\widetilde{u},\varphi_1)\|_{\varphi_1}&= \|R(u_1,v_1,\varphi_1)-R(u_2,v_2,\varphi_1)\|_{\varphi_1} \\ &\leqslant q_0(\delta)\|u_1-u_2\|_{\varphi_0}+ q_3(\delta)\|v_1-v_2\|_{\varphi_0}. \end{aligned}
\end{equation}
\tag{3.78}
$$
As concerns the norms $\|u_1-u_2\|_{\varphi_0}$ and $\|v_1-v_2\|_{\varphi_0}$ involved in (3.78), we have the bound (3.77) for the first of them and
$$
\begin{equation}
\begin{aligned} \, \nonumber \|v_1-v_2\|_{\varphi_0}&=\|\Phi_1(u_1,\varphi_0)- \Phi_2(u_2,\varphi_0)\|_{\varphi_0}\leqslant \rho(\Phi_1,\Phi_2)+L_*(\delta)\|u_1-u_2\|_{\varphi_0} \\ &\leqslant \frac{1}{1-q_1(\delta)q_2(\delta)L_*(\delta)}\, \rho(\Phi_1,\Phi_2), \end{aligned}
\end{equation}
\tag{3.79}
$$
for the second.
Adding inequalities (3.77)–(3.79) and using (3.63) we conclude that where It remains to note that, as $q_*(0)=q_3(0)<1$, we can achieve the inequality $q_*(\delta)<1$ by taking an even smaller $\delta$. $\Box$By this lemma and the contracting mapping principle, under the above assumptions about $\delta$ and $L$ the operator (3.30) has a unique fixed point $\Phi=\Phi_*(u,\varphi_0)$ in $\mathscr{H}$. Now for each $\varphi_0\in E$ we look at the set (3.11) corresponding to this fixed point. We claim that it is the required unstable manifold for $\overline{G}$. The problem reduces to verifying properties (3.17) and (3.19). In this connection we point out that the required invariance (3.17) follows from the inclusion (3.31) for $\Phi=\Phi_*(u,\varphi_0)$ if we take the equality $T(\Phi_*)=\Phi_*$ into account. To prove the second property, fix arbitrary points $\varphi_0,\varphi\in E$ such that $\varphi\in\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$, and consider the relation where $u\in\overline{E}^{\rm \,u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant \delta$. Since we have the inclusions
$$
\begin{equation*}
\overline{G}^{\,-n}(\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)) \subset \overline{\mathcal{F}}^{\rm \,u}_{\delta} (\overline{G}^{\,-n}(\varphi_0)),\qquad n\in \mathbb{N},
\end{equation*}
\notag
$$
which follow from (3.17), for the point (3.80) the iterates $\overline{G}^{\,-n}(\varphi)$ have a representation of the form
$$
\begin{equation}
\overline{G}^{\,-n}(\varphi)=\varphi_{-n}+u_n+\Phi_*(u_n,\varphi_{-n}),\qquad n\in \mathbb{N},
\end{equation}
\tag{3.81}
$$
where
$$
\begin{equation*}
\varphi_{-n}=\overline{G}^{\,-n}(\varphi_0),\qquad u_n\in\overline{E}^{\rm \,u}_{\varphi_{-n}},\quad \|u_n\|_{\varphi_{-n}}\leqslant \delta.
\end{equation*}
\notag
$$
On the other hand (see (3.3), (3.29), (3.81))
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\overline{G}^{\,-n}(\varphi)-\overline{G}^{\,-n}(\varphi_0)\|&= \|u_n+\Phi_*(u_n,\varphi_{-n})\|\leqslant \|u_n\|+\|\Phi_*(u_n,\varphi_{-n})\| \\ \nonumber &\leqslant \|u_n\|_{\varphi_{-n}}+ \|\Phi_*(u_n,\varphi_{-n})\|_{\varphi_{-n}} \\ &\leqslant (1+L_*(\delta))\|u_n\|_{\varphi_{-n}}\leqslant c(1+L_*(\delta))\|u_n\|, \end{aligned}
\end{equation}
\tag{3.82}
$$
so the justification of (3.19) reduces to deducing similar estimates for the sequence $\|u_n\|$, $n\geqslant 1$. An analysis of this sequence is largely based on the representation
$$
\begin{equation}
\overline{G}^{\,-1}(\varphi_0+u+\Phi_*(u,\varphi_0))= \varphi_{-1}+u_*(u,\varphi_0)+\Phi_*(u_*(u,\varphi_0),\varphi_{-1}),
\end{equation}
\tag{3.83}
$$
which follows from (3.32) for $\Phi=\Phi_*$. Here $(u,\varphi_0)$ is an arbitrary element of $U_{\delta}$, $\varphi_{-1}=\overline{G}^{\,-1}(\varphi_0)$, and the vector function $u_*(u,\varphi_0)$ with values in $\overline{E}^{\rm \,u}_{\varphi_{-1}}$ is obtained from $u_{\Phi}(\widetilde{u},\varphi_1)$ (see Lemma 3.2) for $\Phi=\Phi_*$, after the substitution of $(\widetilde{u},\varphi_1)\in U_{\delta}$ by $(u,\varphi_0)\in U_{\delta}$. In fact, drawing on (3.83) we conclude that
$$
\begin{equation}
u_n=u_*(u_{n-1},\varphi_{-(n-1)}),\quad n\geqslant 1,\qquad u_0=u,
\end{equation}
\tag{3.84}
$$
where $u$ is the vector in (3.80). Now combining (3.84) with inequalities (3.3) and (3.64) and the relation $u_*(0,\varphi_0)\equiv 0$ we obtain
$$
\begin{equation}
\nonumber \|u_n\|_{\varphi_{-n}} =\|u_*(u_{n-1},\varphi_{-(n-1)})\|_{\varphi_{-n}}
\end{equation}
\notag
$$
$$
\begin{equation}
\nonumber =\|u_*(u_{n-1},\varphi_{-(n-1)})-u_*(0,\varphi_{-(n-1)})\|_{\varphi_{-n}}
\end{equation}
\notag
$$
$$
\begin{equation}
\leqslant\nu_1\|u_{n-1}\|_{\varphi_{-(n-1)}},\quad \|u_n\|_{\varphi_{-n}}\leqslant \nu_1^n\|u\|_{\varphi_0},
\end{equation}
\tag{3.85}
$$
$$
\begin{equation}
\nonumber \|u_n\| \leqslant \|u_n\|_{\varphi_{-n}}\leqslant \nu_1^n\|u\|_{\varphi_0}= \nu_1^n\|\overline{P}_{\varphi_0}(\varphi-\varphi_0)\|_{\varphi_0}
\end{equation}
\notag
$$
$$
\begin{equation}
\leqslant\nu_1^n c\cdot\sup_{\varphi_0\in E} \|\overline{P}_{\varphi_0}\|_{E\to E}\cdot \|\varphi-\varphi_0\|,
\end{equation}
\tag{3.86}
$$
where
$$
\begin{equation}
\nu_1=\frac{q_1(\delta)}{1-q_1(\delta)q_2(\delta)L_*(\delta)}<1
\end{equation}
\tag{3.87}
$$
by (3.59). Finally, combining the relations (3.82) and (3.85)–(3.87) obtained we arrive at the required estimates (3.19). Thus we have completely verified that the diffeomorphism $\overline{G}$ has Lipschitz unstable local manifolds. 3.3. The smoothness of local manifoldsTo complete the proof of Theorem 3.1 it remains to verify that the vector function $\Phi_*(u,\varphi_0)$ in (3.11), which, we recall, is a fixed point of the operator (3.30), is $C^1$-smooth in $u\in\overline{E}^{\rm\, u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant\delta$, In view of (3.61)–(3.63) and the equality $T(\Phi_*)=\Phi_*$ it is sufficient to show that the vector function $\Phi_*(\widetilde{u},\varphi_1)$ determined by the equation
$$
\begin{equation}
\Phi_*(\widetilde{u},\varphi_1)= R(u,v,\varphi_1)\big|_{u=u_{\Phi_*}(\widetilde{u},\varphi_1),\ v=\Phi_*(u,\varphi_0)},
\end{equation}
\tag{3.88}
$$
is $C^1$-smooth in $\widetilde{u}$. Recall that here $u=u_{\Phi_*}(\widetilde{u},\varphi_1)$ is the solution of the equation
$$
\begin{equation}
u=\Sigma(\widetilde{u},v,\varphi_1)\big|_{v=\Phi_*(u,\varphi_0)},
\end{equation}
\tag{3.89}
$$
$\varphi_0=\overline{G}^{\,-1}(\varphi_1)$, and $\varphi_1\in E$ is an independent variable. In accordance with the general ideas in [34], first we show that the derivative
$$
\begin{equation}
\frac{\partial\Phi_*}{\partial \widetilde{u}}(\widetilde{u},\varphi_1)= \eta_*(\widetilde{u},\varphi_1)\colon\overline{E}^{\rm\,u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1},\qquad (\widetilde{u},\varphi_1)\in U_{\delta}
\end{equation}
\tag{3.90}
$$
(if it exists) satisfies a certain operator equation. Next, applying the contracting mapping principle to this equation we verify that it has a solution $\eta_*(\widetilde{u},\varphi_1)\colon\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1}$ (a so-called formal derivative). Finally, we show that the function $\eta_*(\widetilde{u},\varphi_1)$ constructed in this way is indeed the Fréchet derivative of the vector function $\Phi_*(\widetilde{u},\varphi_1)$. Following this scheme we differentiate equation (3.89) and equality (3.88) with respect to $\widetilde{u}$. Then we see that the derivative (3.90) must satisfy the equation
$$
\begin{equation}
\hspace{-1mm}\eta_*(\widetilde{u},\varphi_1)= [\mathscr{A}(\widetilde{u},\varphi_1)+ \mathscr{B}(\widetilde{u},\varphi_1) \eta_*(u_{\Phi_*}(\widetilde{u},\varphi_1),\varphi_0)] (I-\mathscr{C}(\widetilde{u},\varphi_1))^{-1} \mathscr{D}(\widetilde{u},\varphi_1),
\end{equation}
\tag{3.91}
$$
where $I$ is the identity operator in $\overline{E}^{\rm\, u}_{\varphi_0}$, and the operators $\mathscr{A}$, $\mathscr{B}$, $\mathscr{C}$, and $\mathscr{D}$ are defined by the equalities
$$
\begin{equation}
\mathscr{A}(\widetilde{u},\varphi_1) = \frac{\partial R}{\partial u}(u,v,\varphi_1)\bigg|_{u=u_{\Phi_*} (\widetilde{u},\varphi_1),\ v=\Phi_*(u,\varphi_0)}\colon \overline{E}^{\rm\, u}_{\varphi_0}\to\overline{E}^{\rm\, s}_{\varphi_1},
\end{equation}
\tag{3.92}
$$
$$
\begin{equation}
\mathscr{B}(\widetilde{u},\varphi_1) =\frac{\partial R}{\partial v} (u,v,\varphi_1)\bigg|_{u=u_{\Phi_*}(\widetilde{u},\varphi_1),\ v=\Phi_*(u, \varphi_0)}\colon \overline{E}^{\rm\, s}_{\varphi_0}\to \overline{E}^{\rm\, s}_{\varphi_1},
\end{equation}
\tag{3.93}
$$
$$
\begin{equation}
\mathscr{C}(\widetilde{u},\varphi_1) =\frac{\partial}{\partial v}\, \Sigma(\widetilde{u},v,\varphi_1)\cdot\eta_* (u, \varphi_0)\bigg|_{u=u_{\Phi_*}(\widetilde{u}, \varphi_1),\ v=\Phi_*(u,\varphi_0)}\colon\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0},
\end{equation}
\tag{3.94}
$$
$$
\begin{equation}
\text{and} \qquad \mathscr{D}(\widetilde{u},\varphi_1) =\frac{\partial}{\partial\widetilde{u}}\, \Sigma(\widetilde{u},v,\varphi_1)\bigg|_{u=u_{\Phi_*}(\widetilde{u},\varphi_1), \ v=\Phi_*(u,\varphi_0)}\colon\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, u}_{\varphi_0}.
\end{equation}
\tag{3.95}
$$
Now we consider the existence of a formal derivative. We need the following lemma. Lemma 3.4. Equation (3.91) has a solution $\eta_*(\widetilde{u},\varphi_1)$, which is a bounded linear operator from $\overline{E}^{\rm\, u}_{\varphi_1}$ to $\overline{E}^{\rm\, s}_{\varphi_1}$. This solution is uniformly continuous in $(\widetilde{u},\varphi_1)\in U_{\delta}$ in the uniform operator topology, it depends $2\pi$-periodically on $\varphi_1$ and has the following property: Proof. We show that we can apply the contracting mapping principle to (3.91) in a suitable metric space $\mathscr{H}'$.
We assume that elements of this space $\mathscr{H}'$ are linear bounded operators $\eta(\widetilde{u},\varphi_1)\colon \overline{E}^{\rm\, u}_{\varphi_1}\to\overline{E}^{\rm\, s}_{\varphi_1}$, which are uniformly continuous in $(\widetilde{u},\varphi_1)\in U_{\delta}$ in the uniform operator topology (recall that this means their uniform continuity in the norm of the space $L(E;E)$ of operators $\eta(\widetilde{u},\varphi_1)\overline{P}_{\varphi_1}\colon E\to E$). We also assume that operators in this space depend $2\pi$-periodically on $\varphi_1\in E$ and satisfy the conditions
$$
\begin{equation}
\eta(0,\varphi_1)\equiv 0,\quad \|\eta(\widetilde{u},\varphi_1)\|_{\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1}}\leqslant L_*(\delta),
\end{equation}
\tag{3.97}
$$
which are similar to (3.96).
We define the metric in $\mathscr{H}'$ by:
$$
\begin{equation}
\forall\,\eta_1,\eta_2\in\mathscr{H}'\quad \rho(\eta_1,\eta_2)=\sup_{(\widetilde{u},\varphi_1)\in U_{\delta}} \|\eta_1(\widetilde{u},\varphi_1)- \eta_2(\widetilde{u},\varphi_1)\|_{\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1}}.
\end{equation}
\tag{3.98}
$$
We stress that, in view of (3.97), the supremum in (3.98) is certainly finite. It is also clear that we obtain a complete metric space.
At the next step of the proof of the lemma, we define an operator $T'$ in $\mathscr{H}'$ by where
$$
\begin{equation}
\overline{\eta}(\widetilde{u},\varphi_1)= [\mathscr{A}(\widetilde{u},\varphi_1)+\mathscr{B}(\widetilde{u},\varphi_1) \eta(u_{\Phi_*}(\widetilde{u},\varphi_1),\varphi_0)] (I-\mathscr{C}_{\eta}(\widetilde{u},\varphi_1))^{-1} \mathscr{D}(\widetilde{u},\varphi_1),
\end{equation}
\tag{3.100}
$$
$\mathscr{A}$, $\mathscr{B}$, and $\mathscr{D}$ are the operators (3.92), (3.93), and (3.95), respectively, but in place of (3.94) the operator $\mathscr{C}_{\eta}$ is now defined by
$$
\begin{equation}
\mathscr{C}_{\eta}(\widetilde{u},\varphi_1)= \frac{\partial}{\partial v}\,\Sigma(\widetilde{u},v,\varphi_1)\cdot \eta(u,\varphi_0)\bigg|_{u=u_{\Phi_*}(\widetilde{u},\varphi_1),\ v=\Phi_*(u,\varphi_0)}.
\end{equation}
\tag{3.101}
$$
Our main aim is to show that the operator (3.99), (3.100) takes the space $\mathscr{H}'$ to itself and is contracting there.
First of all, we verify that the operator $I-\mathscr{C}_{\eta}(\widetilde{u},\varphi_1)$ in (3.100) is invertible. To do this we combine conditions (3.97) with the estimates (3.52). Then we obtain sequentially
$$
\begin{equation}
\begin{aligned} \, \|\mathscr{C}_{\eta} (\widetilde{u},\varphi_1)\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0}}&\leqslant q_1(\delta)q_2(\delta)L_*(\delta)<1, \\ \|(I-\mathscr{C}_{\eta} (\widetilde{u},\varphi_1))^{-1}\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0}}&\leqslant \frac{1}{1-q_1(\delta)q_2(\delta)L_*(\delta)}\,. \end{aligned}
\end{equation}
\tag{3.102}
$$
Now we show that $T'(\mathscr{H}')\subset\mathscr{H}'$. We note straight away that from the relations
$$
\begin{equation*}
\mathscr{A}(0,\varphi_1)\equiv \frac{\partial R}{\partial u}(0,0,\varphi_1)\equiv 0
\end{equation*}
\notag
$$
and the first condition in (3.97) we obtain the required identity $\overline{\eta}(0,\varphi_1)\equiv 0$. As regards the second condition in (3.97), to verify it in the case of $\overline{\eta}(\widetilde{u},\varphi_1)$ we use (3.100), the estimates (3.52), (3.66), (3.67), and (3.102), and relations (3.70) and (3.73). Then we verify that
$$
\begin{equation*}
\|\overline{\eta} (\widetilde{u},\varphi_1)\|_{\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1}}\leqslant \frac{q_1(\delta)(q_0(\delta)+q_3(\delta)L_*(\delta))} {1-q_1(\delta)q_2(\delta)L_*(\delta)}=L_*(\delta).
\end{equation*}
\notag
$$
Note also that by (3.100), for $\overline{\eta}(\widetilde{u},\varphi_1)$ we also have the property of $2\pi$-periodicity in $\varphi_1$ and uniform continuity in $(\widetilde{u},\varphi_1)\in U_{\delta}$. This means that the operator $T'$ takes indeed the space $\mathscr{H}'$ to itself. To verify that $T'$ is contracting, fix some $\eta_j(\widetilde{u},\varphi_1)\in\mathscr{H}'$, $j=1,2$, and set $\overline{\eta}_j(\widetilde{u},\varphi_1)=T'(\eta_j)$, $j=1,2$. Then, relying on the relation
$$
\begin{equation*}
(I-\mathscr{C}_{\eta_1})^{-1}-(I-\mathscr{C}_{\eta_2})^{-1}= (I-\mathscr{C}_{\eta_1})^{-1}(\mathscr{C}_{\eta_1}- \mathscr{C}_{\eta_2})(I-\mathscr{C}_{\eta_2})^{-1}
\end{equation*}
\notag
$$
and the estimates (3.52), (3.66), (3.67), (3.102), we deduce in succession
$$
\begin{equation*}
\begin{aligned} \, &\|(I-\mathscr{C}_{\eta_1}(\widetilde{u},\varphi_1))^{-1}- (I-\mathscr{C}_{\eta_2} (\widetilde{u},\varphi_1))^{-1}\|_{\overline{E}^{\rm\, u}_{\varphi_0}\to \overline{E}^{\rm\, u}_{\varphi_0}} \\ &\qquad\leqslant \frac{q_1(\delta)q_2(\delta)}{(1-q_1(\delta)q_2(\delta)L_*(\delta))^2} \rho(\eta_1,\eta_2), \\ &\|\eta_1(\widetilde{u},\varphi_1)- \eta_2(\widetilde{u},\varphi_1)\|_{\overline{E}^{\rm\, u}_{\varphi_1}\to \overline{E}^{\rm\, s}_{\varphi_1}}\leqslant \overline{q}(\delta)\rho(\eta_1,\eta_2), \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber \overline{q}(\delta)&=\frac{q_0(\delta)q_1^2(\delta)q_2(\delta)} {(1-q_1(\delta)q_2(\delta)L_*(\delta))^2}+ \frac{q_1^2(\delta)q_2(\delta)q_3(\delta)L_*(\delta)} {(1-q_1(\delta)q_2(\delta)L_*(\delta))^2} \\ &\qquad+\frac{q_1(\delta)q_3(\delta)}{1-q_1(\delta)q_2(\delta)L_*(\delta)}= \frac{d}{dL}\biggl(\frac{q_1(\delta)(q_0(\delta)+q_3(\delta)L)} {1-q_1(\delta)q_2(\delta)L}\biggr)\bigg|_{L=L_*(\delta)}. \end{aligned}
\end{equation}
\tag{3.103}
$$
On the other hand recall that $L_*(\delta)$ is the least positive root of equation (3.70), and thus $\overline{q}(\delta)<1$. Hence we can use the contracting mapping principle, which shows that the operator (3.99) has a unique fixed point $\eta_*(\widetilde{u},\varphi_1)\in\mathscr{H}'$. We must also add that this fixed point is also a solution of (3.91). $\Box$ Thus we have found a formal derivative of the vector function $\Phi_*(\widetilde{u},\varphi_1)$. Next we verify that this function is indeed continuously differentiable with respect to $\widetilde{u}$ and we have equality (3.90) with $\eta_*(\widetilde{u},\varphi_1)$ equal to the solution of (3.91) found above. To do this we require the function
$$
\begin{equation}
z(\widetilde{u}, \varphi_1)=\varlimsup_{\Delta\widetilde{u}\to 0} \frac{\|\Phi_*(\widetilde{u}+\Delta\widetilde{u},\varphi_1)- \Phi_*(\widetilde{u},\varphi_1)-\eta_*(\widetilde{u},\varphi_1) \Delta\widetilde{u}\|_{\varphi_1}}{\|\Delta\widetilde{u}\|_{\varphi_1}}\,,
\end{equation}
\tag{3.104}
$$
where $\Delta\widetilde{u}\in\overline{E}^{\rm\, u}_{\varphi_1}$ is an increment of the independent variable. We note the following two points straight away. First, the right-hand side in (3.104) is uniformly bounded: it follows from (3.27) and (3.96) that $z(\widetilde{u},\varphi_1)\leqslant 2L_*(\delta)$ for all $(\widetilde{u},\varphi_1)\in U_{\delta}$. Second, we will have established that $\Phi_*(\widetilde{u},\varphi_1)$ is continuously differentiable with respect to $\widetilde{u}$ and (3.90) holds once we will have shown that $z(\widetilde{u},\varphi_1)\equiv 0$. We wish to estimate $z(\widetilde{u},\varphi_1)$ above in terms of the same function $z(u,\varphi_0)$, but for $u=u_{\Phi_*}(\widetilde{u},\varphi_1)$ and $\varphi_0=\overline{G}^{\,-1}(\varphi_1)$. For such an estimate we need the notation
$$
\begin{equation*}
\begin{aligned} \, \Delta\Phi_*(\widetilde{u},\varphi_1)&= \Phi_*(\widetilde{u}+\Delta\widetilde{u},\varphi_1)- \Phi_*(\widetilde{u},\varphi_1), \\ \Delta u&=u_{\Phi_*}(\widetilde{u}+\Delta\widetilde{u},\varphi_1)- u_{\Phi_*}(\widetilde{u},\varphi_1), \\ \Delta\Phi_*(u,\varphi_0)&=\Phi_*(u+\Delta u,\varphi_0)-\Phi_*(u,\varphi_0), \\ \widetilde{z}(u,\varphi_0)&=\Delta\Phi_*(u,\varphi_0)- \eta_*(u,\varphi_0)\Delta u, \\ \widetilde{z}(\widetilde{u},\varphi_1)&=\Delta\Phi_*(\widetilde{u},\varphi_1)- \eta_*(\widetilde{u},\varphi_1)\Delta\widetilde{u}. \end{aligned}
\end{equation*}
\notag
$$
Note that by inequalities (3.27) and (3.64) (for $\Phi=\Phi_*$ and $L=L_*(\delta)$) we have
$$
\begin{equation}
\begin{aligned} \, \|\Delta\Phi_*(u,\varphi_0)\|_{\varphi_0}&\leqslant L_*(\delta)\|\Delta u\|_{\varphi_0}, \\ \|\Delta u\|_{\varphi_0}&\leqslant \frac{q_1(\delta)}{1-q_1(\delta)q_2(\delta)L_*(\delta)} \|\Delta\widetilde{u}\|_{\varphi_1}. \end{aligned}
\end{equation}
\tag{3.105}
$$
Now we proceed as follows. First, on the basis of (3.88) and (3.89), as $\Delta\widetilde{u}\to 0$ we obtain the asymptotic representations
$$
\begin{equation}
\nonumber \widetilde{z}(\widetilde{u},\varphi_1) = R(u+\Delta u,\Phi_*(u+\Delta u,\varphi_0),\varphi_1)- R(u,\Phi_*(u,\varphi_0),\varphi_1)- \eta_*(\widetilde{u},\varphi_1)\Delta\widetilde{u}
\end{equation}
\notag
$$
$$
\begin{equation}
\nonumber =\frac{\partial R}{\partial u}(u,\Phi_*(u,\varphi_0),\varphi_1)\Delta u+ \frac{\partial R}{\partial v}(u,\Phi_*(u,\varphi_0),\varphi_1) \Delta\Phi_*(u, \varphi_0)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+o(\Delta u)+o(\Delta\Phi_*(u,\varphi_0))- \eta_*(\widetilde{u},\varphi_1)\Delta\widetilde{u}
\end{equation}
\tag{3.106}
$$
$$
\begin{equation}
\text{and} \qquad \nonumber \Delta u =\Sigma(\widetilde{u}+ \Delta\widetilde{u},\Phi_*(u+\Delta u,\varphi_0),\varphi_1)- \Sigma(\widetilde{u},\Phi_*(u,\varphi_0),\varphi_1)
\end{equation}
\notag
$$
$$
\begin{equation}
\nonumber =\frac{\partial}{\partial \widetilde{u}}\, \Sigma(\widetilde{u},\Phi_*(u,\varphi_0),\varphi_1)\Delta\widetilde{u}+ \frac{\partial}{\partial v}\, \Sigma(\widetilde{u},\Phi_*(u,\varphi_0),\varphi_1)\Delta\Phi_*(u,\varphi_0)
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+o(\Delta\widetilde{u})+o(\Delta\Phi_*(u,\varphi_0));
\end{equation}
\tag{3.107}
$$
here and below $u=u_{\Phi_*}(\widetilde{u},\varphi_1)$. Next, since
$$
\begin{equation}
\Delta\Phi_*(u,\varphi_0)=\widetilde{z}(u,\varphi_0)+ \eta_*(u,\varphi_0)\Delta u,
\end{equation}
\tag{3.108}
$$
we replace $\Delta\Phi_*(u,\varphi_0)$ by this expression in (3.107) and find $\Delta u$ from the resulting equation. As a result, we obtain
$$
\begin{equation*}
\begin{aligned} \, \Delta u&=(I-\mathscr{C}(\widetilde{u},\varphi_1))^{-1} \biggl(\frac{\partial}{\partial v}\,\Sigma(\widetilde{u}, \Phi_*(u,\varphi_0),\varphi_1)\widetilde{z}(u,\varphi_0) \\ &\qquad+\frac{\partial}{\partial \widetilde{u}}\,\Sigma(\widetilde{u}, \Phi_*(u, \varphi_0),\varphi_1)\Delta\widetilde{u}\biggr)+ o(\Delta\widetilde{u})+o(\Delta\Phi_*(u,\varphi_0)), \end{aligned}
\end{equation*}
\notag
$$
where $\mathscr{C}(\widetilde{u}, \varphi_1)$ is the operator in (3.94). In turn, plugging this expression and (3.108) into (3.106) and taking (3.91) into account we verify that
$$
\begin{equation}
\begin{aligned} \, \nonumber \widetilde{z}(\widetilde{u},\varphi_1)&=\biggl[\frac{\partial R}{\partial v} (u,\Phi_*(u,\varphi_0),\varphi_1) \\ \nonumber &\qquad+\biggl(\frac{\partial R}{\partial u}(u,\Phi_*(u,\varphi_0),\varphi_1)+ \frac{\partial R}{\partial v}(u,\Phi_*(u,\varphi_0),\varphi_1) \eta_*(u,\varphi_0)\biggl) \\ \nonumber &\qquad\times(I-\mathscr{C}(\widetilde{u},\varphi_1))^{-1} \frac{\partial}{\partial v}\, \Sigma(\widetilde{u},\Phi_*(u,\varphi_0),\varphi_1)\biggr] \widetilde{z}(u,\varphi_0) \\ &\qquad+o(\Delta\widetilde{u})+o(\Delta u)+o(\Delta\Phi_*(u,\varphi_0)). \end{aligned}
\end{equation}
\tag{3.109}
$$
Using formula (3.109) we can rather easily establish the required upper bound for (3.104). In fact, of the basis of (3.52), (3.66), (3.67), (3.102), and (3.105) we conclude that
$$
\begin{equation*}
\begin{aligned} \, \frac{\|\widetilde{z}(\widetilde{u},\varphi_1)\|_{\varphi_1}} {\|\Delta\widetilde{u}\|_{\varphi_1}}&\leqslant \biggl(q_3(\delta)+ \frac{(q_0(\delta)+q_3(\delta)L_*(\delta))q_1(\delta)q_2(\delta)} {1-q_1(\delta)q_2(\delta)L_*(\delta)}\biggr) \\ &\qquad\times\frac{\|\widetilde{z}(u,\varphi_0)\|_{\varphi_0}} {\|\Delta u\|_{\varphi_0}}\cdot \frac{\|\Delta u\|_{\varphi_0}}{\|\Delta\widetilde{u}\|_{\varphi_1}}+o(1)\\ &\leqslant \overline{q}(\delta) \frac{\|\widetilde{z}(u,\varphi_0)\|_{\varphi_0}}{\|\Delta u\|_{\varphi_0}} +o(1), \end{aligned}
\end{equation*}
\notag
$$
where $\overline{q}(\delta)$ is the constant in (3.103). Hence, taking the limit as $\Delta\widetilde{u}\to 0$ we obtain
$$
\begin{equation}
z(\widetilde{u},\varphi_1)\leqslant\overline{q}(\delta) z(u,\varphi_0)\big|_{u=u_{\Phi_*}(\widetilde{u},\varphi_1)}.
\end{equation}
\tag{3.110}
$$
It is easy to show using (3.110) that in fact $z(\widetilde{u},\varphi_1)\equiv 0$. Namely, fix an arbitrary point $(\widetilde{u},\varphi_1)\in U_{\delta}$ and set
$$
\begin{equation}
\widetilde{u}_0=\widetilde{u}\quad\text{and}\quad \widetilde{u}_k=u_{\Phi_*}(\widetilde{u}_{k-1},\varphi_{-(k-2)}),\quad k\geqslant1,
\end{equation}
\tag{3.111}
$$
where, as usual, $\varphi_{-(k-2)}=\overline{G}^{\,-(k-1)}(\varphi_1)$. Note also that by the known properties of $u_{\Phi_*}(\widetilde{u},\varphi_1)$ (see (3.57)) we have
$$
\begin{equation}
\widetilde{u}_k\in\overline{E}^{\rm\, u}_{\varphi_{-(k-1)}}\quad\text{and}\quad \|\widetilde{u}_k\|_{\varphi_{-(k-1)}}\leqslant \delta,\qquad k\geqslant 1.
\end{equation}
\tag{3.112}
$$
Taking (3.111) and (3.112) into account, we conclude from (3.110) that
$$
\begin{equation*}
\begin{aligned} \, z(\widetilde{u},\varphi_1)&\leqslant \overline{q}(\delta)z(\widetilde{u}_1,\varphi_0)\leqslant \overline{q}^2(\delta)z(\widetilde{u}_2,\varphi_{-1})\leqslant\cdots \leqslant\overline{q}^k(\delta)z(\widetilde{u}_k,\varphi_{-(k-1)}) \\ &\leqslant 2L_*(\delta)\overline{q}^k(\delta). \end{aligned}
\end{equation*}
\notag
$$
As $\overline{q}(\delta)<1$ and $k$ is arbitrary, this implies directly the required idenity $z(\widetilde{u},\varphi_1)\equiv 0$. Thus we have shown that the diffeomorphism $\overline{G}$ has smooth unstable local manifolds $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)$, $\varphi_0\in E$. Next, performing all the above constructions for the diffeomorphism $\overline{G}^{\,-1}$ we see that $\overline{G}$ also has similar smooth local stable manifolds $\overline{\mathcal{F}}^{\rm \,s}_{\delta}(\varphi_0)$, $\varphi_0\in E$. This completes the proof of Theorem 3.1, which is an analogue of the celebrated Hadamard–Perron theorem. 3.4. The existence of invariant foliationsAs usual, we call partitions $\mathcal{F}^{\rm \,u}$ and $\mathcal{F}^{\rm \,s}$ of the original manifold $\mathbb{T}^{\infty}$ an unstable and a stable invariant foliation for the hyperbolic diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ if the following conditions are satisfied: 1) the elements $\mathcal{F}^{\rm \,u}(\varphi_0)$ and $\mathcal{F}^{\rm \,s}(\varphi_0)$ of $\mathcal{F}^{\rm \,u}$ and $\mathcal{F}^{\rm \,s}$, respectively, that contain an arbitrary fixed point $\varphi_0\in\mathbb{T}^{\infty}$ are $C^1$-smooth submanifolds immersed in $\mathbb{T}^{\infty}$ (they are called global leaves of the foliations $\mathcal{F}^{\rm \,u}$ and $\mathcal{F}^{\rm \,s}$ at $\varphi_0$); moreover, it is assumed that there exists a universal (independent of $\varphi_0$) sufficiently small value of $\delta>0$ such that for
$$
\begin{equation*}
\begin{aligned} \, B_{\delta}(\varphi_0)=\operatorname{pr}\bigl[\operatorname{pr}^{-1}(\varphi_0)+ u+v\colon u&\in\overline{E}^{\rm \, u}_{\operatorname{pr}^{-1}(\varphi_0)},\ \|u\|_{\operatorname{pr}^{-1}(\varphi_0)}\leqslant \delta, \\ v&\in\overline{E}^{\rm \,s}_{\operatorname{pr}^{-1}(\varphi_0)},\ \|v\|_{\operatorname{pr}^{-1}(\varphi_0)}\leqslant \delta\bigr] \end{aligned}
\end{equation*}
\notag
$$
the connected components of the intersections $\mathcal{F}^{\rm \,u}(\varphi_0)\cap B_{\delta}(\varphi_0)$ and $\mathcal{F}^{\rm \,s}(\varphi_0)\cap B_{\delta}(\varphi_0)$ containing $\varphi_0$ coincide with the local manifolds $\mathcal{F}^{\rm \,u}_{\delta}(\varphi_0)$ and $\mathcal{F}^{\rm \,s}_{\delta}(\varphi_0)$, respectively (see (3.22)); 2) the following invariance properties hold:
$$
\begin{equation}
\forall\,\varphi_0\in\mathbb{T}^{\infty}\quad G(\mathcal{F}^{\rm \,u}(\varphi_0))=\mathcal{F}^{\rm \,u}(G(\varphi_0))\quad\text{and}\quad G(\mathcal{F}^{\rm \,s}(\varphi_0))=\mathcal{F}^{\rm \,s}(G(\varphi_0)).
\end{equation}
\tag{3.113}
$$
The following theorem is the central result in this section and the whole paper. Theorem 3.2. Each hyperbolic diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ admits a stable and an unstable invariant foliation. Proof. As in the proof of Theorem 3.1, it is sufficient to verify only that the foliation $\mathcal{F}^{\rm \,u}$ exists. As concerns $\mathcal{F}^{\rm \,s}$, we can turn to the inverse map $G^{-1}$ and repeat for it all arguments used in the proof of the existence of $\mathcal{F}^{\rm \,u}$.
First we show that, given a global lifting $\overline{G}$ of an arbitrary hyperbolic diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$, there exists an unstable invariant foliation $\overline{\mathcal{F}}^{\rm \,u}$. To do this, using the methods presented in [3] we introduce an equivalence relation in $E$ as follows. We say that two points $\overline{\varphi}$ and $\overline{\overline{\varphi}}$ in $E$ are equivalent if there exists a chain of local manifolds $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_k)$, $k=0,1,\dots,n$, such that
$$
\begin{equation}
\begin{gathered} \, \overline{\varphi}\in\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_0),\quad \overline{\overline{\varphi}}\in \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_n), \\ \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_k)\cap \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_{k+1})\ne \varnothing,\qquad k=0,1,\dots,n-1 \end{gathered}
\end{equation}
\tag{3.114}
$$
(we call it a joining chain). Next we denote the set of points in $E$ that are equivalent to the fixed point $\varphi_0$ by $\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$.
Note that by the above definitions, for each $\overline{\varphi}\in\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$ we have $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})\subset \overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$. Moreover, by Theorem 3.1 and Corollary 3.1.1 the leaf $\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$ coincides locally (in a neighbourhood of each point $\overline{\varphi}$ on it) with the local manifold $\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\overline{\varphi})$. Thus, the set $\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$ is an embedded $C^1$-smooth submanifold. On the other hand, it is obvious that two leaves $\overline{\mathcal{F}}^{\rm \,u}(\overline{\varphi})$ and $\overline{\mathcal{F}}^{\rm \,u}(\overline{\overline{\varphi}})$ are either disjoint or coincide, and therefore the system
$$
\begin{equation}
\overline{\mathcal{F}}^{\rm \,u}= \{\overline{\mathcal{F}}^{\rm \,u}(\varphi_0),\,\varphi_0\in E\}
\end{equation}
\tag{3.115}
$$
is a foliation on $E$.
Now we verify that the foliation (3.115) is invariant, that is, we have the equalities
$$
\begin{equation}
\forall\,\varphi_0\in E\quad \overline{G}(\overline{\mathcal{F}}^{\rm \,u}(\varphi_0))= \overline{\mathcal{F}}^{\rm \,u}(\overline{G}(\varphi_0)),
\end{equation}
\tag{3.116}
$$
which are similar to (3.113). In this connection we look at formula (3.32) for $\Phi=\Phi_*$ and $\overline{\Phi}=\Phi_*$, where we recall that $\Phi_*(u,\varphi_0)$ is a fixed point of (3.30). We treat $u\in\overline{E}^{\rm \,u}_{\varphi_0}$, $\|u\|_{\varphi_0}\leqslant\delta$, as an independent variable and $\widetilde{u}\in\overline{E}^{\rm \,u}_{\varphi_1}$ as a function of $u$. Then from (3.29) and (3.33) we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber \|\widetilde{u}\|_{\varphi_1}&= \|\overline{P}_{\varphi_1}(\overline{G}(\varphi_0+u+ \Phi_*(u,\varphi_0))-\overline{G}(\varphi_0))\|_{\varphi_1} \\ \nonumber &\leqslant c\cdot\sup_{\varphi_1\in E}\|\overline{P}_{\varphi_1}\|_{E\to E} \cdot\sup_{\varphi\in E}\|D\overline{G}(\varphi)\|_{E\to E}\cdot \|u+\Phi_*(u,\varphi_0)\| \\ \nonumber &\leqslant c\cdot\sup_{\varphi_1\in E}\|\overline{P}_{\varphi_1}\|_{E\to E} \cdot\sup_{\varphi\in E}\|D\overline{G}(\varphi)\|_{E\to E}\cdot (\|u\|_{\varphi_0}+\|\Phi_*(u,\varphi_0)\|_{\varphi_0}) \\ &\leqslant c(1+L_*(\delta))\sup_{\varphi_1\in E} \|\overline{P}_{\varphi_1}\|_{E\to E}\cdot \sup_{\varphi\in E}\|D\overline{G}(\varphi)\|_{E\to E}\cdot \delta\overset{\rm def}{=}\delta', \end{aligned}
\end{equation}
\tag{3.117}
$$
where $c$ is the constant in (3.3) and (3.4). In its turn, this means that
$$
\begin{equation}
\overline{G}(\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0))\subset \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\overline{G}(\varphi_0))\quad \forall\,\varphi_0\in E,
\end{equation}
\tag{3.118}
$$
where $\delta'$ is the quantity in (3.117).
On the basis of (3.17) and (3.118) we easily establish the required equalities (3.116). First we prove that
$$
\begin{equation}
\overline{G}(\overline{\mathcal{F}}^{\rm \,u}(\varphi_0))\subset \overline{\mathcal{F}}^{\rm \,u}(\overline{G}(\varphi_0))\quad \forall\,\varphi_0\in E.
\end{equation}
\tag{3.119}
$$
To do this fix some $\overline{\varphi}\in\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$. Then by definition (see (3.114)) there exist points $\theta_k$ in $E$, $k=0,1,\dots,n$, such that
$$
\begin{equation*}
\begin{gathered} \, \overline{\varphi}\in\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_0),\quad \varphi_0\in\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_n), \\ \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_k)\cap \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_{k+1})\ne \varnothing,\qquad k=0,1,\dots,n-1. \end{gathered}
\end{equation*}
\notag
$$
Now, taking the inclusion (3.118) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \overline{G}(\overline{\varphi})&\in \overline{G}(\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_0))\subset \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\overline{G}(\theta_0)),\quad \overline{G}(\varphi_0)\in \overline{G}(\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_n))\subset \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\overline{G}(\theta_n)), \\ \varnothing&\ne \overline{G}(\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_k)\cap \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_{k+1}))\subset \overline{G}(\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_k))\cap \overline{G}(\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\theta_{k+1})) \\ &\subset \overline{\mathcal{F}}^{\rm \,u}_{\delta'}(\overline{G}(\theta_k)) \cap \overline{\mathcal{F}}^{\rm \,u}_{\delta'} (\overline{G}(\theta_{k+1})),\qquad k=0,1,\dots,n-1. \end{aligned}
\end{equation*}
\notag
$$
Thus, the points $\overline{G}(\overline{\varphi})$ and $\overline{G}(\varphi_0)$ are equivalent, so that $\overline{G}(\overline{\varphi})\in \overline{\mathcal{F}}^{\rm\, u}(\overline{G}(\varphi_0))$ and, as $\overline{\varphi}\in\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$ can be arbitrary, we have (3.119).
To obtain the reverse inclusions
$$
\begin{equation*}
\overline{G}(\overline{\mathcal{F}}^{\rm \,u}(\varphi_0))\supset \overline{\mathcal{F}}^{\rm \,u}(\overline{G}(\varphi_0))\quad \forall\,\varphi_0\in E
\end{equation*}
\notag
$$
we establish the property
$$
\begin{equation}
\overline{G}^{\,-1}(\overline{\mathcal{F}}^{\rm \,u}(\varphi_0))\subset \overline{\mathcal{F}}^{\rm \,u}(\overline{G}^{\,-1}(\varphi_0))\quad \forall\,\varphi_0\in E,
\end{equation}
\tag{3.120}
$$
which is equivalent to these inclusions. To prove it we use the relations
$$
\begin{equation*}
\overline{G}^{\,-1}(\overline{\mathcal{F}}^{\rm \,u}_{\delta}(\varphi_0)) \subset\overline{\mathcal{F}}^{\rm \,u}_{\delta} (\overline{G}^{\,-1}(\varphi_0))\quad \forall\,\varphi_0\in E,
\end{equation*}
\notag
$$
which are equivalent to (3.17), and repeat the above arguments for $\overline{G}$ replaced by $\overline{G}^{\,-1}$. As a result, we verify (3.120). In combination with (3.119) this is equivalent to the condition of invariance (3.116).
Returning to the original diffeomorphism we set
$$
\begin{equation}
\mathcal{F}^{\rm\, u}=\bigl\{\mathcal{F}^{\rm\, u}(\varphi_0)= \operatorname{pr}\bigl[\,\overline{\mathcal{F}}^{\rm \,u} (\operatorname{pr}^{-1}(\varphi_0))\bigr], \ \varphi_0\in\mathbb{T}^{\infty}\bigr\}.
\end{equation}
\tag{3.121}
$$
Bearing in mind the properties of the global leaves $\overline{\mathcal{F}}^{\rm \,u}(\varphi_0)$, $\varphi_0\in E$, which we have established already, it is easy to see that (3.121) defines the required unstable invariant foliation for $G$. $\Box$ The next result is related to global analogues of (3.24) and (3.25). Before stating it, we define the so-called intrinsic metrics on the global leaves $\mathcal{F}^{\rm\, u}(\varphi_0)$ and $\mathcal{F}^{\rm\, s}(\varphi_0)$. Namely, for any points $\overline{\varphi}$, $\overline{\overline{\varphi}}\in\mathcal{F}^{\rm\, u}(\varphi_0)$ set
$$
\begin{equation}
d^{\rm\,u}(\overline{\varphi},\overline{\overline{\varphi}})= \inf_{\{\theta\}}L(\theta).
\end{equation}
\tag{3.122}
$$
Here $\{\theta\}$ is the set of rectifiable continuous curves $\theta(t)$, $0\leqslant t\leqslant 1$, $\theta(0)=\overline{\varphi}$, $\theta(1)=\overline{\overline{\varphi}}$, that lie in $\mathcal{F}^{\rm\, u}(\varphi_0)$ entirely, and $L(\theta)$ is the length of such a curve. We stress that, as each leaf $\mathcal{F}^{\rm\, u}(\varphi_0)$ is clearly path connected, the set $\{\theta\}$ is certainly non-empty. On each stable leaf $\mathcal{F}^{\rm\, s}(\varphi_0)$ the intronsic metric $d^{\rm\,s}(\overline{\varphi}, \overline{\overline{\varphi}})$ is defined similarly to (3.122). Theorem 3.3. For all $n\in\mathbb{N}$, Proof. First we make a useful observation. It follows from the construction of global leaves $\mathcal{F}^{\rm \,u}(\varphi_0)$, $\varphi_0\in E$, that for any sufficiently close points $\overline{\varphi}$ and $\overline{\overline{\varphi}}$ in $\mathcal{F}^{\rm\, u}(\varphi_0)$ we have $\overline{\overline{\varphi}}\in \mathcal{F}^{\rm\, u}_{\delta}(\overline{\varphi})$, where we recall that $\mathcal{F}^{\rm\, u}_{\delta}(\overline{\varphi})$ is the local manifold in (3.22). Taking this into account we fix an arbitrary rectifiable continuous curve $\theta(t)$, $0\leqslant t\leqslant 1$, lying on $\mathcal{F}^{\rm\, u}(\varphi_0)$ and connecting $\varphi_0$ with $\varphi$. Next we consider the set $\mathscr{T}$ of partitions $0=t_0<t_1<\cdots<t_m=1$ of the interval $0\leqslant t\leqslant 1$. We assume that these partitions are sufficiently fine so that
Using the inclusions (3.125) we can verify the global bounds (3.123) with the help of their local versions (3.24). In fact, it is easy to observe that
$$
\begin{equation}
\begin{aligned} \, \nonumber \rho(G^{-n}(\varphi),G^{-n}(\varphi_0))&\leqslant \sum_{k=0}^{m-1}\rho\bigl(G^{-n}(\theta(t_{k+1})),G^{-n}(\theta(t_{k}))\bigr) \\ &\leqslant r_1\nu_1^n\sum_{k=0}^{m-1}\rho(\theta(t_{k}),\theta(t_{k+1})). \end{aligned}
\end{equation}
\tag{3.126}
$$
Now taking the supremum over partitions in $\mathscr{T}$ and then the infimum over the curves $\theta(t)$ on the right-hand side of (3.126), we obtain the required inequalities (3.123). The case (3.124) is treated in a similar way. Theorem 3.3 is proved. It is a characteristic feature of Anosov diffeomorphisms on a finite-dimensional torus $\mathbb{T}^m$, $m\geqslant 2$, that any two global leaves $\mathcal{F}^{\rm\, u}(\overline{\varphi})$ and $\mathcal{F}^{\rm\, s}(\overline{\overline{\varphi}})$, $\overline{\varphi},\overline{\overline{\varphi}}\in\mathbb{T}^m$, have a non-empty intersection. This also holds in the infinite-dimensional case. Namely, we have the following result. Theorem 3.4. Let $G$ be a hyperbolic diffeomorphism in the class $\operatorname{Diff}(\mathbb{T}^{\infty})$. Then for any points $x,y\in\mathbb{T}^{\infty}$ Proof. As usual, we start by establishing the property analogous to (3.127) for a global lifting $\overline{G}$ of $G$, that is, we show that
$$
\begin{equation}
\forall\,\overline{\varphi},\overline{\overline{\varphi}}\in E\quad \overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm\, s}(\overline{\overline{\varphi}})\ne\varnothing.
\end{equation}
\tag{3.128}
$$
In turn, we can show that the above intersection is non-empty following the same scheme as in the finite-dimensional case (see [4]). Namely, we fix some $\overline{\overline{\varphi}}\in E$ and verify that
Note that equality (3.129) solves the problem of a non-empty intersection (3.128): if it holds, then for each $\overline{\varphi}\in E$ there exists $\psi_0\in\overline{\mathcal{F}}^{\rm\, s}(\overline{\overline{\varphi}})$ such that $\overline{\mathcal{F}}^{\rm\, u}(\psi_0)= \overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})$ and $\psi_0\in\overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})\cap \overline{\mathcal{F}}^{\rm\, s}(\overline{\overline{\varphi}})$. We see that justifying (3.128) reduces to the proof of (3.129). We will establish the latter equality once we will have shown that, together with any point $\overline{\varphi}$, the set $\widetilde{E}$ in (3.129) contains an entire open ball $O(\overline{\varphi},r)\subset E$ of radius $r>0$ independent of $\overline{\varphi}$. To prove this we consider the special neighbourhoods $H_{\delta}(\varphi_0)$, which for each $\varphi_0\in E$ are defined by
$$
\begin{equation}
H_{\delta}(\varphi_0)= \bigcup_{\psi\in \overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi_0)} \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi).
\end{equation}
\tag{3.130}
$$
We show that such a neighbourhood must contain a ball $O(\varphi_0,r)\subset E$ of radius $r=\operatorname{const}>0$ independent of $\varphi_0$.
Taking account of (3.11), (3.12), and the fact that
$$
\begin{equation}
\Phi_*(u,\varphi_0)=o(u),\quad u\to 0,\quad\text{and}\quad \Psi_*(v,\varphi_0)=o(v),\quad v\to 0,
\end{equation}
\tag{3.131}
$$
are uniformly infinitesimal quantities independently of $\varphi_0$, from (3.130) we go over to a simplified set $\widetilde{H}_{\delta}(\varphi_0)$ by dropping the corresponding nonlinearities in the formulae for $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$ and $\overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi_0)$. We can write this set in the following form:
$$
\begin{equation}
\widetilde{H}_{\delta}(\varphi_0)=\bigl\{\varphi=\varphi_0+u+v\colon u\in\overline{E}^{\rm\,u}_{\varphi_0+v},\ \|u\|_{\varphi_0+v}\leqslant\delta,\ v\in\overline{E}^{\rm\,s}_{\varphi_0},\ \|v\|_{\varphi_0}\leqslant\delta\bigr\}.
\end{equation}
\tag{3.132}
$$
It is also clear that if we show that $O(\varphi_0,r)\subset\widetilde{H}_{\delta}(\varphi_0)$ for some $r>0$, then, since $\Phi_*(u,\varphi_0)$ and $\Psi_*(v,\varphi_0)$ are small (see (3.131)), we also obtain a similar result for the original set (3.130).
It follows from (3.132) that
$$
\begin{equation}
\overline{P}_{\varphi_0}(\varphi-\varphi_0)=\overline{P}_{\varphi_0}u=u+ (\overline{P}_{\varphi_0}-\overline{P}_{\varphi_0+v})u.
\end{equation}
\tag{3.133}
$$
Now, to express $u$ from (3.133) we verify that the operator $I+(\overline{P}_{\varphi_0}-\overline{P}_{\varphi_0+v})$ is invertible in $E$. However, this holds because
$$
\begin{equation}
\begin{gathered} \, \nonumber \|\overline{P}_{\varphi_0}-\overline{P}_{\varphi_0+v}\|_{E\to E}\leqslant \overline{\theta}_1(\delta)\overset{\rm def}{=} \sup_{\substack{\varphi_0\in E,\,v\in\overline{E}^{\rm\,s}_{\varphi_0}:\\ \|v\|_{\varphi_0}\leqslant\delta}}\|\overline{P}_{\varphi_0}- \overline{P}_{\varphi_0+v}\|_{E\to E}\to 0,\qquad \delta\to 0, \\ \bigl\|(I+(\overline{P}_{\varphi_0}- \overline{P}_{\varphi_0+v}))^{-1}\bigr\|_{E\to E}\leqslant \frac{1}{1-\overline{\theta}_1(\delta)}\,. \end{gathered}
\end{equation}
\tag{3.134}
$$
Combining (3.133) and (3.134) and assuming a priori that $\|\varphi-\varphi_0\|<r$, we arrive at the chain of inequalities
$$
\begin{equation}
\begin{aligned} \, \nonumber \|u\|_{\varphi_0+v}&\leqslant c\|u\|=c\bigl\|(I+(\overline{P}_{\varphi_0}- \overline{P}_{\varphi_0+v}))^{-1} \overline{P}_{\varphi_0}(\varphi-\varphi_0)\bigr\| \\ \nonumber &\leqslant\frac{c}{1-\overline{\theta}_1(\delta)}\cdot \sup_{\varphi_0\in E}\|\overline{P}_{\varphi_0}\|_{E\to E}\cdot \|\varphi-\varphi_0\| \\ &<\frac{c}{1-\overline{\theta}_1(\delta)}\cdot \sup_{\varphi_0\in E}\|\overline{P}_{\varphi_0}\|_{E\to E}\cdot r, \end{aligned}
\end{equation}
\tag{3.135}
$$
where $c$ is the constant in (3.3) and (3.4). In a similar way, starting from the formula
$$
\begin{equation*}
\overline{Q}_{\varphi_0+v}(\varphi-\varphi_0)=\overline{Q}_{\varphi_0+v}v=v+ (\overline{Q}_{\varphi_0+v}-\overline{Q}_{\varphi_0})v
\end{equation*}
\notag
$$
we obtain in succession
$$
\begin{equation}
\begin{gathered} \, \nonumber \|\overline{Q}_{\varphi_0+v}-\overline{Q}_{\varphi_0}\|_{E\to E}\leqslant \overline{\theta}_2(\delta)\overset{\rm def}{=} \sup_{\substack{\varphi_0\in E,\,v\in\overline{E}^{\rm\,s}_{\varphi_0}:\\ \|v\|_{\varphi_0}\leqslant\delta}}\|\overline{Q}_{\varphi_0}- \overline{Q}_{\varphi_0+v}\|_{E\to E}, \to 0,\qquad \delta\to 0, \\ \nonumber \bigl\|(I+(\overline{Q}_{\varphi_0}- \overline{Q}_{\varphi_0+v}))^{-1}\bigr\|_{E\to E}\leqslant \frac{1}{1-\overline{\theta}_2(\delta)}\,, \\ \|v\|_{\varphi_0}<\frac{c}{1-\overline{\theta}_2(\delta)}\cdot \sup_{\varphi_0\in E}\|\overline{Q}_{\varphi_0}\|_{E\to E}\cdot r. \end{gathered}
\end{equation}
\tag{3.136}
$$
It remains to observe that, in view of (3.135) and (3.136), we certainly have the required inclusion $O(\varphi_0,r)\subset\widetilde{H}_{\delta}(\varphi_0)$ if $r$ satisfies the conditions
$$
\begin{equation*}
\frac{c}{1-\overline{\theta}_1(\delta)}\cdot\sup_{\varphi_0\in E} \|\overline{P}_{\varphi_0}\|_{E\to E}\cdot r<\delta \quad\text{and}\quad \frac{c}{1-\overline{\theta}_2(\delta)}\cdot \sup_{\varphi_0\in E}\|\overline{Q}_{\varphi_0}\|_{E\to E}\cdot r<\delta.
\end{equation*}
\notag
$$
Thus we have shown the existence of $r>0$, such that each special neighbourhood (3.130) contains a ball $O(\varphi_0, r)\subset E$. We can verify that $\widetilde{E}=E$ on this basis by taking a certain geometric approach. The idea of this approach is as follows. Fix some point $\overline{\varphi}\in\widetilde{E}$. By the formula for $\widetilde{E}$ (see (3.129)) there exists $\psi_0\in\overline{\mathcal{F}}^{\rm\, s}(\overline{\overline{\varphi}})$ such that $\overline{\mathcal{F}}^{\rm\, u}(\psi_0)= \overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})$. Now we consider an arbitrary continuous curve $\varphi(t)$, $0\leqslant t\leqslant 1$, on the leaf $\overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})$ such that $\varphi(0)=\psi_0$ and $\varphi(1)=\overline{\varphi}$. Finally, we introduce the continuous family of special neighbourhoods
$$
\begin{equation}
H_{\delta}(\varphi(t)),\qquad 0\leqslant t\leqslant 1.
\end{equation}
\tag{3.137}
$$
It follows from the explicit form of $H_{\delta}(\varphi_0)$ (see (3.130)) and the obvious equality $\overline{\mathcal{F}}^{\rm\, s}(\psi_0)= \overline{\mathcal{F}}^{\rm\, s}(\overline{\overline{\varphi}})$ that $H_{\delta}(\varphi(0))\subset\widetilde{E}$. As
$$
\begin{equation}
\varphi(t)\in \overline{\mathcal{F}}^{\rm\, u}(\psi_0)= \overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi}),
\end{equation}
\tag{3.138}
$$
formula (3.137) provides a rule for extending continuously, for each $t\in (0,1]$, the family of manifolds $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$, $\psi\in\overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi(0))$, to the family $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$, $\psi\in\overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi(t))$. In other words, for each manifold $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$ in the first family there exists a chain of the form (3.114) that connects it with some local manifold in the second family, and vice versa.
In fact, consider a sufficiently fine partition $0=t_0<t_1<\cdots<t_m=1$ of the interval $0\leqslant t\leqslant 1$ such that
$$
\begin{equation}
\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\varphi(t_k))\cap \operatorname{int}\overline{\mathcal{F}}^{\rm\, u}_{\delta} (\varphi(t_{k+1}))\ne\varnothing,\qquad k=0,1,\dots,m-1
\end{equation}
\tag{3.139}
$$
(here $\mathrm{int}$ means the same as in Corollary 3.1.2; the existence of such a partition is ensured by (3.138)). Now fix an arbitrary local manifold $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$, $\psi\in\overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi(t_0))$, and note that, as $\Phi_*(u,\varphi_0)$ in (3.11) is uniformly continuous, it is close to $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\varphi(t_0))$ (in the Hausdorff metric). Hence it follows from (3.139) that $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)\cap H_{\delta}(\varphi(t_1))$ is non-empty. However, by Corollary 3.1.2 the manifold $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)$ can intersect only one unstable local manifold coming into $H_{\delta}(\varphi(t_1))$. Thus there exists a unique $\psi_1\in\overline{\mathcal{F}}^{\rm\, s}_{\delta}(\varphi(t_1))$ such that $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi)\cap \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi_1)\ne\varnothing$. In a similar way, by looking at the next pair of values $t=t_1$, $t=t_2$, we find a manifold $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi_2) \subset H_{\delta}(\varphi(t_2))$ such that $\overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi_1)\cap \overline{\mathcal{F}}^{\rm\, u}_{\delta}(\psi_2)\ne\varnothing$, and so on.
It follows from these geometric constructions that $H_{\delta}(\varphi(t))\in \widetilde{E}$ for all $0\leqslant t\leqslant 1$. This means that the family of special neighbourhoods (3.137) makes up a ‘tunnel’ in the set $\widetilde{E}$ that connects the points $\psi_0$ and $\overline{\varphi}$ in $\widetilde{E}$. In particular, approaching the end of the ‘tunnel’, that is, setting $t=1$, we conclude that $H_{\delta}(\overline{\varphi})\subset\widetilde{E}$. Thus it is immediate that $O(\overline{\varphi},r)\subset\widetilde{E}$ and therefore $\widetilde{E}=E$. Summarizing, we note that we have verified (3.129) and so have established (3.128). We must also add that in the case of a finite-dimensional torus $\mathbb{T}^m$, $m\geqslant 2$, the intersection (3.128) consists of one point (see [4]). However, in the infinite- dimensional case we can only prove that this intersection is non-empty. Finally, we verify the required property (3.127). To do this we fix arbitrary points $x,y\in\mathbb{T}^{\infty}$ and select $\overline{\varphi},\overline{\overline{\varphi}}\in E$ such that $x=\operatorname{pr}(\overline{\varphi})$ and $y=\operatorname{pr}(\overline{\overline{\varphi}})$. Then by (3.128) we have
$$
\begin{equation*}
\varnothing\ne\operatorname{pr}[\overline{\mathcal{F}}^{\rm\, u} (\overline{\varphi})\cap\overline{\mathcal{F}}^{\rm\, s} (\overline{\overline{\varphi}})]\subset \operatorname{pr}[\overline{\mathcal{F}}^{\rm\, u}(\overline{\varphi})]\cap \operatorname{pr}[\overline{\mathcal{F}}^{\rm\, s} (\overline{\overline{\varphi}})]= \mathcal{F}^{\rm\, u}(x)\cap\mathcal{F}^{\rm\, s}(y).
\end{equation*}
\notag
$$
Theorem 3.4 is proved. ConclusionIn this paper we have endeavoured to give a systematic presentation of the foundations of the hyperbolic theory on an infinite-dimensional torus. In this connection we have thoroughly discussed the definitions of an integer lattice, an integer torus, a tangent space, liftings (both local and global ones), a differential, a diffeomorphism, and so on. We have also introduced and investigated the special class $\operatorname{Diff}(\mathbb{T}^{\infty})$ of diffeomorphisms and have managed to establish a number of foundational results, including a criterion of hyperbolicity, the $C^1$-roughness of the property of hyperbolicity, the Hadamard–Perron theorem, and a result on the existence of a stable and an unstable invariant foliation. We must also stress that these results have been established for hyperbolic diffeomorphisms of a torus $\mathbb{T}^{\infty}$ under some minimal additional assumptions (1.21). This is particularly important because in the infinite-dimensional case even some simplest properties of hyperbolic diffeomorphisms, such as the boundedness away from zero of the angle between a stable and an unstable subspace, uniform boundedness of projections, and so on, do not hold automatically: they must be proved, and the corresponding proofs are significantly based on the assumptions (1.21) involved in the definition of the class $\operatorname{Diff}(\mathbb{T}^{\infty})$. Without these assumptions we cannot establish such results. In this connection it is natural to ask the following question: can we extend the class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ so as to preserve for it all results mentioned above. We do not know the answer yet. However, whatever it can be, one should bear in mind that there can also exist classes of diffeomorphisms other than $\operatorname{Diff}(\mathbb{T}^{\infty})$ for which similar results also hold. In other words, various versions of hyperbolic theory on the infinite-dimensional torus $\mathbb{T}^{\infty}$ are possible. One such version was presented in [21]. Another block of unsolved problems is related to the topological conjugacy of an arbitrary hyperbolic diffeomorphism $G$ in $\operatorname{Diff}(\mathbb{T}^{\infty})$ and its linear part, that is, the linear diffeomorphism (2.5). By contrast to the finite-dimensional torus $\mathbb{T}^{m}$, $m\geqslant 2$, where such a conjugacy is known [4], we cannot establish such a result for $\mathbb{T}^{\infty}$ It is not even clear whether the fact that a diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ is hyperbolic ensures that the corresponding linear automorphism (2.5) is too. The related question of the structure stability of hyperbolic diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ is also open so far. It is interesting to note that, nevertheless, these problems of topological conjugacy and structural stability have a partial solution. More precisely, they can be solved for a class $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$, which is more narrow than $\operatorname{Diff}(\mathbb{T}^{\infty})$. We let $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ consist of the diffeomorphisms $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ satisfying the following additional conditions. First, the linear operator $\Lambda$ involved in (1.24) gives rise to a hyperbolic automorphism (2.5). Second, the vector function $g(\varphi)$ in (1.24) satisfies the assumptions of Theorem 1.3. Namely, we assume that it is bounded (1.55) and the operator (1.56) is completely continuous in $E$. Third, Conditions 2.1–2.3 are satisfied for $n_0=1$, $E_1(\varphi)=E_1$, and $E_2(\varphi)=E_2$, where $E_1$ and $E_2$ are the subspaces in the decomposition (2.7) corresponding to the operator (2.5). We stress that by Theorem 2.1 each diffeomorphism in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ is hyperbolic. Theorem 1. Each diffeomorphism $G$ in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ is topologically conjugate to its linear part (2.5). Under certain weaker assumptions, but only in the special case of $\mathbb{T}^{\infty}=\ell_{\infty}/2\pi\mathbb{Z}^{\infty}$, where $\mathbb{Z}^{\infty}$ is the integer lattice (1.5), the proof of this theorem was given in [21]. However, the constructions in [21] are also valid for an arbitrary torus $\mathbb{T}^{\infty}$. It turns out that for diffeomorphisms in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ we also have structural stability. Before formulating the corresponding result we refine the relevant definitions. We consider the Banach space $\mathscr{H}^1_{\rm per}(E)$ of vector functions $g(\varphi)\in C^1_{\rm per}(E)$ such that the operators (1.56) are completely continuous. We define the norm $\|g\|_{\mathscr{H}^1_{\rm per}}$ by (2.42) as before. Definition 1. We say that a diffeomorphism (1.24) in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ is $C^1$-structurally stable if there exists $\varepsilon>0$ such that for each function $\Delta(\varphi)\in\mathscr{H}^1_{\rm per}(E)$, $\|\Delta\|_{\mathscr{H}^1_{\rm per}}<\varepsilon$, the corresponding perturbed map is topologically conjugate to the unperturbed map $G$. Definition 2. A diffeomorphism (1.24) in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ is said to be $C^1$-strongly structurally stable if, first, it is $C^1$-structurally stable, and, second, there exists $\varepsilon>0$ such that for all $\Delta(\varphi)\in\mathscr{H}^1_{\rm per}(E)$, $\|\Delta\|_{\mathscr{H}^1_{\rm per}}<\varepsilon$, there is a family of diffeomorphisms $\tau_{\Delta}\colon\mathbb{T}^{\infty}\to \mathbb{T}^{\infty}$ with the following properties: Here $\operatorname{id}$ is the identity operator on $\mathbb{T}^{\infty}$ and the convergence is uniform. Drawing on the results in [21] again, we arrive at the following. Theorem 2. Each diffeomorphism $G$ in $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ has the property of $C^1$-strong structural stability. We add in conclusion that the class $\widetilde{\operatorname{Diff}}(\mathbb{T}^{\infty})$ certainly contains the example (2.53)–(2.56) considered above, provided that (2.96) is satisfied. 
Citation in format AMSBIB
\Bibitem{GlyKol22} |
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