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On the measure of the KAM-tori in a neighbourhood of a separatrix A. G. Medvedev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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     § 1. IntroductionConsider a Hamiltonian system $(M,\omega,\mathcal{H}_0)$, where $(M,\omega)$ is a real analytic symplectic manifold of dimension $2n$ (the phase space) and $\mathcal{H}_0$ is a real analytic Hamiltonian function. Assume that the system is Liouville integrable. Then each compact nonsingular common level of the first integrals is a finite union of invariant Lagrangian $n$-tori, and the motion along these tori is quasiperiodic. A foliation by such tori arises in the phase space $M$, but this foliation virtually always has singularities. Consider an invariant domain $D \subset M$ such that the foliation by tori has no singularities in $D$. Then there exists a coordinate system $(I,\varphi)$ in $D$, where ${I \in B \subset \mathbb{R}^n}$ and $\varphi \in \mathbb{T}^n=\mathbb{R}^n / 2 \pi \mathbb{Z}^n$ (action-angle variable), such that $\omega=\sum_{j=1}^n dI_j \wedge d\varphi_j$ and the function $\mathcal{H}_0$ depends only on the action variables: $\mathcal{H}_0=\mathcal{H}_0(I)$. Thus, the invariant $n$-tori look like
$$
\begin{equation*}
\mathbb{T}_{I^0}^n=\{(I,\varphi)\colon I=I^0 \in B\}.
\end{equation*}
\notag
$$
On such a torus Hamilton’s equations are as follows:
$$
\begin{equation*}
\dot \varphi=\nu(I^0), \qquad \nu=\mathcal{H}'_{0I}(I^0).
\end{equation*}
\notag
$$
The vector $\nu$ is called the frequency vector. We perturb the system by adding a small real analytic perturbation to $\mathcal{H}_0$:
$$
\begin{equation}
\mathcal{H}=\mathcal{H}_0(I)+\varepsilon \widetilde{\mathcal{H}}(I,\varphi,\varepsilon), \qquad \varepsilon \ll 1.
\end{equation}
\tag{1.1}
$$
Below it will be convenient to assume that $\varepsilon \geqslant 0$. Theorem 1 ([1]–[3]). Assume that the closure $\overline{D}$ is compact and the unperturbed system is nondegenerate: Then most invariant $n$-tori $\mathbb{T}^n_{I^0}$ do not disappear after the perturbation, but, slightly deformed, persist in the system $(M,\omega,\mathcal{H})$. Let $C \subset M$ be the set of points not lying on invariant $n$-tori of the system $(D,\omega,\mathcal{H})$. Theorem 2 ([4]–[7]). Assume that $\overline{D}$ is compact and Then the measure of the set $C \cap D$ is at most $c_3 \sqrt \varepsilon$, where the constant $c_3$ depends on $c_1$ and $c_2$. Only nonresonant tori survive the perturbation, namely, ones on which the unperturbed frequencies satisfy $\langle \nu, k \rangle \neq 0$ for each $k \in \mathbb{Z}^n \setminus \{0\}$. Here $\langle \,\cdot\,{,}\,\cdot\, \rangle$ is the standard scalar product. In fact, we require that the frequencies satisfy certain more stringent Diophantine conditions, but we discuss these technical aspects in what follows. The unperturbed resonant tori accumulate towards the resonance surfaces
$$
\begin{equation*}
R_k=\{(I,\varphi) \in D \times \mathbb{T}^n\colon \langle \nu(I),k\rangle=0\}, \qquad k \in \mathbb{Z}^n \setminus \{0\}.
\end{equation*}
\notag
$$
In small neighbourhoods of the surfaces $R_k$ so-called secondary invariant $n$-tori arise after perturbations. Their characteristic feature is their nonexistence for $\varepsilon=0$. However, using the methods of the KAM-theory (developed by Kolmogorov, Arnold and Moser) not only the existence of such tori can be established, but also their measure in the phase space $(M,\omega,\mathcal{H})$ of the system can be estimated: see [8] and [9]. Let $C_* \subset C$ be the set of points obtained by the removal of the points on secondary tori from $C$. Conjecture 1 ([10]). Under certain natural assumptions the measure of the set $C_* \cap D$ has the estimate $c_4 \varepsilon$, where $c_4$ is a positive constant independent of $\varepsilon$. A weak version of this conjecture for systems with Hamiltonian $\mathcal{H}_0=\frac{1}{2}|I|^2+ f(\varphi)$ was proved in [8] and [11]. Theorem 3 (Biasco and Chierchia). Under certain natural assumptions the set $C_* \cap D$ has measure at most $c_5 \varepsilon |{\ln \varepsilon}|^a$, where $a$ and $c_5$ are positive constants independent of $\varepsilon$. In this paper we consider the so-called a priori unstable case, when in the domain $D \subset M$ under consideration the foliation of the phase space by invariant Lagrangian $n$-tori is degenerate on a $(2n-1)$-dimensional singular submanifold $\mathbb{W}$ formed by the asymptotic manifolds of hyperbolic $(n-1)$-tori. We consider the case when $D=D' \times \mathbb{T}^{n-1} \times D''$, where $D' \subset \mathbb{R}^{n-1}$ and $D'' \subset \mathbb{R}^2$, and coordinates $(y,x,p,q)$ on $D$ are chosen so that $y \in D'$, $x \in \mathbb{T}^{n-1}$, $(p,q) \in D''$,
$$
\begin{equation}
\mathcal{H}_0=\mathcal{H}_0(y,p,q)\quad\text{and} \quad \omega=dy \wedge dx+dp \wedge dq.
\end{equation}
\tag{1.2}
$$
We assume that $\mathcal{H}_0$ is a real analytic function. The system $(D,\omega,\mathcal{H}_0)$ is Liouville integrable: as a full system of first integrals we can take $y_1, \dots, y_{n-1}$ and $ \mathcal{H}_0$ (of course, provided that $\mathcal{H}_0$ depends on $(p,q)$ in a nontrivial way). Since the $y$-variables do not change during the motion, it is useful to look at the family of systems $D'', dp \wedge dq, \mathcal{H}_0)$ with one degree of freedom, in which $y$ is a parameter. Assume that for each $y \in D'$ the function $\mathcal{H}_0$ has a nondegenerate saddle critical point $(p^0(y),q^0(y))$, where $p^0$ and $q^0$ are real analytic functions. Below we assume without loss of generality that $p^0= q^0=0$ because this can easily be attained by means of a canonical change of variables on $D$. Thus, the foliation of $D$ by invariant $n$-tori is degenerate on the singular submanifold
$$
\begin{equation*}
\mathbb{W}=\{(y,x,p,q) \in D\colon \mathcal{H}_0(y,p,q)=\mathcal{H}_0(y,0,0)\}.
\end{equation*}
\notag
$$
We assume in what follows that the foliation in $D$ has no other singularities, and the nondegeneracy condition $\det \mathcal{H}_0(y,0,0) \neq 0$ holds in $\overline{D'}$. We also assume that $\mathbb{W}$ is connected. This can always be achieved by considering the connected component containing the points on This manifold $\mathbf T$ is foliated by the hyperbolic $(n-1)$-tori
$$
\begin{equation*}
{\mathbb{T}}^{n-1}_{y^0}=\{(y,x,p,q) \in \mathbb{W}\colon p=q=0, \,y=y^0\}.
\end{equation*}
\notag
$$
At points in $\mathbf T$, $\mathbb{W}$ has a singularity of the type of self-intersection. Points in $\mathbb{W} \setminus \mathbf T$ lie on asymptotic manifolds of the hyperbolic tori ${\mathbb{T}}^{n-1}_{y^0}$. Similar systems also arise in the problem of evaluating the measure of the secondary tori. In fact, in a neighbourhood of a resonance surface $R_k$ the function $\mathcal{H}$ can be reduced to the form
$$
\begin{equation*}
\mathcal{H}=\Lambda(y)+\varepsilon \biggl(\frac{1}{2}a(y) p^2+u(y,q)\biggr) +\varepsilon \widetilde{\mathcal{H}}(y,x,p,q), \qquad \omega=dy \wedge dx+\sqrt{\varepsilon}\, dp \wedge dq,
\end{equation*}
\notag
$$
where $a$ and $\Lambda$ are some functions and $\frac{1}{2}a(y) p^2+u(y,q)$ is a ‘pendulum’ with potential $u$ depending periodically on $q \in \mathbb{T}^1$. This potential $u$ has a local maximum at $\overline{u}=\overline{u}(y)$, so for each $y$ the pendulum has a hyperbolic equilibrium. Then
$$
\begin{equation*}
\mathbb{W}=\biggl\{(y,x,p,q) \in D\colon \frac{1}{2}a(y) p^2+u(y,q) =\overline{u}(y)\biggr\}.
\end{equation*}
\notag
$$
Now we state our main result. Theorem 4. There exists a neighbourhood $U \subset D$ of the submanifold $\mathbb{W}$ such that the measure of the set $C \cap U$ is at most $c \sqrt \varepsilon$, where the constant $c$ and the neighbourhood $U$ are determined by the function $\mathcal{H}_0$ and the dimension $n$. The neighbourhood $U$ is independent of $\varepsilon$. In the domain $D \setminus U$, provided that the unperturbed system is nondegenerate, we can use Theorem 2. Thus, Theorem 4 means that the upper estimate of the measure of $C \cap D$ as a quantity of order $\sqrt{\varepsilon}$ also holds in the presence of a singular manifold $\mathbb{W}$. § 2. Angle-action variables in a neighbourhood of $ {\mathbb{W}}$For each $y \in D'$ the system $(D'', dp \wedge dq, \mathcal{H}_0)$ has one hyperbolic point. Under the assumption that the common level sets of the first integrals are compact we conclude that the asymptotic curves coming out of a hyperbolic point form two homoclinic loops $\Gamma_1$ and $\Gamma_2$, which partition $D''$ into three connected components: $D''=D''_1 \cup D''_2 \cup D''_3$. Then $D=D_1 \cup D_2 \cup D_3$, where
$$
\begin{equation*}
D_i=\bigcup_{y \in D'} \{y\} \times \mathbb{T}^{n-1} \times D''_i(y), \qquad i=1,2,3.
\end{equation*}
\notag
$$
The codimension-one submanifold $\mathbb{W}$ is a direct product of the union of these loops by the torus $\mathbb{T}^{n-1}$. It partitions $D$ into connected components $D_1$, $D_2$ and $D_3$. We can reduce $\mathcal{H}_0$ to a normal form in a neighbourhood of $\mathbf T$. Lemma 1. There exists a neighbourhood $U_{\mathbf T} \subset D$ of $\mathbf T$ and a canonical change of coordinates $(y,x,p,q) \mapsto (\widehat y, \widehat x, \widehat p, \widehat q)$ in this neighbourhood such that (1) $y=\widehat y$, (2) $\mathbf T=\{(\widehat y,\widehat x,\widehat p,\widehat q) \in \mathbb{W}\colon \widehat p=\widehat q=0\}$, (3) in $U_{\mathbf T}$ $\mathcal{H}_0$ has the form Proof. Consider the system $(D'', dp \wedge dq, \mathcal{H}_0(y_0,p,q))$, where we treat $y_0\,{\in}\, D'$ as a parameter. By a theorem of Moser (see [12]; the real analytic dependence on a multidimensional parameter $y$ must be added to Moser’s proof) there exists a neighbourhood $\widehat U_{y_0}$ of the hyperbolic equilibrium $p=0$, $q=0$ in which we can find a canonical change of variables $(p,q) \mapsto (\widehat p,\widehat q)$ specified by a generating function $\widehat s=\widehat s(y_0, \widehat p, q)$ such that in the new coordinates the Hamiltonian $\mathcal{H}_0$ is reduced to the normal form: Let $U_{\mathbf T}=\bigcup_{y \in D'} D' \times \mathbb{T}^{n-1} \times \widehat U_{y}$. The required change of variables is defined by the generating function The function $\alpha$ is real analytic. Because the hyperbolic equilibrium is nondegenerate, $\alpha$ is distinct from zero. The proof is complete. We choose the domain $D_i$ in which $\widehat p>0$ and $\widehat q>0$. Assume that this is the domain $D_1$ and it is bounded by a homoclinic loop $\Gamma_1$ Figure 1. In Theorem 4 we replace $D$ by $D_1$ and prove the theorem for it. In the other domains the proof is similar. Now we introduce action-angle variables. Lemma 2. In a neighbourhood $U_{\mathbb{W}} \,{\subset}\, D_1$ of the submanifold $\mathbb{W}$ there exists a canonical real-analytic change of variables $( y, x, p, q) \mapsto (\widetilde y,\widetilde x,\widetilde I,\widetilde \varphi)$ such that the Hamiltonian $\mathcal{H}_0$ has the following form in the new coordinates: where $I_0>0$. Furthermore, in the new variables $U_{\mathbb{W}}=D' \times (0,I_0) \times \mathbb{T}^{n}$. The variables $(\widetilde I,\widetilde \varphi)$ are action-angle variables for $\mathcal{H}_0(\widetilde y,\cdot\,{,}\,\cdot\,)$. As $\widetilde I \to 0$, the point approaches $\mathbb{W}$. The following formula holds: where $F$ is real analytic in $(0,0,0,\widetilde y)$. Proof. As in the proof of Lemma 1, consider the system $(D'', d p \wedge d q, \mathcal{H}_0( y_0, p, q))$ depending on the parameter $ y_0 \in D'$. It was proved in [13], Ch. 9, as well as in [14] and [8], that in a neighbourhood $\widetilde U_{y_0}$ of the homoclinic loop $\Gamma_1$ there exists a canonical change of variables $(p,q) \mapsto (\widetilde I,\widetilde \varphi)$ such that where $h$ is defined by (2.3). Assume that the change $(p,q) \mapsto (\widetilde I,\widetilde \varphi)$ is specified by a generating function $\widetilde s(y_0,\widetilde I,q)$. Let $U_{\mathbb{W}}= \bigcup_{y \in D'} D' \times \mathbb{T}^{n-1} \times \widetilde U_{y}$; then the required change of variables is specified by the generating function The proof is complete. For compactness we drop tildes over the variables $( y, x, I, \varphi)$. Consider the complex $a$-neighbourhood of the domain $ D' \times (0,I_0)$, where $a \in \mathbb{R}$, $a> 0$:
$$
\begin{equation*}
\begin{aligned} \, U_{a}(D' \times (0,I_0)) &=\bigl\{(y+\eta,I+\kappa)\colon (y,I) \in D' \times (\sqrt \varepsilon\, a,I_0), \\ &\qquad |\eta|<\sqrt{\varepsilon}\, a, \,|\kappa|<\sqrt{\varepsilon}\, a,\, \eta \in \mathbb{C}^{n-1}, \kappa \in \mathbb{C}\bigr\} \end{aligned}
\end{equation*}
\notag
$$
and the $b$-neighbourhood of $\mathbb{T}^{n}$:
$$
\begin{equation*}
\mathbb{T}^{n}_b=\bigl\{(x+\zeta,\varphi+\xi)\colon (x,\varphi) \in \mathbb{T}^{n},\,|\zeta|<b,\, |\xi|<b,\, \zeta \in \mathbb{C}^{n-1},\, \xi \in \mathbb{C}\bigr\}.
\end{equation*}
\notag
$$
Consider a small real-analytic perturbation of the function $\mathcal{H}_0$:
$$
\begin{equation}
\mathcal{H}=\Lambda(y)+h(y,I)+\varepsilon \widetilde{\mathcal{H}}(y,x,I,\varphi).
\end{equation}
\tag{2.4}
$$
For some $\widetilde a>0$ and $\widetilde b>0$ the function $\widetilde{\mathcal{H}}$ is real analytic in $U_{\widetilde a}(D' \times (0,I_0)) \times \mathbb{T}^{n}_{\widetilde b}$ because so is the composition of changes of variables, the nondegeneracy conditions
$$
\begin{equation}
\underline{c}_{\,\Lambda}<|{\det \Lambda''_{yy}}|<\overline{c}_{\Lambda}, \qquad \underline{c}_{\,\alpha}<|\alpha|<\overline{c}_{\alpha},
\end{equation}
\tag{2.5}
$$
$$
\begin{equation}
|\Lambda|<c_{\Lambda}, \qquad |\Lambda'_y|<c_{\Lambda}, \qquad |\Lambda''_{yy}|<c_{\Lambda}, \qquad |\alpha'_{y}|<c_{\alpha}\quad\text{and} \quad |\alpha''_{yy}|<c_{\alpha}
\end{equation}
\tag{2.6}
$$
are satisfied, and the following estimate holds: where $\widetilde s$, $\underline{c}_{\,\Lambda}$, $\overline{c}_{\Lambda}$, $\underline{c}_{\,\alpha}$, $\overline{c}_{\alpha}$, $c_{\Lambda}$ and $c_{\alpha}$ are positive constants. As the norm of a matrix or a vector we take the greatest modulus of its entries. Let $\delta$ be a positive number such that $\delta^2<I_0$ and $\delta<1$. Consider the unions of intervals
$$
\begin{equation*}
\bigcup_{m=1}^{+\infty} [\delta^{2(m+1)},\delta^{2m}] \subset (0,I_0)
\end{equation*}
\notag
$$
and of the corresponding domains $\Delta^{(m)}=D' \times [\delta^{2(m+1)},\delta^{2m}]$ and $\mathbb{D}^{(m)}=\Delta^{(m)} \times \mathbb{T}^n$. The plan of the proof of Theorem 4 is as follows. We consider a finite set of domains $\mathbb{D}^{(m)}$, $m=1, \dots, [\frac{1}{4} \log_{\delta} \varepsilon]$, and then step away from $\mathbb{W}$ at a distance of order $\sqrt \varepsilon$, estimate the measure of the complement to tori $C \cap \mathbb{D}^{(m)}$ and show that the total measure is of order $\sqrt{\varepsilon}$. Theorem 5. There exist $\delta>0$, a small $\varepsilon_0>0$ and a positive constant $c_{\mathbb{D}}$ such that for all $0<\varepsilon<\varepsilon_0$ and all positive integers $m \leqslant [\frac{1}{4} \log_{\delta} \varepsilon]$, for the Hamiltonian system $(\mathbb{D}^{(m)}, \omega, \mathcal{H})$ the set $C \cap \mathbb{D}^{(m)}$ has measure at most $c_{\mathbb{D}} \sqrt{\varepsilon}\, m \delta^m$. We set $U=\bigcup_{m=1}^{+\infty} \mathbb{D}^{(m)}$. Then we deduce Theorem 4 from the inequality $\sum_{m=1}^{+\infty} m \delta^{m}<+\infty$ and Theorem 5. The parameter $\delta$ determines the width of the neighbourhood $U$. It must be taken sufficiently small so that certain nondegeneracy conditions, which we discuss in the next subsection, hold in $U$. 2.1. The KAM procedureBelow we assume by default that $m \leqslant [\frac{1}{4} \log_{\delta} \varepsilon]$. For $a \in \mathbb{R}$, $a > 0$, we consider the complex $a$-neighbourhood of the domain $\Delta^{(m)}$:
$$
\begin{equation*}
\Delta^{(m)}_{a} =\bigl \{(y+\eta,I+\kappa)\colon (y,I) \in \Delta^{(m)},\,|\eta|< \sqrt{\varepsilon}\, a, \,|\kappa|<\sqrt{\varepsilon}\, m \delta^m a, \,\eta \in \mathbb{C}^n, \, \kappa \in \mathbb{C} \bigr\}.
\end{equation*}
\notag
$$
For functions $f\colon\Delta^{(m)} \mapsto \mathbb{R}$, $g\colon\mathbb{T}^{n} \mapsto \mathbb{R}$ and $u\colon\Delta^{(m)} \times \mathbb{T}^{n} \mapsto \mathbb{R}$ that have real analytic extensions to the neighbourhoods $\Delta^{(m)}_{a}$, $\mathbb{T}^{n}_b$ and $\Delta^{(m)}_{a} \times \mathbb{T}^{n}_b$, $a>0$, $b>0$, we define their norms by
$$
\begin{equation}
|f|_a=\sup_{z \in \Delta^{(m)}_{a}} |f(z)|, \qquad |g|_b=\sup_{\phi \in \mathbb{T}^{n}_b} |g(z)|
\end{equation}
\tag{2.8}
$$
and
$$
\begin{equation}
|u|_{a,b}=\sup_{z \in \Delta^{(m)}_{a} \times \mathbb{T}^{n}_b} |u(z)|.
\end{equation}
\tag{2.9}
$$
Lemma 3. There exist positive constants $\delta$, $\underline{c}_{\,h}$, $\overline{c}_h$ and $ \varepsilon_1 $ such that for all ${\varepsilon \in (0,\varepsilon_1)}$ the following inequalities hold in $\Delta^{(m)}_{\widetilde a}$: For the proof, see § 3.1. Set
$$
\begin{equation}
J=\begin{pmatrix} \Lambda''_{yy}+h''_{yy}+\widehat h''_{yy} & h''_{Iy}+\widehat h''_{Iy}\\ h''_{yI}+\widehat h_{yI} & h''_{II}+\widehat h_{II} \end{pmatrix}.
\end{equation}
\tag{2.13}
$$
Lemma 4. There exist positive constants $\delta,\underline{c}_{\,J}, \overline{c}_J, \widehat c$ and $ \varepsilon_1 $ such that, first, the estimates from Lemma 3 hold and, second, for each function $\widehat h=\widehat h (y,I)$ with derivatives satisfying the inequalities the following estimates hold: For the proof, see § 3.1. Let $\sigma>0$, $\sigma<a$ and $\widetilde \sigma>0$, $\widetilde \sigma<b$. Then the derivatives of the functions $f$ and $g$ in $\Delta^{(m)}_{a-\sigma}$ and $\mathbb{T}^{n}_{b- \widetilde \sigma}$, respectively, satisfy the Cauchy estimates
$$
\begin{equation}
\biggl|\frac{ d f} {dz}\biggr|_{a-\sigma} \leqslant \frac{|f|_a}{\sigma}\quad\text{and} \quad \biggl|\frac{ d g} {d \phi}\biggr|_{b-\widetilde \sigma} \leqslant \frac{|g|_b}{\widetilde \sigma}.
\end{equation}
\tag{2.16}
$$
Let $g$ be expressed by a Fourier series:
$$
\begin{equation*}
g(\phi)=\sum_{K \in \mathbb{Z}^{n}} g^{K} e^{i\langle K, \phi \rangle}.
\end{equation*}
\notag
$$
Then its Fourier coefficients satisfy the estimates For $N \in \mathbb{N}$ we define the $N$-projection and the mean value $\langle \cdot \rangle_{z}$ by
$$
\begin{equation}
\Pi_N g(\phi)=\sum_{0<|K| \leqslant N} g^{K} e^{i\langle K, \phi \rangle}\quad\text{and} \quad \langle g \rangle_{\phi}=g^{0}.
\end{equation}
\tag{2.18}
$$
For the proofs of estimates (2.16) and (2.17) and Lemma 5, see [9]. We fix some $m$ and go over to the construction of the KAM-procedure. We set
$$
\begin{equation}
H_0=\mathcal{H}, \qquad \widehat h_0=\langle \widetilde{\mathcal{H}} \rangle_{x,\varphi}, \qquad \widehat H_{0}=\widetilde{\mathcal{H}} -\langle \widetilde{\mathcal{H}}\rangle_{x,\varphi},
\end{equation}
\tag{2.19}
$$
$$
\begin{equation}
a_0=\frac{\widetilde a}4, \qquad b_0=\widetilde b, \qquad s_0=\frac{16 \widetilde s}{\widetilde a^2}, \qquad \Delta_0= \Delta^{(m)}\quad\text{and} \quad \mathbb{D}_0=\mathbb{D}^{(m)},
\end{equation}
\tag{2.20}
$$
and write out the original Hamiltonian (2.4):
$$
\begin{equation}
H_0=\Lambda(y)+h(y,I)+\varepsilon \widehat h_0(y,I)+\varepsilon \widehat H_0(y,x,I,\varphi), \qquad \langle \widehat H_0 \rangle_{x, \varphi}=0.
\end{equation}
\tag{2.21}
$$
It follows from (2.7) and (2.16) that
$$
\begin{equation}
|\varepsilon \widehat h'_{0I}|_{a_0}<\sqrt{\varepsilon} \frac{s_0}{m\delta^m}, \qquad |\varepsilon \widehat h'_{0y}|_{a_0}<\sqrt{\varepsilon} s_0,
\end{equation}
\tag{2.22}
$$
$$
\begin{equation}
|\varepsilon \widehat h''_{0Iy}|_{a_0}<\frac{s_0}{m\delta^m}, \qquad |\varepsilon \widehat h''_{0II}|_{a_0}<\frac{s_0}{m^2 \delta^{2m}}\quad\text{and} \quad |\varepsilon \widehat h''_{0yy}|_{a_0}< s_0.
\end{equation}
\tag{2.23}
$$
We assume that $k \in \mathbb{N} \cup \{0\}$. The KAM procedure is a sequence of changes of variables $\{\Phi_k\}$, a sequence of Hamiltonians $\{H_k\}$, and nested sequences of domains $\{\mathbb{D}_k \}=\{\Delta_k \times \mathbb{T}^{n} \}$ and the corresponding complex neighbourhoods $\{U_{a_{k},b_{k}}(\mathbb{D}_k)\}$ on which these maps and Hamiltonians are defined:
$$
\begin{equation}
\begin{gathered} \, \notag \Phi_k\colon U_{a_{k+1},b_{k+1}}(\mathbb{D}_{k+1}) \to U_{a_{k},b_k}(\mathbb{D}_{k}), \qquad H_{k+1}=H_k \circ \Phi_k, \\ H_{k}=\Lambda(y)+h(y,I)+\varepsilon \widehat h_k (y,I)+\varepsilon \widehat H_{k}(y,x,I,\varphi), \qquad \langle \widehat H_{k} \rangle_{x,\varphi}=0. \end{gathered}
\end{equation}
\tag{2.24}
$$
Let $\{\sigma_k\}$ and $\{\widetilde \sigma_k\}$ be two decreasing sequences. Then
$$
\begin{equation}
a_{k+1}=a_k-2 \sigma_k, \qquad b_{k+1}=b_k-2\widetilde \sigma_k,
\end{equation}
\tag{2.25}
$$
$$
\begin{equation}
a_{0}=2 \sum_{k=0}^{+\infty} \sigma_k\quad\text{and} \quad b_{0}=2 \sum_{k=0}^{+\infty} \widehat \sigma_k.
\end{equation}
\tag{2.26}
$$
The inductive assumptions. Let $\{s_k\}$ be a sequence tending to zero, and let
$$
\begin{equation}
|\widehat H_k|_{a_k} \leqslant s_k\quad\text{and} \quad |\widehat h_{k+1}-\widehat h_{k}|_{a_{k+1}} \leqslant s_{k+1}.
\end{equation}
\tag{2.27}
$$
Assume that for $k>0$
$$
\begin{equation}
\frac{s_k}{\sigma^2_k} \leqslant \frac{s_0}{2^{k+2}}.
\end{equation}
\tag{2.28}
$$
Now using induction we establish the following estimates:
$$
\begin{equation}
|\varepsilon \widehat h'_{kI}|_{a_k}<\sqrt{\varepsilon} \frac{(2-2^{-k-1})s_0}{m\delta^m}, \qquad |\varepsilon \widehat h'_{ky}|_{a_k}< \sqrt{\varepsilon} (2-2^{-k-1})s_0,
\end{equation}
\tag{2.29}
$$
$$
\begin{equation}
|\varepsilon \widehat h''_{kIy}|_{a_k}<\frac{(2-2^{-k-1})s_0}{m\delta^m}, \qquad |\varepsilon \widehat h''_{kII}|_{a_k}<\frac{(2-2^{-k-1})s_0}{m^2 \delta^{2m}}
\end{equation}
\tag{2.30}
$$
and
$$
\begin{equation}
|\varepsilon \widehat h''_{kyy}|_{a_k}<(2-2^{-k-1})s_0.
\end{equation}
\tag{2.31}
$$
For $k=0$ inequalities (2.29)–(2.31) follow from (2.22) and (2.23). Let (2.29)–(2.31) hold for some $k>0$. Then from (2.16), since $\varepsilon \widehat h_{k+1}=\varepsilon \widehat h_{k}+\varepsilon ( \widehat h_{k+1}-\widehat h_{k})$, taking (2.28) into account we obtain
$$
\begin{equation}
|\varepsilon h'_{k+1I}|_{a_{k+1}} <\sqrt{\varepsilon}\biggl(\frac{(2-2^{-k-1})s_0}{m\delta^m}+\frac{s_{k+1}}{m \delta^m\sigma_k}\biggr) \leqslant \sqrt{\varepsilon}\frac{(2-2^{-k-2})s_0}{m\delta^m},
\end{equation}
\tag{2.32}
$$
$$
\begin{equation}
|\varepsilon h'_{k+1y}|_{a_{k+1}} <\sqrt{\varepsilon} (2-2^{-k-1})s_0 +\sqrt{\varepsilon} \frac{s_{k+1}}{\sigma_k} \leqslant \sqrt{\varepsilon} (2-2^{-k-2})s_0,
\end{equation}
\tag{2.33}
$$
$$
\begin{equation}
|\varepsilon h''_{k+1Iy}|_{a_{k+1}} <\frac{(2-2^{-k-1})s_0}{m\delta^m}+\frac{s_{k+1}}{m \delta^m\sigma^2_k} \leqslant \frac{(2-2^{-k-2})s_0}{m\delta^m},
\end{equation}
\tag{2.34}
$$
$$
\begin{equation}
|\varepsilon h''_{k+1II}|_{a_{k+1}} <\frac{(2-2^{-k-1})s_0}{m^2\delta^{2m}}+ \frac{s_{k+1}}{m^2 \delta^{2m} \sigma^2_k} \leqslant \frac{(2-2^{-k-2})s_0}{m^2\delta^{2m}}
\end{equation}
\tag{2.35}
$$
and
$$
\begin{equation}
|\varepsilon h''_{k+1yy}|_{a_{k+1}} < (2-2^{-k-1})s_0 +\frac{s_{k+1}}{\sigma^2_k} \leqslant (2-2^{-k-2})s_0.
\end{equation}
\tag{2.36}
$$
2.2. ResonancesWe define the domains $\Delta_{k}$. Let $\{N_k\}$, $N_{-1}=0$, be an increasing sequence of integers and $\{\lambda_{k}\}$ be a decreasing sequence of real numbers. Let the map $j\colon \mathbb{Z}^{n} \setminus \{0\} \mapsto \mathbb{N} \cup \{0\}$ be defined by the following rule: We write out the frequency vector:
$$
\begin{equation*}
\nu_k(y,I) = \begin{pmatrix} \Lambda'_{y}(y)+h'_y(y,I)+\varepsilon \widehat {h_k'}_y(y,I) \\ {h'}_I(y,I)+\varepsilon \widehat {h_k'}_I(y,I) \end{pmatrix}.
\end{equation*}
\notag
$$
For a nontrivial integer vector $K \in \mathbb{Z}^{n}$ we define the resonance zones
$$
\begin{equation}
Q^{(K)}_{k}=\biggl\{(y,I) \in \Delta_{k}\colon |\langle \nu_k(y,I), K \rangle | \leqslant \frac{\lambda_{k} \sqrt{\varepsilon}}{m \delta^m}\biggr\}, \qquad |K| \leqslant N_k;
\end{equation}
\tag{2.38}
$$
then we set by definition
$$
\begin{equation}
\Delta_{k+1}=\Delta_{k} \setminus \bigcup_{|K| \leqslant N_k} U_{a_k}(Q^{(K)}_k).
\end{equation}
\tag{2.39}
$$
2.3. KAM: the change of variablesWe define the change of variables $\Phi_k$: $(x,y,I,\varphi) \mapsto (\widehat x, \widehat y,\widehat I,\widehat \varphi)$ in terms of the generating function $S$:
$$
\begin{equation*}
\begin{gathered} \, \widehat S=\widehat y x+\widehat I \varphi+\sqrt{\varepsilon} \, m \delta^m S_k(\widehat y,\widehat I,x,\varphi), \\ y=\widehat y+\sqrt{\varepsilon} \, m \delta^m S_{kx}', \qquad I=\widehat I+\sqrt{\varepsilon} \, m \delta^m S_{k\varphi}', \\ x=\widehat x-\sqrt{\varepsilon} \, m \delta^m S_{k \widehat y}', \qquad \varphi=\widehat \varphi- \sqrt{\varepsilon} \, m \delta^m S_{k\widehat I}'. \end{gathered}
\end{equation*}
\notag
$$
A necessary condition for the change of variables to be well defined is that we stay within the neighbourhoods under consideration:
$$
\begin{equation}
m \delta^m|S_{kx}'| \leqslant \sigma_k, \qquad |S_{k\varphi}'| \leqslant \sigma_k, \qquad \sqrt \varepsilon \, m \delta^m |S_{ky}'| \leqslant \widehat \sigma_k, \qquad \sqrt \varepsilon \, m \delta^m |S_{kI}'| \leqslant \widehat \sigma_k.
\end{equation}
\tag{2.40}
$$
We express $\widehat H_{k}$ and $S_k$ by Fourier series:
$$
\begin{equation}
\widehat H_{k}=\sum_{(p,l) \in \mathbb{Z}^{n} \setminus \{0\}} \widehat H_{k,(p,l)}(\widehat y,\widehat I) e^{i p x +il\varphi}\quad\text{and} \quad S_{k}=\sum_{(p,l) \in \mathbb{Z}^{n} \setminus \{0\}} S_{k,(p,l)}(\widehat y,\widehat I) e^{i p x +il\varphi}.
\end{equation}
\tag{2.41}
$$
Substituting the change of variables into the Hamiltonian yields
$$
\begin{equation*}
\begin{aligned} \, H_k &= \Lambda\Bigl(\widehat y+\sqrt{\varepsilon} \, m \delta^m S_{kx}'\Bigr) +h\Bigl(\widehat y+\sqrt{\varepsilon} \, m \delta^m S_{kx}',\widehat I+\sqrt{\varepsilon} \, m \delta^m S_{k\varphi}'\Bigr) \\ &\qquad +\varepsilon \widehat h_k(\widehat y+\sqrt{\varepsilon} \, m \delta^m S_{kx}',\widehat I+ \sqrt{\varepsilon} \, m \delta^m S_{k\varphi}') \\ &\qquad +\varepsilon \widehat H_{k}(\widehat y+\sqrt{\varepsilon} \, m \delta^m S_{kx}',x,\widehat I+ \sqrt{\varepsilon} \, m \delta^m S_{k\varphi}',\varphi) \\ &= \Lambda(\widehat y)+h(\widehat I,\widehat y)+\varepsilon \widehat h_k(\widehat I,\widehat y)+\sqrt{\varepsilon} \, m \delta^m \langle \Lambda'_{y}(\widehat y), S_{kx}'\rangle \\ &\qquad +\sqrt{\varepsilon} \, m \delta^m h'_{I}(\widehat I,\widehat y) S_{k\varphi}'+\sqrt{\varepsilon} \, m \delta^m \langle h'_{y}(\widehat I,\widehat y), S_{kx}' \rangle \\ &\qquad +\sqrt{\varepsilon} \, m \delta^m \varepsilon \widehat h'_{kI}(\widehat I,\widehat y) S_{k\varphi}'+ \sqrt{\varepsilon} \, m \delta^m \varepsilon \langle \widehat h'_{ky}(\widehat I,\widehat y), S_{kx}' \rangle \\ &\qquad +\varepsilon \widehat H^{(1)}_{k}+\varepsilon R^{(1)}_{k}+\varepsilon R^{(2)}_{k}+\varepsilon R^{(3)}_{k}+\varepsilon R^{(4)}_{k}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \widehat H_k^{(1)} &=\sum_{|(p,l)| \leqslant N_k} \widehat H_{k,(p,l)}(\widehat y,\widehat I) e^{i p x +il\varphi}, \\ R^{(1)}_{k} &= \widehat H_k(y,x,I,\varphi)-\widehat H^{(1)}_{k}(\widehat y,x,\widehat I,\varphi), \\ R^{(2)}_{k} &= \frac{1}{\varepsilon} \bigl( \Lambda(y)-\Lambda(\widehat y)-\sqrt{\varepsilon} \, m \delta^m \Lambda'_{y}(\widehat y) S_{kx}' \bigr), \\ R^{(3)}_{k} &= \frac{1}{\varepsilon}\bigl( h(y,I)-h(\widehat y,\widehat I)- \sqrt{\varepsilon} \, m \delta^m h'_{I}(\widehat y,\widehat I) S_{k\varphi}'-\sqrt{\varepsilon} \, m \delta^m \langle h'_{y}(\widehat y,\widehat I), S_{kx}'\rangle \bigr) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
R^{(4)}_{k} = \widehat h_k(y,I)-\widehat h_k(\widehat y,\widehat I)- \sqrt{\varepsilon} \, m \delta^m \widehat h'_{kI}(\widehat y,\widehat I) S_{k\varphi}'-\sqrt{\varepsilon} \, m \delta^m \langle \widehat h'_{ky}(\widehat y,\widehat I), S_{kx}'\rangle.
\end{equation*}
\notag
$$
We find $S_k$ from the homology equation
$$
\begin{equation}
\langle \Lambda'_{y}+h'_{y}+\varepsilon \widehat h'_{ky},S_{kx}'\rangle+(h'_{I}+\varepsilon \widehat h'_{kI}) S_{k\varphi}'+\frac{\sqrt{\varepsilon}}{m\delta^m} \widehat H^{(1)}_{k}=0.
\end{equation}
\tag{2.42}
$$
We plug (2.41) into (2.42) and equate the like coefficients:
$$
\begin{equation}
S_{k,(p,l)}=\frac{i \sqrt{\varepsilon}}{m\delta^m} \, \frac{\widehat H^{(1)}_{k,(p,l)}}{\langle \Lambda'_{y}+h'_{y}+\varepsilon \widehat h'_{ky},p \rangle+(h'_{I}+\varepsilon \widehat h'_{kI}) l}.
\end{equation}
\tag{2.43}
$$
It follows from (2.17) and (2.27) that
$$
\begin{equation}
| \widehat H^{(1)}_{k,(p,l)}|_{a_k} \leqslant s_k \exp(-\widehat \sigma_k n |(p,l)|).
\end{equation}
\tag{2.44}
$$
Taking (2.38) into account we obtain
$$
\begin{equation}
| S_{k,(p,l)}|_{a_k} \leqslant \frac{s_k \exp(-\widehat \sigma_k n |(p,l)|)}{\lambda_{j((p,l))}}.
\end{equation}
\tag{2.45}
$$
We find an estimate for $S_k$:
$$
\begin{equation}
| S_{k}|_{a_k-\sigma_k,b_k-\widehat \sigma_k} \leqslant L_k s_k\quad\text{and} \quad L_k=\sum_{j=0}^k \frac{N^{n}_j}{\lambda_{j}} \exp(-\widehat \sigma_k n N_{j-1}).
\end{equation}
\tag{2.46}
$$
For (2.40) to hold it is sufficient that
$$
\begin{equation}
L_k s_k \leqslant \widehat \sigma_k \sigma_k\quad\text{and} \quad \sqrt \varepsilon \, m \delta^m L_k s_k \leqslant \widehat \sigma^2_k.
\end{equation}
\tag{2.47}
$$
Now we estimate $R^{(1)}_{k}$, $R^{(2)}_{k}$, $R^{(3)}_{k}$ and $R^{(4)}_{k}$. The term $R^{(1)}_{k}$:
$$
\begin{equation*}
\begin{aligned} \, R^{(1)}_{k} &=\widehat H_k(y,x,I,\varphi)-\widehat H^{(1)}_{k}(\widehat y,x,\widehat I,\varphi)=\widehat H_k(y,x,I,\varphi)-\widehat H^{(1)}_{k}(y,x,I,\varphi) \\ &\qquad + \widehat H^{(1)}_{k}(y,x,I,\varphi)-\widehat H^{(1)}_{k}(\widehat y,x,\widehat I,\varphi)= R^{(1,1)}_{k}+{R}^{(1,2)}_{k}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
R^{(1,1)}_{k} = \widehat H_{k}(y,x,I,\varphi)-\widehat H^{(1)}_{k}( y,x, I,\varphi)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
R^{(1,2)}_{k} = \widehat H^{(1)}_{k}(y,x,I,\varphi)-\widehat H^{(1)}_{k}(\widehat y,x,\widehat I,\varphi).
\end{equation*}
\notag
$$
We can estimate $R^{(1,1)}_{k}$ by taking Lemma 5 into account:
$$
\begin{equation*}
|R^{(1,1)}_{k}|_{a_k-\sigma_k,b_k-\widehat \sigma_k} \leqslant \frac{C_{\Pi}}{\widehat \sigma_k} \biggl(N_k+ \frac{1}{\widehat \sigma_k}\biggr)^n \exp(-N_k \widehat \sigma_k)s_k.
\end{equation*}
\notag
$$
The term $R^{(1,2)}_{k}$:
$$
\begin{equation*}
\begin{aligned} \, |{R}^{(1,2)}_{k}|_{a_k-\sigma_k,b_k-\widehat \sigma_k} &\leqslant (n-1) \sqrt \varepsilon \, m \delta^m|\widehat H'^{(1)}_{ky}|_{a_k-\sigma_k} |S'_{kx}|_{b_k-\widehat \sigma_k} \\ &\qquad+\sqrt \varepsilon \, m \delta^m|\widehat H'^{(1)}_{kI}|_{a_k-\sigma_k} |S'_{k \varphi}|_{b_k- \widehat \sigma_k} \\ &\leqslant \bigl((n-1) m \delta^m+1\bigr) L_k \frac{s_k^2}{\sigma_k \widehat \sigma_k}. \end{aligned}
\end{equation*}
\notag
$$
The term $R^{(2)}_{k}$:
$$
\begin{equation*}
|R^{(2)}_{k}|_{a_k,b_k-\widehat \sigma_k} \leqslant (n-1)^2 m^2 \delta^{2m} |\Lambda''_{yy}|_{a_k} |S'^2_{kx}|_{b_k-\widehat \sigma_k} \leqslant m^2 \delta^{2m} (n-1)^2 c_{\Lambda} \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2.
\end{equation*}
\notag
$$
The term $R^{(3)}_{k}$:
$$
\begin{equation*}
\begin{aligned} \, &|R^{(3)}_{k}|_{a_k,b_k-\widehat \sigma_k} \leqslant (n-1)^2 m^2 \delta^{2m} |h''_{yy}|_{a_k} |S'^2_{kx}|_{b_k-\widehat \sigma_k} \\ &\qquad\qquad+(n-1) m^2 \delta^{2m} |h''_{yI}|_{a_k} |S'_{kx}|_{b_k-\widehat \sigma_k} |S'_{k\varphi}|_{b_k- \widehat \sigma_k} + m^2 \delta^{2m} |h''_{II}|_{a_k} |S'_{k\varphi}|_{b_k-\widehat \sigma_k} \\ &\qquad\leqslant \overline{c}_h \bigl( (n-1)^2 m \delta^{4m}+(n-1) m \delta^{2m}+1 \bigr) \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2. \end{aligned}
\end{equation*}
\notag
$$
The term $R^{(4)}_{k}$:
$$
\begin{equation*}
\begin{aligned} \, &|R^{(4)}_{k}|_{a_k-2\sigma_k,b_k-\widehat \sigma_k} \leqslant (n-1)^2 m^2 \delta^{2m} \varepsilon |\widehat h''_{kyy}|_{a_k-2 \sigma_k} |S'^2_{kx}|_{b_k-\widehat \sigma_k} \\ &\qquad\qquad + (n-1) m^2 \delta^{2m} \varepsilon |\widehat h''_{kyI}|_{a_k-2 \sigma_k} |S'_{kx}|_{b_k-\widehat \sigma_k} |S'_{k\varphi}|_{b_k-\widehat \sigma_k} \\ &\qquad\qquad + m^2 \delta^{2m} \varepsilon |\widehat h''_{kII}|_{a_k-2 \sigma_k} |S'^2_{k\varphi}|_{b_k- \widehat \sigma_k} \\ &\qquad\leqslant 2 s_0 \bigl((n-1)^2 m^2 \delta^{2m}+(n-1) m \delta+1\bigr) \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2. \end{aligned}
\end{equation*}
\notag
$$
Set
$$
\begin{equation}
\begin{gathered} \, \notag R_k=R^{(1)}_{k}+R^{(2)}_{k}+R^{(3)}_{k}+R^{(3)}_{k}+R^{(4)}_{k}, \\ \widehat h_{k+1}=\widehat h_{k}+\langle R_k \rangle_{x, \varphi}, \qquad \widehat H_{k+1}=R_k-\langle R_k \rangle_{x, \varphi}. \end{gathered}
\end{equation}
\tag{2.48}
$$
The coefficients $m \delta^m$, $m^2 \delta^{2m}$, $m \delta^{4m}$ and $m \delta^{2m}$ do not exceed 1 for $\delta<0.5$. Therefore,
$$
\begin{equation}
|R_{k}|_{a_k-2\sigma_k,b_k-\widehat \sigma_k} \,{\leqslant}\, \frac{C_{\Pi}}{\widehat \sigma_k} \biggl(N_k+ \frac{1}{\widehat \sigma_k}\biggr)^n \exp(-N_k \widehat \sigma_k)s_k+n \frac{L_k s_k^2}{\sigma_k \widehat \sigma_k}+C_R \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2,
\end{equation}
\tag{2.49}
$$
where $C_R=(\overline{c}_{h}+2 s_0) (n^2+n)$. Then the condition for the convergence of the KAM procedure is
$$
\begin{equation}
\frac{C_{\Pi}}{\widehat \sigma_k} \biggl(N_k+\frac{1}{\widehat \sigma_k}\biggr)^n \exp(-N_k \widehat \sigma_k) s_k+n \frac{L_k s_k^2}{\sigma_k \widehat \sigma_k}+C_R \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2 \leqslant s_{k+1}.
\end{equation}
\tag{2.50}
$$
2.4. An estimate for the measure of the resonance statesNow we show that instead of (2.39) we can write
$$
\begin{equation}
\Delta_{k+1}=\Delta_{k} \setminus \bigcup_{N_{k-1}<|K| \leqslant N_k} U_{a_k}(Q^{(K)}_k).
\end{equation}
\tag{2.51}
$$
Proof. Assume the contrary, and let $(\widehat y, \widehat I) \in Q^{(K)}_{k+1} \subset \Delta_{k+1}$. Then
$$
\begin{equation*}
\begin{aligned} \, |\langle \nu_{k}(\widehat y, \widehat I), K \rangle | &\leqslant |\langle \nu_{k+1}(\widehat y,\widehat I), K \rangle |+|\langle \nu_{k+1}(\widehat y,\widehat I)-\nu_{k}(\widehat y,\widehat I), K \rangle | \\ &\leqslant \frac{\lambda_{k+1} \sqrt{\varepsilon}}{m\delta^m}+\varepsilon | {{\widehat h'}_{k+1I}}(\widehat y,\widehat I)-{{\widehat h'}_{kI}}(\widehat y,\widehat I)|_{a_{k+1}} N_{k} \\ &\qquad + (n-1) \varepsilon | {{\widehat h'}_{k+1y}}(\widehat y,\widehat I)-{{\widehat h'}_{ky}}(\widehat y,\widehat I)|_{a_{k+1}} N_{k} \\ &\leqslant \frac{\lambda_{k+1} \sqrt{\varepsilon}}{m\delta^m}+n \frac{\sqrt{\varepsilon}\, s_{k+1}}{m\delta^m \sigma_{k+1}} N_{k}. \end{aligned}
\end{equation*}
\notag
$$
Now in view of (2.52)
$$
\begin{equation*}
|\langle \nu_k(\widehat y,\widehat I), K \rangle | \leqslant \frac{\lambda_{k} \sqrt{\varepsilon}}{m \delta^m}.
\end{equation*}
\notag
$$
On the other hand it follows from (2.39) that $\Delta_{k+1} \cap U_{a_k}(Q^{(K)}_k)=\varnothing$. Then taking (2.40) into account we arrive at a contradiction: The proof is complete. Let $\mu=\mu(\Theta)$ denote the standard Lebesgue measure of the measurable set $\Theta$. We find an estimate for $\mu\bigl(U_{a_k}(Q^{(K)}_k)\bigr)$. Let $|\widetilde{y}| \leqslant \sqrt{\varepsilon}\, a_k$ and $|\widetilde{I}| \leqslant \sqrt{\varepsilon} \, m \delta^m a_k$. Set
$$
\begin{equation*}
d\,\nu_k=\nu_k(y+\widetilde{y},I+\widetilde{I})-\nu_k(y,I);
\end{equation*}
\notag
$$
then
$$
\begin{equation*}
\begin{aligned} \, |\langle d\,\nu_k, K \rangle|_{a_k} &\leqslant \sqrt{\varepsilon} \bigl( |\Lambda''_{yy}|_{a_k}+ |h''_{yy}|_{a_k}+2m \delta^m|h''_{Iy}|_{a_k}+m \delta^m |h''_{II}|_{a_k}+\varepsilon |\widehat h''_{kyy}|_{a_k} \\ &\qquad +2 \varepsilon m \delta^m |\widehat h''_{kyI}|_{a_k}+\varepsilon m \delta^m |\widehat h''_{kII}|_{a_k} \bigr) n |K| a_k \\ &\leqslant \sqrt{\varepsilon} \biggl( c_{\Lambda}+\frac{\delta^{2m}\overline{c}_h}{m}+2 \delta^m \overline{c}_h+\frac{\overline{c}_h}{m \delta^m}+6 s_0+\frac{2 s_0}{m \delta^m} \biggr)n N_k a_k. \end{aligned}
\end{equation*}
\notag
$$
Let $C_{\nu}=c_{\Lambda}+8s_0+4 \overline{c}_h$. Then
$$
\begin{equation}
|\langle d\,\nu_k, K \rangle|_{a_k} \leqslant C_{\nu} N_k a_k \frac{\sqrt{\varepsilon}}{m\delta^m}.
\end{equation}
\tag{2.53}
$$
Now we define $\Omega^{(K)}_k$:
$$
\begin{equation}
\Omega^{(K)}_k=\biggl\{(y,I) \in U_{a_k}(\Delta_{k})\colon |\langle \nu_k(y,I), K \rangle | \leqslant (\lambda_{k}+C_{\nu} N_k a_k)\frac{\sqrt{\varepsilon}}{m\delta^m}\biggr\}.
\end{equation}
\tag{2.54}
$$
It follows from (2.53) that Consider the frequency mapping
$$
\begin{equation*}
\nu_{k}\colon (y,I) \mapsto \bigl( \Lambda'_{y}(y)+h'_{y}(y,I)+\varepsilon \widehat h'_{ky}(y,I), h'_{I}(y,I)+\varepsilon \widehat h'_{kI}(y,I) \bigr).
\end{equation*}
\notag
$$
Its Jacobian matrix is
$$
\begin{equation}
J_k=\begin{pmatrix} \Lambda''_{yy}+h''_{yy}+\varepsilon \widehat h''_{kyy} & h''_{Iy}+\varepsilon \widehat h''_{kIy} \\ h''_{kyI}+\varepsilon \widehat h_{yI} & h''_{II}+\varepsilon \widehat h_{kII} \end{pmatrix}.
\end{equation}
\tag{2.56}
$$
From (2.6), (2.10) and (2.29) we obtain the estimates
$$
\begin{equation}
|\Lambda'_{y}+h'_{y}+\varepsilon \widehat h'_{ky}|_{a_k} < c_{\Lambda}+\overline{c}_{h}+2 \sqrt{\varepsilon} s_0
\end{equation}
\tag{2.57}
$$
and
$$
\begin{equation}
| h'_{I}+\varepsilon \widehat h'_{kI}|_{a_k} < \frac{1}{m}(\overline{c}_{h}+2 s_0).
\end{equation}
\tag{2.58}
$$
Set $c_{\nu}=\max\bigl(c_{\Lambda}+\overline{c}_{h}+2 \sqrt{\varepsilon} s_0,(\overline{c}_{h}+2 s_0)/m\bigr)$. Let $\Xi_k \subset \mathbb{R}^n$ denote the image of the set $U_{a_k}(\Delta_{k})$ under $\nu_k$. Let $(\overline{\nu}_1, \dots,\overline{\nu}_n)$ be the Cartesian coordinate system in the frequency space $\mathbb{R}^n$. Then (2.57) and (2.58) yield the following result. Lemma 7. The embedding $\Xi_k \subset \Pi$ holds, where $\Pi$ is the cube in $\mathbb{R}^n$ defined by We look at the inequality in (2.54) and estimate the measure of the corresponding region in $\Xi_k$. In $\mathbb{R}^n$ we consider the hyperplanes
$$
\begin{equation*}
\langle \overline{\nu}_k, K \rangle=(\lambda_{k}+C_{\nu} N_k a_k)\frac{\sqrt{\varepsilon}}{m\delta^m}\quad\text{and} \quad \langle \overline{\nu}_k, K \rangle= -(\lambda_{k}+C_{\nu} N_k a_k)\frac{\sqrt{\varepsilon}}{m\delta^m},
\end{equation*}
\notag
$$
where $\overline{\nu}_k=(\overline{\nu}_{k1}, \dots,\overline{\nu}_{kn})$ are the coordinates in the frequency space. The distance $d_{\nu_k}$ between these hyperplanes has the estimate
$$
\begin{equation}
d_{\nu_k} \leqslant 4 \frac{(\lambda_{k}+C_{\nu} N_k a_k)\sqrt{\varepsilon}}{(N_{k-1}+ 1)m\delta^m}.
\end{equation}
\tag{2.60}
$$
Let $\Sigma_k$ denote the part of $\mathbb{R}^n$ enclosed between the hyperplanes. Then It follows from (2.60) and Lemma 7 that
$$
\begin{equation}
\mu(\Omega^{(K)}_k)<4 \sqrt{n} \, c_{\nu}\frac{(\lambda_{k}+C_{\nu} N_k a_k)\sqrt{\varepsilon}}{(N_{k-1}+1)m\delta^m}.
\end{equation}
\tag{2.61}
$$
Note that inequalities (2.30) and Lemma 4 yield a bound on the determinant of $J_k$:
$$
\begin{equation}
\frac{\underline{c}_{\,J}}{m^2 \delta^{2m}}<|{\det J_k}|_{a_k}< \frac{\overline{c}_J}{m^2 \delta^{2m}}.
\end{equation}
\tag{2.62}
$$
Then from (2.55) we obtain
$$
\begin{equation*}
\mu \bigl( U_{a_k}(Q^{(K)}_k) \bigr) \leqslant \mu \bigl(\nu^{-1}_k(\Omega^{(K)}_k)\bigr) \leqslant 4 \sqrt{n} \, c_{\nu}\frac{(\lambda_{k}+C_{\nu} N_k a_k)\sqrt{\varepsilon}}{(N_{k-1}+ 1)\underline{c}_{\,J}}m \delta^m.
\end{equation*}
\notag
$$
Now,
$$
\begin{equation*}
\mu \biggl(\bigcup_{N_{k-1}<|K| \leqslant N_k} U_{a_k}(Q^{(K)}_k) \biggr) \leqslant 4 \sqrt{n} \, n!\, c_{\nu}\frac{(\lambda_{k}+C_{\nu} N_k a_k)\sqrt{\varepsilon}}{(N_{k-1}+1)\underline{c}_{\,J}}m \delta^m N_k^{n-1}(N_k-N_{k-1}).
\end{equation*}
\notag
$$
Finally, we have
$$
\begin{equation}
\mu(\Delta_0 \setminus \Delta_{+\infty}) \leqslant \sum_{k=0}^{+\infty} C_{\mu} (\lambda_{k}+N_k a_k) \frac{N_k^{n-1}(N_k-N_{k-1})}{N_{k-1}+1} m\delta^m \sqrt{\varepsilon},
\end{equation}
\tag{2.63}
$$
where $C_{\mu}=4 \sqrt{n}\, n!\, (c_{\nu}/\underline{c}_{\,J}) \max(C_{\nu},1)$. 2.5. The sequences $\sigma_k$, $\widetilde \sigma_k$, $s_k$, $N_k$ and $\lambda_k$To complete the proofs we must define the sequences $\sigma_k$, $\widetilde \sigma_k$, $s_k$, $N_k$ and $\lambda_k$ so as to satisfy inequalities (2.28), (2.47), (2.50) and (2.52) and make the series in (2.63) convergent. We define these sequences by the formulae
$$
\begin{equation}
\sigma_k=\beta_{\sigma} 2^{-(2n+1)k}, \qquad \beta_{\sigma}=a_0 \biggl(\frac{2^{2n+1}-1}{2^{2n+2}}\biggr), \qquad \widehat \sigma_k=b_0 2^{-k-2},
\end{equation}
\tag{2.64}
$$
$$
\begin{equation}
N_k=\beta_N 2^{2k}, \qquad \lambda_{k}=\beta_{\lambda} 2^{-2nk}\quad\text{and} \quad s_k=s_0 e^{-\beta_s k- 2^k},
\end{equation}
\tag{2.65}
$$
where we specify the constants $\beta_N, \beta_{\lambda}$ and $ \beta_s$ in what follows. It is easy to see that
$$
\begin{equation*}
a_0=2 \sum_{k=0}^{\infty} \sigma_k, \qquad b_0=2\sum_{k=0}^{\infty} \widehat \sigma_k
\end{equation*}
\notag
$$
and also We estimate the series in (2.63). Note that
$$
\begin{equation*}
\frac{N_k^{n-1}(N_k-N_{k-1})}{N_{k-1}+1}<3 N_k^{n-1}.
\end{equation*}
\notag
$$
Set $\beta_{\lambda}=1$. Then the series in (2.63) is convergent and
$$
\begin{equation}
\mu(\Delta_0 \setminus \Delta_{+\infty})\mkern-3mu \leqslant \mkern-3mu \sum_{k=0}^{+\infty} 3 C_{\mu} (\lambda_{k}N_k^{n-1}+N^n_k a_k) m\delta^m \sqrt{\varepsilon}\mkern-2mu<\mkern-2mu6 C_{\mu} \beta^n_N (1+a_0) m\delta^m \sqrt{\varepsilon}.
\end{equation}
\tag{2.67}
$$
Substituting (2.64) and (2.65) into (2.28) we see that the condition
$$
\begin{equation}
\frac{1}{\beta_{\sigma}}\, e^{(2n+3-\beta_s)k-2^k} \leqslant 1, \qquad k>0,
\end{equation}
\tag{2.68}
$$
is sufficient for (2.28) to hold. We substitute (2.64) and (2.65) into (2.46):
$$
\begin{equation}
\begin{aligned} \, \notag L_k &=\sum_{j=0}^k \frac{N^{n}_j}{\lambda_{j}} \exp(-\widehat \sigma_k n N_{j-1}) =\beta^n_N+ \sum_{j=1}^k \beta^n_N 2^{4nj}\exp(-nb_0 \beta_N 2^{j-4}) \\ & <\beta^n_N+\sum_{j=1}^k \beta^n_N 2^{4nj-nb_0 \beta_N 2^{j-4}}. \end{aligned}
\end{equation}
\tag{2.69}
$$
For
$$
\begin{equation}
\beta_N \geqslant \biggl[\frac{16(4n+1)}{nb_0}\biggr]+1
\end{equation}
\tag{2.70}
$$
it follows from (2.69) that Next we substitute (2.64) and (2.65) into (2.50) and estimate each term:
$$
\begin{equation}
\begin{split} &\frac{C_{\Pi}}{\widehat \sigma_k} \biggl(N_k+\frac{1}{\widehat \sigma_k}\biggr)^n \exp(-N_k \widehat \sigma_k)s_k \\ &\qquad\leqslant \frac{4 s_0 C_{\Pi}}{b_0}\biggl(\beta_N+\frac{4}{b_0}\biggr)^n \exp\biggl((2n+1-\beta_s)k-\biggl(\frac{\beta_N b_0}{4}+1\biggr)2^k\biggr), \end{split}
\end{equation}
\tag{2.72}
$$
$$
\begin{equation}
\begin{gathered} \, n \frac{L_k s_k^2}{\sigma_k \widehat \sigma_k} \leqslant \frac{8 s^2_0 n \beta^n_N}{\beta_{\sigma} b_0}\exp((2n+2-2\beta_s)k-2^{k+1}), \\ C_R \biggl(\frac{L_k s_k}{\widehat \sigma_k}\biggr)^2 \leqslant \frac{64 s_0^2 C_R \beta^{2n}_N}{b_0^2} \exp((2-2\beta_s)k-2^{k+1}). \end{gathered}
\end{equation}
\tag{2.73}
$$
To prove (2.50) it is sufficient to verify the inequalities
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, &\frac{4C_{\Pi}}{b_0}\biggl(\beta_N+\frac{4}{b_0}\biggr)^n \exp\biggl((2n+1-\beta_s)k-\biggl(\frac{\beta_N b_0}{4}+1\biggr)2^k\biggr) \\ &\qquad\leqslant \frac{1}{3} \exp(-\beta_s(k+1)-2^{k+1}), \end{aligned} \\ \frac{8 s_0 n \beta^n_N}{\beta_{\sigma} b_0}\exp((2n+2-2\beta_s)k-2^{k+1}) \leqslant \frac{1}{3} \exp(-\beta_s(k+1)-2^{k+1}) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{64 s_0 C_R \beta^{2n}_N}{b_0^2} \exp((2-2\beta_s)k-2^{k+1}) \leqslant \frac{1}{3} \exp(-\beta_s(k+1)-2^{k+1}).
\end{equation*}
\notag
$$
We transform them as follows:
$$
\begin{equation}
\frac{12 C_{\Pi}}{b^{n+1}_0}(\beta_N b_0+4)^n \exp\biggl(\beta_s+(2n+1)k-\biggl(\frac{\beta_N b_0}{4}- 1\biggr)2^k\biggr) \leqslant 1
\end{equation}
\tag{2.74}
$$
and
$$
\begin{equation}
\begin{gathered} \, \begin{gathered} \, \frac{24 s_0 n \beta^n_N}{\beta_{\sigma} b_0} \exp(\beta_s+(2n+2-\beta_s)k-2^{k}) \leqslant 1, \\ \frac{192 s_0 C_R \beta^{2n}_N}{b_0^2} \exp(\beta_s+(2-\beta_s)k) \leqslant 1. \end{gathered} \end{gathered}
\end{equation}
\tag{2.75}
$$
Inequalities (2.74) and (2.75) must hold for all $k \geqslant 0$. Substituting (2.64) and (2.65) into (2.47) we obtain the following sufficient condition for (2.47) to hold:
$$
\begin{equation}
\begin{gathered} \, \frac{4 s_0 \beta^n_N}{b_0 \beta_s} \exp((2n+2-\beta_s)k-2^k) \leqslant 1, \\ \frac{16 \sqrt{\varepsilon} \, m \delta^m \beta^n_N s_0}{b^2_0} \exp((2-\beta_s)k-2^k) \leqslant 1, \\ k \geqslant 0. \end{gathered}
\end{equation}
\tag{2.76}
$$
Substituting (2.64) and (2.65) into (2.52) we obtain the following sufficient condition for (2.52) to hold:
$$
\begin{equation}
\frac{2 n s_0 \beta_N}{\beta_{\sigma}} \exp((2n+1-\beta_s)(k+1)+2k-2^{k+1}) \leqslant 1, \qquad k>0.
\end{equation}
\tag{2.77}
$$
Let us verify (2.74)–(2.77). Set $\beta_s=3$. Lemma 8. There exists $\beta_N$ depending on $b_0, \beta_{\sigma}, n$ and $ C_{\Pi}$ such that inequality (2.74) holds for all $k \geqslant 0$. Proof. Note that if $\beta_N>(16(2n+4))/b_0$, then
$$
\begin{equation*}
\begin{aligned} \, &\frac{12 C_{\Pi}}{b^{n+1}_0}(\beta_N b_0+4)^n \exp\biggl(\beta_s+(2n+1)k-\biggl(\frac{\beta_N b_0}{4}- 1\biggr)2^k\biggr) \\ &\qquad <\frac{32^n 12 C_{\Pi}}{b^{n+1}_0}\biggl(\frac{\beta_N b_0}{16}\biggr)^n \exp\biggl(- \frac{\beta_N b_0}{16}2^k\biggr). \end{aligned}
\end{equation*}
\notag
$$
It is known that Hence for sufficiently large $\widehat \beta_N$, for $\beta_N=\widehat \beta_N$ we have The proof is complete. For $k>2n+5$ the exponentials in (2.75)–(2.77) are at most 1. Let
$$
\begin{equation*}
s_0=\biggl(e^{2n+5} \max\biggl(\frac{24 n \beta^n_N}{\beta_{\sigma} b_0}, \frac{192 C_R \beta^{2n}_N}{b_0^2},\frac{4 \beta^n_N}{b_0 \beta_s}, \frac{16 \beta^n_N}{b^2_0},\frac{2 n \beta_N}{\beta_{\sigma}}\biggr)\biggr)^{-1}.
\end{equation*}
\notag
$$
In view of (2.20) this equality ensures that (2.75)–(2.77) hold and completes the proof of the convergence of the KAM-procedure.
§ 3. Technical lemmas3.1. The proof of Lemma 3From Lemma 2 we obtain formulae for the derivatives of $h$:
$$
\begin{equation}
\frac{\partial h}{\partial I}=\frac{\alpha(y)}{\log |I|} R\biggl(\frac{1}{\log |I|},\frac{\log |{\log |I|}|}{\log |I|},I,y\biggr), \qquad R(0,0,0,y)=1,
\end{equation}
\tag{3.1}
$$
and
$$
\begin{equation}
\frac{\partial^2 h}{\partial I^2}=- \frac{\alpha(y)}{I \log^2 |I|} G\biggl(\frac{1}{\log |I|},\frac{\log |{\log |I|}|}{\log |I|},I,y\biggr), \qquad G(0,0,0,y)=1,
\end{equation}
\tag{3.2}
$$
where the functions $G$ and $R$ are real analytic at $(0,0,0,y)$. It follows from the equalities
$$
\begin{equation*}
\lim_{I \to 0} \frac{1}{\log |I|}=0\quad\text{and} \quad \lim_{I \to 0} \frac{\log |{\log |I|}|}{\log |I|}=0
\end{equation*}
\notag
$$
that there exist $ \delta_1 \in (0, \sqrt I_0)$ and $\varepsilon_1>0$ such that for all $\varepsilon \in (0,\varepsilon_1)$ in ${U_{\widetilde a}(D \times (0,\delta_1))}$ the functions $F$, $R$ and $G$ are analytic and we have
$$
\begin{equation}
\frac{1}{2}<|F|<\frac{3}{2}, \qquad \frac{1}{2}<|R|<\frac{3}{2},
\end{equation}
\tag{3.3}
$$
$$
\begin{equation}
\frac{1}{2}<|G|<\frac{3}{2}, \qquad |F'_{y}|<1, \qquad |F''_{yy}|<1\quad\text{and} \quad |R'_y|<1.
\end{equation}
\tag{3.4}
$$
Now,
$$
\begin{equation}
\frac{\partial h}{\partial y}=\frac{I}{\log |I|} (\alpha'_{y} F+\alpha F'_y), \qquad \frac{\partial^2 h}{\partial I \partial y}=\frac{\alpha'_y R+\alpha R'_y}{\log |I|}
\end{equation}
\tag{3.5}
$$
and
$$
\begin{equation}
\frac{\partial^2 h}{\partial^2 y}=\frac{I}{\log |I|} (\alpha''_{yy} F+2 \alpha'_{y} F'_y+\alpha F''_{yy}).
\end{equation}
\tag{3.6}
$$
Let $\delta_1$ be sufficiently small so that $1/\log |I|$, $I/\log |I|$ and $1/(I \log^2 |I|)$ are monotone on the interval $(0, \delta_1)$. Set $\delta=\delta_1$. If $\varepsilon_1$ is small, then for $\varepsilon \in (0,\varepsilon_1)$ we have
$$
\begin{equation}
\frac{1}{4(m+1)|{\log \delta}|}<\biggl|\frac{1}{\log |I|}\biggr|_{\widetilde a} < \frac{1}{m |{\log \delta}|}, \qquad \biggl|\frac{I}{\log |I|}\biggr|_{\widetilde a}< \frac{\delta^{2m}}{m |{\log \delta}|}
\end{equation}
\tag{3.7}
$$
and
$$
\begin{equation}
\frac{\delta^{2(m+1)}}{2(m+1) |{\log \delta}|} <\biggl|\frac{I}{\log |I|}\biggr|_{\widetilde a}<\frac{\delta^{2m}}{m |{\log \delta}|}.
\end{equation}
\tag{3.8}
$$
It follows from (3.3)–(3.8) that
$$
\begin{equation*}
\begin{gathered} \, \frac{\delta^{2(m+1)}\underline{c}_{\,\alpha}}{8(m+1) |{\log \delta}|}< |h|_{\widetilde a}<\frac{3\delta^{2m} \overline{c}_{\alpha}}{2m |{\log \delta}|}, \qquad \frac{\underline{c}_{\,\alpha}}{8(m+1) |{\log \delta}|}<|h'_I|_{\widetilde a}< \frac{3\overline{c}_{\alpha}}{2m |{\log \delta}|}, \\ \frac{\underline{c}_{\,\alpha}}{8m^2 \delta^{2m} |{\log \delta}|}< |h''_{II}|_{\widetilde a}<\frac{3\overline{c}_{\alpha}}{2(m+1)^2 \delta^{2(m+1)} |{\log \delta}|}, \\ |h''_{yI}|_{\widetilde a}<\frac{2 c_{\alpha}+\overline{c}_{\alpha}}{m |{\log \delta}|}, \qquad |h''_{yy}|_{\widetilde a}<\frac{\delta^{2m}(4 c_{\alpha}+ \overline{c}_{\alpha})}{m |{\log \delta}|}\quad\text{and} \quad |h'_{y}|_{\widetilde a}<\frac{\delta^{2m}(2 c_{\alpha}+\overline{c}_{\alpha})}{m |{\log \delta}|}. \end{gathered}
\end{equation*}
\notag
$$
We set
$$
\begin{equation*}
\underline{c}_{\,h}=\frac{\delta^2 \underline{c}_{\,\alpha}}{16 |{\log \delta}|}\quad\text{and} \quad \overline{c}_h=\frac{4 c_{\alpha}+2 \overline{c}_{\alpha}}{\delta^2}.
\end{equation*}
\notag
$$
The proof is complete. 3.2. The proof of Lemma 4The first assertion of the lemma, that Lemma 3 holds for $\delta$ and $\varepsilon_1$ under consideration, follows from the proof of Lemma 3. Let us write out the determinant of $J$:
$$
\begin{equation*}
\det J=\det \bigl(\Lambda''_{yy}+h''_{yy}+\widehat h''_{yy} \bigr)(h''_{II}+\widehat h''_{II})+ \sum_{i=1}^{n-1} (-1)^{1+i} (h''_{y_iI}+\widehat h''_{y_iI}) M_i,
\end{equation*}
\notag
$$
where $M_i$ is the corresponding minor. Now we show that if we take $\delta$ and $\widehat c$ sufficiently small but independent of $\varepsilon$, then the lemma follows from (2.10), (2.11) and (2.14). The determinant of a sum of two matrices $\Lambda''_{yy}$ and $h''_{yy}+\widehat h''_{yy}$ is equal to the sum of the determinants of all matrices obtained by taking some rows (or columns) from $\Lambda''_{yy}$ and the others from $h''_{yy}+\widehat h_{(yy)}$:
$$
\begin{equation*}
\det \bigl(\Lambda''_{yy}+h''_{yy}+\widehat h''_{yy} \bigr)=\det \Lambda''_{yy}+\sum_{i= 2}^{(2(n-1))!/(n-1)!}B_{i},
\end{equation*}
\notag
$$
where $B_i$ is the determinant of the matrix formed by rows of $\Lambda''_{yy}$ and $h''_{yy}+ \widehat h''_{yy}$ and containing at least one row of the matrix $h''_{yy}+\widehat h''_{yy}$. Consequently,
$$
\begin{equation*}
\sum_{i=2}^{(2(n-1))!/(n-1)!} |B_i|_{\widetilde a}<\frac{(2(n-1))!}{(n-2)!} c_{\Lambda}^{n-2} \biggl(\frac{\delta^{2m}\overline{c}_h}{m}+\widehat c \biggr).
\end{equation*}
\notag
$$
For small $\delta$ and $\widehat c$ we have
$$
\begin{equation}
\frac{\underline{c}_{\,\Lambda}}{2}<\bigl|\det (\Lambda''_{yy}+h''_{yy}+\widehat h''_{yy} )\bigr|_{\widetilde a}<\frac{3 \underline{c}_{\,\Lambda}}{2}.
\end{equation}
\tag{3.9}
$$
For small $\widehat c$ we have
$$
\begin{equation}
\frac{\underline{c}_{\,h}}{2m^2 \delta^{2m}}<|h''_{II}+\widehat h''_{II}|_{\widetilde a} <\frac{3\overline{c}_{h}}{2m^2 \delta^{2m}},
\end{equation}
\tag{3.10}
$$
and therefore
$$
\begin{equation}
\frac{\underline{c}_{\,h}\, \underline{c}_{\,\Lambda}}{4m^2 \delta^{2m}} <\bigl|\det \bigl(\Lambda''_{yy}+h''_{yy}+\widehat h''_{yy} \bigr)(h''_{II}+\widehat h''_{II})\bigr|_{\widetilde a} <\frac{9\overline{c}_{h}\overline{c}_{\Lambda}}{4m^2 \delta^{2m}}.
\end{equation}
\tag{3.11}
$$
We find an estimate for $M_i$:
$$
\begin{equation}
\begin{gathered} \, \notag |M_i|_{\widetilde a}<(n-1)!\,c^{n-2}_{\Lambda} \biggl(\frac{\overline{c}_{h}}{m}+\frac{\widehat c}{m \delta^m}\biggr), \\ \biggl|\sum_{i=1}^{n-1} (-1)^{1+i} (h''_{y_iI}+\widehat h''_{y_i I}) M_i\biggr|_{\widetilde a} <n!\,c^{n-1}_{\Lambda} \biggl(\frac{\overline{c}_{h}}{m}+\frac{\widehat c}{m \delta^m}\biggr)^2. \end{gathered}
\end{equation}
\tag{3.12}
$$
For small $\delta$ and $\widehat c$ we have
$$
\begin{equation*}
n!\,c^{n-1}_{\Lambda} \biggl(\frac{\overline{c}_{h}}{m}+\frac{\widehat c}{m \delta^m}\biggr)^2< \frac{\underline{c}_{\,h}\, \underline{c}_{\,\Lambda}}{8m^2 \delta^{2m}}.
\end{equation*}
\notag
$$
Set $\underline{c}_{\,J}=\underline{c}_{\,h} \, \underline{c}_{\,\Lambda}/8$, $\overline{c}_J=3 \overline{c}_{h}\overline{c}_{\Lambda}$. The lemma is proved. AcknowledgementThe author is grateful to Dmitry Valer’evich Treschev for useful discussions and the comments he made.
Citation in format AMSBIB
\Bibitem{Med24} |
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