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Normalization flow in the presence of a resonance D. V. Treschev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
§ 1. IntroductionConsider a Hamiltonian system with $n$ degrees of freedom in a neighbourhood of an elliptic singular point. In the linear approximation, the dynamics is determined by the Hamiltonian equations
$$
\begin{equation*}
\begin{gathered} \, \dot x = \partial_y H_2, \quad \dot y = - \partial_x H_2, \qquad x = (x_1,\dots,x_n), \quad y = (y_1,\dots,y_n), \\ H_2 = \sum_{j=1}^n \frac{\omega_j}{2} (y_j^2 + x_j^2). \end{gathered}
\end{equation*}
\notag
$$
The real numbers $\omega_1,\dots,\omega_n$ form the frequency vector $\omega\in\mathbb{R}^n$. According to Birkhoff [1], it is convenient to use the complex coordinates
$$
\begin{equation*}
(z,\overline z) = (z_1,\dots,z_n,\overline z_1,\dots,\overline z_n) \in \mathbb{C}^{2n}, \qquad z_j = \frac{y_j+ix_j}{\sqrt 2}, \quad \overline z_j = \frac{y_j-ix_j}{\sqrt 2}.
\end{equation*}
\notag
$$
Below, all the phase variables including $x$ and $y$ may be complex. Hence the overbar does not mean necessarily complex conjugation. In the other words, the variables $z$ and $\overline z$ are assumed to be independent. The variables $(z,\overline z)$ are canonical, that is, for any two smooth functions $F = F(z,\overline z), G = G(z,\overline z)$, the Poisson bracket has the form
$$
\begin{equation*}
\{F,G\}=i\sum_{j=1}^n \bigl(\partial_{\overline z_j} F \, \partial_{z_j} G - \partial_{z_j} F\, \partial_{\overline z_j} G \bigr).
\end{equation*}
\notag
$$
The same equation determines the bracket $\{\,,\}$ on the space of formal power series in $z$ and $\overline z$. The function $H_2$ takes the form The Hamiltonian function is assumed to be $H_2 + \widehat H$,
$$
\begin{equation*}
\widehat H= \sum_{|k|+|\overline k|\geqslant 3} H_{k,\overline k} z^k\overline z^{\,\overline k}.
\end{equation*}
\notag
$$
Here, $k,\overline k\in\mathbb{Z}_+^n$, $\mathbb{Z}_+ :=\{0,1,\dots\}$ are multiindices and $|k| = |k_1|+\dots+|k_n|$ is the $l^1$-norm. Below, we use the shorter notation
$$
\begin{equation}
\mathbf{z} = (z,\overline z)\in\mathbb{C}^{2n}, \quad \mathbf{k} = (k,\overline k)\in\mathbb{Z}_+^{2n}, \quad \mathbf{z}^\mathbf{k} = z^k\overline z^{\,\overline k}, \qquad \widehat H = \sum_{|\mathbf{k}|\geqslant 3} \widehat H_\mathbf{k} \mathbf{z}^\mathbf{k} .
\end{equation}
\tag{1.1}
$$
The Hamiltonian equations are as follows:
$$
\begin{equation}
\dot z = i\, \partial_{\overline z} (H_2 + \widehat H), \qquad \dot{\overline z} = - i\, \partial_z (H_2 + \widehat H).
\end{equation}
\tag{1.2}
$$
More generally, $\dot F = \{\widehat H,F\}$ for any function (or formal power series) $F = F(\mathbf{z})$. According to the theory of normal forms, the function $\widehat H$ may be simplified by passage to another coordinate system, [1]. The monomial $\widehat H_\mathbf{k}\mathbf{z}^\mathbf{k}$ in expansion (1.1) is called resonant if $\langle\omega,\overline k - k\rangle = 0$, where $\langle\,,\rangle$ is the standard inner product in $\mathbb{R}^n$. The integer number $|\overline k - k|$ is called the order of the resonance. Any integer vector $\mathbf{k}=(k,\overline k)$ which determines a resonant monomial $\mathbf{z}^\mathbf{k}$ satisfies $\overline k - k \in\mathbb{L}_\omega$, where
$$
\begin{equation}
\mathbb{L}_\omega = \{q\in\mathbb{Z}^n \colon \langle\omega,q\rangle = 0 \}.
\end{equation}
\tag{1.3}
$$
If $\langle\omega,q\rangle = 0$, $q\in\mathbb{Z}^n$ implies $q=0$, the frequency vector $\omega$ is said to be non-resonant. By using an appropriate coordinate change, one gets rid of any finite set of non-resonant monomials. To eliminate all non-resonant monomials, we have to use coordinate changes in the form of, in general, divergent power series. The Hamiltonian function $N$ obtained as a result of this (formal) normalization is called the normal form of the original Hamiltonian $H_2+\widehat H$. The normal form is unique as a formal series. The problem of convergence/divergence of the normalizing transformation under the assumption of analyticity of $\widehat H$ is central in the theory. H. Eliasson attracted attention of specialists to another (harder) problem: convergence/divergence of the normal form. If the normalization converges and the lattice $\mathcal{L}_\omega$ is at most 1-dimensional, then the system is locally completely integrable. Various versions of the inverse statement are proved in [19], [7], [4], [8]. Another corollary from convergence of the normalization is Lyapunov stability of the equilibrium position in the case of a non-resonant frequency vector.1 The papers [5], [9] contain examples of real-analytic Hamiltonians $H_2 + \widehat H$ such that the origin is Lyapunov unstable in system (1.2) although in the linear approximation the system is obviously stable. Convergence of the normal form does not imply convergence of the normalization. But it has interesting dynamical consequences: the measure of the set covered by KAM-tori turns out to be noticeably bigger than in the case when the normal form diverges [10]. Convergence of the normalizing transformation is an exceptional phenomenon in any reasonable sense. At the moment, this exceptionality is known in terms of Baire category [14] and $\Gamma$-capacity [12], [10]. Explicit examples of real-analytic systems with divergent normal form can be found in [6], [20], [5]. The case of the “trivial” normal form $N = H_2$ is special. According to the Bruno–Rüssmann theorem [2], [13], see also [16], if $N = H_2$ and $\omega$ satisfies some (rather weak) Diophantine conditions,2 then the normalization converges. The normalizing change of variables is traditionally constructed as a composition of an infinite sequence of coordinate changes which normalize the Hamiltonian function up to a remainder of a higher and higher degree [1]. In further works (see for example [15]), this change of coordinates is represented as a formal series in $z$ and $\overline z$. We propose another approach to normalization. Let $\mathcal{F}$ be the space of all power series in the variables $z$ and $\overline z$. Sometimes, we refer to elements of $\mathcal{F}$ as functions although they are only formal power series. Let $\mathcal{F}_\diamond\subset\mathcal{F}$ be the subspace of series which start from terms of degree at least 3. In this paper, assuming that $H_0$ is fixed, we study various normalization flows $\phi_\xi^\delta$ on the space $\mathcal{F}_\diamond$, $\delta\in [0,+\infty)$. Any shift
$$
\begin{equation*}
H_2 + \widehat H \mapsto H_2 + \phi_\xi^\delta(\widehat H), \qquad \widehat H\in\mathcal{F}_\diamond,
\end{equation*}
\notag
$$
is a transformation of the Hamiltonian function $H_2 + \widehat H$ according to a certain (depending on $\delta$ and on the initial Hamiltonian $H_2 + \widehat H$) canonical change of variables. Any flow $\phi_\xi^\delta$ is determined by a certain ODE in $\mathcal{F}_\diamond$,
$$
\begin{equation}
\partial_\delta H = - \{\xi H, H_2 + H\}, \qquad H|_{\delta=0} = \widehat H.
\end{equation}
\tag{1.4}
$$
Here, $\xi$ is a linear operator on $\mathcal{F}_\diamond$. In fact, (1.4) is an initial value problem (IVP) for a differential equation presented in the form of Lax L-A pair. Such systems are usually considered in the theory of integrable systems. However, integrability of system (1.4) whatever it means seems to be of no use to us. Let $\mathcal{N}_\diamond\subset\mathcal{F}_\diamond$ be the subspace which consists of series which contain only resonant monomials. We will choose operators $\xi$ such that the space $\mathcal{N}_\diamond$ is invariant with respect to $\phi^\delta_\xi$ and any point of $\mathcal{N}_\diamond$ is fixed. We proposed such an approach in [18]. In this paper, we present further results. In particular, here we do not assume that $\omega$ is a non-resonant vector. In § 3.1, we define the operator $\xi=\xi_*$. We represent system (1.4) in the form of an infinite ODE system (3.9) for the coefficients $H_\mathbf{k}$. A more convenient equivalent form of it is system (3.11). Because of a special “nilpotent” structure of system (3.11), the existence and uniqueness of a solution for the corresponding IVP for any initial condition turns out to be a simple fact (§ 4.1). In the product topology, the solution tends to the normal form as $\delta\to +\infty$. We are particularly interested in the restriction of $\phi^\delta_{\xi_*}$ to the subspace $\mathcal{A}\subset\mathcal{F}$ of analytic Hamiltonian functions. In § 4.2, we prove that, for any $\widehat H\in\mathcal{A}$, the solution $H=H(\mathbf{z},\delta)$ also lies in $\mathcal{A}$ for any $\delta\geqslant 0$. However, the polydisc of analyticity generically shrinks when $\delta$ grows. A rough lower estimate for its (poly)radius gives a quantity of order $1/(1+\delta)$ (Theorem 4.2). In § 5, we assume that the normal form $H_2 + N_\diamond$ of the Hamiltonian $H_2\,{+}\,\widehat H$ satisfies the equation $N_\diamond = O_r(\mathbf{z})$, $r\geqslant 3$. For example, generically3 $r=4$. The case $r>4$ is not generic but may happen for some values of parameters in multiparametric families of Hamiltonians. Assuming that $\widehat H\in\mathcal{A}$, $\omega$ satisfies the Bruno Condition, and $N_\diamond=O_r(\mathbf{z})$, Theorem 5.1 says that $H(\,{\cdot}\,,\delta)$ stays in $\mathcal{A}$ and the polydisc of analyticity has radius of order (at least) $(1+\delta)^{-1/(r-2)}$, $\delta>0$. In particular, generically this radius $\sim (1+\delta)^{-1/2}$. This improves the corresponding estimate $(1+\delta)^{-1}$ in [18]. If $\omega$ is collinear to a vector with rational components (codimention one resonance) then Theorem 5.1 implies Corollary 5.1. This corollary says that there exists a change of variables which reduces in a polydisc of radius $\sim (1+\delta)^{-1/(r-2)}$ the Hamiltonian to the form $H_2 + G^0 + G^*$, where $G^0\in\mathcal{N}_\diamond$ and $|G^*|\sim e^{-c\delta}$, $c>0$. Theorem 6.1 from § 6 is an auxiliary statement used in the proof of Theorem 5.1. However, it probably has an independent value. It has the same assumptions: $\widehat H\in \mathcal{A}$, $\omega$ satisfies the Bruno Condition, $N_\diamond = O_r(\mathbf{z})$. Then there exists a change of the variables which reduces the Hamiltonian function to the form $H_2\,{+}\, G$, $G = O_r(\mathbf{z})$. Theorem 6.1 presents upper estimates for Taylor coefficients of the function $G$ in terms of analytic properties of $\widehat H$. As a corollary, we obtain the Bruno–Sigel theorem about convergence of the normalization in the case of the trivial normal form (Corollary 6.1). Some technical results concerning majorants, Bruno sequences, etc., are given in § 7. § 2. Basic constructionFor any $q = (q_1,\dots,q_n)\in\mathbb{Z}^n$, let $|q|=|q_1|+\dots+|q_n|$ be its $l^1$-norm. For any $k,\overline k\in\mathbb{Z}_+^n$ we put
$$
\begin{equation*}
\mathbf{k}=(k,\overline k)\in\mathbb{Z}_+^{2n},\qquad \mathbf{k}' = \overline k - k\in\mathbb{Z}^n , \qquad \mathbf{k}^* = (\overline k,k), \qquad |\mathbf{k}| = |k| + |\overline k|.
\end{equation*}
\notag
$$
Then $\mathbf{k}^*-\mathbf{k}=(\mathbf{k}',-\mathbf{k}')$. In particular, $\mathbf{k}'=0$ if and only if $\mathbf{k}=\mathbf{k}^*$. We will use the notation
$$
\begin{equation*}
\mathbb{Z}_\diamond^{2n} = \{\mathbf{k}\in\mathbb{Z}_+^{2n} \colon |\mathbf{k}|\geqslant 3\}.
\end{equation*}
\notag
$$
2.1. SpacesLet $\mathcal{F}$ be the vector space of all series
$$
\begin{equation}
H = \sum_{\mathbf{k}\in\mathbb{Z}_+^{2n}} H_{\mathbf{k}} \mathbf{z}^\mathbf{k}, \qquad H_{\mathbf{k}}\in\mathbb{C}.
\end{equation}
\tag{2.1}
$$
Series (2.1) are assumed to be formal, that is, there is no restriction on the values of the coefficients $H_{\mathbf{k}}$. So, $\mathcal{F}$ coincides with the ring $\mathbb{C}[[z_1,\dots,z_n,\overline z_1,\dots,\overline z_n]]$. The Poisson bracket defines on $\mathcal{F}$ the structure of a Lie algebra. Below, we use on $\mathcal{F}$ the product topology, that is, a sequence $H^{(1)},H^{(2)},\ldots\in\mathcal{F}$ is said to be convergent if, for any $\mathbf{k}\in\mathbb{Z}_+^{2n}$, the sequence of coefficients $H^{(1)}_{\mathbf{k}},H^{(2)}_{\mathbf{k}},\dots$ converges. For any $H$ satisfying (2.1), we define $p_{\mathbf{k}}(H) = H_{\mathbf{k}}$. So $p_{\mathbf{k}} \colon \mathcal{F}\to\mathbb{C}$ is a canonical projection corresponding to $\mathbf{k}\in\mathbb{Z}_+^{2n}$. Suppose $F\in\mathcal{F}$ depends on a parameter $\delta\in I$, where $I\subset\mathbb{R}$ is an interval. In other words, we consider a map
$$
\begin{equation*}
f \colon I\to\mathcal{F}, \quad I\ni\delta\mapsto F(\,{\cdot}\,,\delta).
\end{equation*}
\notag
$$
We say that $F$ is smooth in $\delta$ if all the maps $p_\mathbf{k}\circ f$ are smooth. Let $\mathcal{F}_r\subset\mathcal{F}$ be the space of “real” series,
$$
\begin{equation*}
\mathcal{F}_r = \{H\in\mathcal{F} \colon \overline H_{\mathbf{k}} = H_{\mathbf{k}^*} \text{ for any } \mathbf{k}\in\mathbb{Z}_+^n\}.
\end{equation*}
\notag
$$
We define $\mathcal{A}\subset\mathcal{F}$ as the space of analytic series:
$$
\begin{equation*}
\mathcal{A} = \{H\in\mathcal{F} \colon \text{there exist }c, a\text{ such that }|H_{\mathbf{k}}| \leqslant c e^{a|\mathbf{k}|}\text{ for any }\mathbf{k}\in\mathbb{Z}_+^{2n}\}.
\end{equation*}
\notag
$$
The product topology may be restricted from $\mathcal{F}$ to $\mathcal{A}$, but, being a scale of Banach spaces, $\mathcal{A}$ may be endowed with a more natural topology. We have $\mathcal{A} = \bigcup_{\rho>0}\mathcal{A}^\rho$, where $\mathcal{A}^\rho$ is a Banach space with the norm $\|\,{\cdot}\,\|_\rho$,
$$
\begin{equation}
\|H\|_\rho = \sup_{\mathbf{z}\in D_\rho} |H(\mathbf{z})|, \qquad D_\rho = \{\mathbf{z}\in\mathbb{C}^{2n} \colon |z_j|<\rho,\, |\overline z_j|<\rho,\, j=1,\dots,n\}.
\end{equation}
\tag{2.2}
$$
We have $\mathcal{A}^{\rho'}\subset\mathcal{A}^\rho$, $\|\,{\cdot}\,\|_\rho \leqslant \|\,{\cdot}\,\|_{\rho'}$ for any $0<\rho<\rho'$. Proof. The proof is standard. It follows from the Cauchy formula: for any positive $\rho_0<\rho$ and any $H\in\mathcal{A}^\rho$
$$
\begin{equation*}
H_{\mathbf{k}} = \frac1{(2\pi i)^{2n}} \oint dz_1 \dots \oint dz_n \oint d\overline z_1 \dots \oint \frac{H(\mathbf{z})\, d\overline z_n}{\mathbf{z}^{\mathbf{k}+\mathbf{1}}},
\end{equation*}
\notag
$$
where $\mathbf{1}=(1,\dots,1)\in\mathbb{Z}_+^{2n}$ and the integration is performed along the circles
$$
\begin{equation*}
|z_1|=\rho_0,\quad \dots,\quad |z_n|=\rho_0,\qquad |\overline z_1|=\rho_0,\quad \dots,\quad |\overline z_n|=\rho_0.
\end{equation*}
\notag
$$
This implies $|H_\mathbf{k}|\leqslant c\rho_0^{-|\mathbf{k}|}$. Since $\rho_0<\rho$ is arbitrary, we obtain (2.3). The lemma is proved. Corollary 2.1. The topology on $\mathcal{A}^\rho$ by the norm $\|\,{\cdot}\,\|_\rho$ is stronger than the product topology induced from $\mathcal{F}$, that is, if a sequence $\{H^{(j)}\}$ converges in $(\mathcal{A}^\rho, \|\,{\cdot}\,\|_\rho)$, then it converges in $\mathcal{F}$. We define the resonant sublattice (1.3). The number of generators in the group $(\mathbb{L}_\omega,+)$ is said to be the rank of $\mathbb{L}_\omega$. In particular, if $\operatorname{rank}\mathbb{L}_\omega=0$ (equivalently, $\mathbb{L}_\omega=\{0\}$), then the frequency vector is called non-resonant. If $\operatorname{rank}\mathbb{L}_\omega=1$, we have a simple resonance, if $\operatorname{rank}\mathbb{L}_\omega>1$, the resonance is multiple, if $\operatorname{rank}\mathbb{L}_\omega=n-1$, then we have a codimension one resonance. In the latter case, all trajectories of the linearized system
$$
\begin{equation*}
\dot z = i\, \partial_{\overline z} H_2, \qquad \dot {\overline z} = i\, \partial_z H_2
\end{equation*}
\notag
$$
are periodic. Let $\mathcal{N}\subset\mathcal{F}$ be the space of “normal forms”,
$$
\begin{equation*}
\mathcal{N} =\{H\in\mathcal{F} \colon H_{\mathbf{k}} \ne 0 \text{ implies } \mathbf{k}'\in\mathbb{L}_\omega\}.
\end{equation*}
\notag
$$
It is easy to check that $\mathcal{N}$ is a subalgebra in $\mathcal{F}$ with respect to the operations of multiplication and the Poisson bracket. We define the subspace (the subring) $\mathcal{F}_\diamond\subset\mathcal{F}$ of the series
$$
\begin{equation}
H = \sum_{\mathbf{k}\in\mathbb{Z}_\diamond^{2n}} H_{\mathbf{k}} \mathbf{z}^\mathbf{k}, \qquad H_{\mathbf{k}}\in\mathbb{C},
\end{equation}
\tag{2.4}
$$
that is $H=O_3(\mathbf{z})$ for any $H\in\mathcal{F}_\diamond$. By definition, we also put $\mathcal{N}_\diamond = \mathcal{N}\cap\mathcal{F}_\diamond$.
§ 3. Normalization flowConsider the change of variables in the form of a shift
$$
\begin{equation}
\mathbf{z}=(z,\overline z) \mapsto \mathbf{z}_\delta=(z_\delta,\overline z_\delta)
\end{equation}
\tag{3.1}
$$
along solutions of the Hamiltonian system4 with Hamiltonian $F=F(\mathbf{z},\delta)=O_3(\mathbf{z})$ and independent variable $\delta$,
$$
\begin{equation}
z' = i \, \partial_{\overline z} F, \quad \overline z^{\,\prime} = - i \, \partial_z F, \qquad (\,{\cdot}\,)' = \frac{d}{d\delta}.
\end{equation}
\tag{3.2}
$$
Suppose the function $H_2+\widehat H$ expressed in the variables $\mathbf{z}_\delta$ takes the form $H_2+H$,
$$
\begin{equation}
H_2(\mathbf{z}) + \widehat H(\mathbf{z}) = H_2(\mathbf{z}_\delta) + H(\mathbf{z}_\delta,\delta).
\end{equation}
\tag{3.3}
$$
Differentiating equation (3.3) with respect to $\delta$, we obtain
$$
\begin{equation*}
\partial_\delta H = -\{F,H_2+H\}, \qquad H|_{\delta=0} = \widehat H.
\end{equation*}
\notag
$$
The main idea of the continuous averaging is to take $F$ in the form $\xi H$, where $\xi$ is a certain operator5 depending on the problem we deal with. We now obtain an initial value problem in $\mathcal{F}_\diamond$,
$$
\begin{equation}
\partial_\delta H = - \{\xi H,H_2 + H\}, \qquad H|_{\delta=0} = \widehat H.
\end{equation}
\tag{3.4}
$$
3.1. Operator $\xi_*$To explain the idea, we start with the “simplest” operator $\xi=\xi_*$. We put
$$
\begin{equation}
\sigma_q = \operatorname{sign} (\langle\omega,q\rangle), \quad \omega_q = |\langle\omega,q\rangle|, \qquad q\in\mathbb{Z}^n.
\end{equation}
\tag{3.5}
$$
For any $H$ satisfying (2.4), we put
$$
\begin{equation}
\xi_* H = - i \sum_{\mathbf{k}\in\mathbb{Z}^{2n}_\diamond} \sigma_{\mathbf{k}'} H_{\mathbf{k}} \mathbf{z}^\mathbf{k} = i (H^- - H^+), \qquad H^\pm = \sum_{\pm\sigma_{\mathbf{k}'} > 0} H_{\mathbf{k}} \mathbf{z}^\mathbf{k}.
\end{equation}
\tag{3.6}
$$
We also set
$$
\begin{equation*}
H^0 = \sum_{\langle\omega,\mathbf{k}'\rangle = 0} H_{\mathbf{k}} \mathbf{z}^\mathbf{k}.
\end{equation*}
\notag
$$
So, $H = H^- + H^0 + H^+$. We thus have a more detailed form of the IVP (3.4)
$$
\begin{equation}
\begin{gathered} \, \partial_\delta H = v_0(H) + v_1(H) + v_2(H), \qquad H|_{\delta=0} = \widehat H, \\ \nonumber v_0 = - \{\xi_* H,H_2\}, \qquad v_1 = - \{\xi_* H,H^0\}, \qquad v_2 = - \{\xi_* H,H^- + H^+\}. \end{gathered}
\end{equation}
\tag{3.7}
$$
Informal explanation for the choice (3.6) of the operator $\xi=\xi_*$ is as follows. Removing the non-linear terms $v_1+v_2$ in (3.7), we obtain the equation $\partial_\delta H = v_0(H)$ or, in more detail
$$
\begin{equation}
\partial_\delta H_{\mathbf{k}} = -\omega_{\mathbf{k}'} H_{\mathbf{k}}, \qquad H_{\mathbf{k}}|_{\delta=0} = \widehat H_{\mathbf{k}}
\end{equation}
\tag{3.8}
$$
which can be easily solved,
$$
\begin{equation*}
H_{\mathbf{k}} = e^{-\omega_{\mathbf{k}'}\delta} \widehat H_{\mathbf{k}} .
\end{equation*}
\notag
$$
So, the solution $H$ of the truncated problem (3.8) tends to $H^0\in\mathcal{N}_\diamond$ as $\delta\to +\infty$. Let $e_j=(0,\dots,1,\dots,0)$ be the $j$th unit vector in $\mathbb{Z}_+^n$ and let $\mathbf{e}_j=(e_j,e_j)\in\mathbb{Z}_+^{2n}$. Equation (3.7) written for each Taylor coefficient $H_{\mathbf{k}}$ has the form
$$
\begin{equation}
\begin{aligned} \, \partial_\delta H_{\mathbf{k}}&= -\omega_{\mathbf{k}'} H_{\mathbf{k}} + v_{1,\mathbf{k}}(H) + v_{2,\mathbf{k}}(H),\qquad H_\mathbf{k}|_{\delta=0} = \widehat H_\mathbf{k}, \end{aligned}
\end{equation}
\tag{3.9}
$$
$$
\begin{equation}
\begin{aligned} \, v_{1,\mathbf{k}}&= - \sum_{j=1}^n \, \sum_{\sigma_{\mathbf{m}'}=0,\, \mathbf{l}+\mathbf{m}-\mathbf{k}=\mathbf{e}_j} \sigma_{\mathbf{k}'} (\overline l_j m_j - l_j\overline m_j) H_{\mathbf{l}} H_{\mathbf{m}}, \\ \nonumber v_{2,\mathbf{k}} &= 2 \sum_{j=1}^n \, \sum_{\sigma_{\mathbf{l}'} < 0 < \sigma_{\mathbf{m}'},\, \mathbf{l}+\mathbf{m}-\mathbf{k} = \mathbf{e}_j} (\overline l_j m_j - l_j\overline m_j) H_{\mathbf{l}} H_{\mathbf{m}}. \end{aligned}
\end{equation}
\tag{3.10}
$$
By using the change of variables
$$
\begin{equation*}
H_{\mathbf{k}} = \mathcal{H}_{\mathbf{k}} e^{-\omega_{\mathbf{k}'} \delta},
\end{equation*}
\notag
$$
we reduce equations (3.9) to the form
$$
\begin{equation}
\begin{aligned} \, \partial_\delta\mathcal{H}_{\mathbf{k}} &= v_{1,\mathbf{k}}(\mathcal{H}) + \mathbf{v}_{2,\mathbf{k}}(\mathcal{H}), \qquad \mathcal{H}_{\mathbf{k}}|_{\delta=0} = \widehat H_{\mathbf{k}}, \end{aligned}
\end{equation}
\tag{3.11}
$$
$$
\begin{equation}
\begin{aligned} \, \mathbf{v}_{2,\mathbf{k}} &= 2 \sum_{j=1}^n \, \sum_{\sigma_{\mathbf{l}'} < 0 < \sigma_{\mathbf{m}'},\, \mathbf{l}+\mathbf{m}-\mathbf{k} = \mathbf{e}_j} (\overline l_j m_j - l_j\overline m_j)\mathcal{H}_{\mathbf{l}}\mathcal{H}_{\mathbf{m}} e^{- \omega_{\mathbf{l}',\mathbf{m}'} \delta}, \\ \nonumber \omega_{\mathbf{l}',\mathbf{m}'} &= \omega_{\mathbf{l}'}+\omega_{\mathbf{m}'}-\omega_{\mathbf{l}'+\mathbf{m}'}. \end{aligned}
\end{equation}
\tag{3.12}
$$
Remark 3.1. (1) The functions $v_{1,\mathbf{k}}$ and $v_{2,\mathbf{k}}$ are quadratic polynomials in the variables $H_{\mathbf{k}}$. The functions $\mathbf{v}_{2,\mathbf{k}}$ are quadratic polynomials in $\mathcal{H}_{\mathbf{k}}$ with coefficients depending on $\delta$. (2) The polynomial $v_{1,\mathbf{k}}$ vanishes if $\mathbf{k}'\in\mathbb{L}_\omega$. (3) The polynomials $v_{1,\mathbf{k}}, v_{2,\mathbf{k}}$ and $\mathbf{v}_{2,\mathbf{k}}$ do not depend on the variables $H_\mathbf{s},\mathcal{H}_{\mathbf{s}}$, for any $\mathbf{s}\in\mathbb{Z}_\diamond^{2n}$ such that $|\mathbf{s}| > |\mathbf{k}|-2$. (4) The polynomials $v_{2,\mathbf{k}}$ and $\mathbf{v}_{2,\mathbf{k}}$ do not depend on the variables $H_\mathbf{s},\mathcal{H}_{\mathbf{s}}$ for any $\mathbf{s}\in\mathbb{Z}_\diamond^{2n}$ such that $0\leqslant\sigma_{\mathbf{k}'} \langle\omega,\mathbf{s}'\rangle\leqslant\omega_{\mathbf{k}'}$. (5) For any $\sigma_{\mathbf{l}'} < 0 < \sigma_{\mathbf{m}'}$, the quantity $\omega_{\mathbf{l}',\mathbf{m}'}$ is positive, § 4. Resonant normalization exists4.1. Formal aspectLet $I\subset\mathbb{R}$ be an interval containing the point $0$. We say that the path $\gamma\colon I\to\mathcal{F}_\diamond$ is a solution of system (3.4) on the interval $I$ if $\gamma(0)=\widehat H$ and the coefficients $H_{\mathbf{k}}(\delta) = p_{\mathbf{k}}\gamma(\delta)$ satisfy (3.9). If the solution $\gamma \colon I\to\mathcal{F}_\diamond$ exists and is unique, then, for any $\delta\in I$, we put $\gamma(\delta)=\phi^\delta_{\xi_*}(\widehat H)$. Definition 4.1. The ordinary differential equation on $\mathcal{F}$ has a nilpotent form if $F'_\mathbf{k}=\Phi_\mathbf{k}$ for any $\mathbf{k}\in\mathbb{Z}_\diamond^{2n}$, where $\Phi_\mathbf{k} = p_\mathbf{k}\circ\Phi$ is a function of the variables $F_\mathbf{m}$, $|\mathbf{m}|<|\mathbf{k}|$. According to Remark 3.1, (3), system (3.11) has a nilpotent form. In particular, $\partial_\delta\mathcal{H}_{\mathbf{k}} = 0$ for $|\mathbf{k}|=3$. Theorem 4.1. For any $\widehat H\,{\in}\,\mathcal{F}_\diamond$ and any $\delta\,{\in}\,\mathbb{R}$, the element $H(\,{\cdot}\,,\delta)\,{=}\,\phi^\delta_{\xi_*}(\widehat H)\,{\in}\,\mathcal{F}$ is well-defined. For any $\mathbf{k}\in\mathbb{Z}_\diamond^{2n}$, the function $\mathcal{H}_\mathbf{k}(\delta) = H_\mathbf{k}(\delta) e^{\omega_{\mathbf{k}'}\delta}$ equals where $P_\mathbf{k}$ is a polynomial in $\widehat H_\mathbf{m}$, $|\mathbf{m}|<|\mathbf{k}|$ with coefficients in the form of (finite) linear combinations of terms $\delta^s e^{-\nu\delta}$, $s\in\mathbb{Z}_+$. Here, $\nu\geqslant 0$, and, moreover, if the term $\delta^s e^{-\nu\delta}$ with $\nu=0$ appears in a coefficient of $P_\mathbf{k}$, $\mathbf{k}'\in\mathbb{L}_\omega$, then $s=0$ in this term. Proof. For $|\mathbf{k}|=3$, we have $\mathcal{H}_\mathbf{k}(\delta)=\widehat H_\mathbf{k}$. Suppose equations (4.1) hold for all vectors $\mathbf{k}\in\mathbb{Z}_\diamond^{2n}$ with $|\mathbf{k}|<K$. We take any $\mathbf{k}\in\mathbb{Z}^{2n}_\diamond$ with $|\mathbf{k}|=K$. By (3.11)
$$
\begin{equation*}
\mathcal{H}_\mathbf{k}=\widehat H_\mathbf{k} + I_1 + I_2, \qquad I_1 = \int_0^\delta v_{1,\mathbf{k}}(\mathcal{H}(\lambda))\, d\lambda, \quad I_2 = \int_0^\delta \mathbf{v}_{2,\mathbf{k}}(\mathcal{H}(\lambda),\lambda)\, d\lambda.
\end{equation*}
\notag
$$
By using (3.10), (3.12), and the induction assumption, we obtain (4.1).
If $\mathbf{k}'\in\mathbb{L}_\omega$, then $v_{1,\mathbf{k}}=0$, while each term in $\mathbf{v}_{2,\mathbf{k}}$ has the form $\lambda^s e^{-\lambda\delta}$ with $\lambda>0$. This implies the last statement of Theorem 4.1. The theorem is proved. Corollary 4.1. The limit $\lim_{\delta\to +\infty} \phi^\delta(\widehat H)$ exists in the product topology on $\mathcal{F}_\diamond$ and lies in $\mathcal{N}_\diamond$. Indeed, by Theorem 4.1 for any $\mathbf{k}\in\mathbb{Z}_\diamond^{2n}$, the limit
$$
\begin{equation*}
\lim_{\delta\to +\infty} H_{\mathbf{k}}(\delta) = H_{\mathbf{k}}(+\infty)
\end{equation*}
\notag
$$
exists. The convergence is exponential and $H_{\mathbf{k}}(+\infty)$ vanishes if $\mathbf{k}'\not\in\mathbb{L}_\omega$. This is equivalent to the existence of $\lim_{\delta\to +\infty}\phi^\delta_{\xi_*}(\widehat H)=\phi^{+\infty}_{\xi_*}(\widehat H)$ together with the fact that $\phi^{+\infty}_{\xi_*}(\widehat H)\in\mathcal{N}_\diamond$. Corollary 4.2. Suppose $\widehat H\in\mathcal{F}_r\cap\mathcal{F}_\diamond$. Then $\phi^\delta_{\xi_*}(\widehat H)\in\mathcal{F}_r\cap\mathcal{F}_\diamond$ for any $\delta\geqslant 0$. Indeed, the required identities $\overline H_{\mathbf{k}}(\delta) = H_{\mathbf{k}^*}(\delta)$, $\delta\geqslant 0$, can be proved by induction on $|\mathbf{k}|$ by using equation (3.11). 4.2. Analytic aspectTheorem 4.2. Suppose $\widehat H\in\mathcal{A}^\rho\cap\mathcal{F}_\diamond$, $\|\widehat H\|_\rho=h\rho^3$. Then, for any $\delta\geqslant 0$, where for some constants $A,B>0$ depending on $\rho$, $h$, and $n$. More precisely, Proof. We use the majorant method. We recall the definitions and basic facts concerning majorants in § sec:maj. We have The function $f(\zeta)=O_3(\zeta)$, a majorant for the initial condition, will be chosen later.
Consider together with (3.11) the majorant system
$$
\begin{equation}
\partial_\delta\mathbf{H}_{\mathbf{k}} = \mathbf{V}_{\mathbf{k}}, \qquad \mathbf{H}_{\mathbf{k}}|_{\delta=0} = \widehat{\mathbf{H}}_{\mathbf{k}},
\end{equation}
\tag{4.3}
$$
$$
\begin{equation}
\mathbf{V}_{\mathbf{k}} = 2 \sum_{j=1}^n \, \sum_{\mathbf{l}+\mathbf{m}-\mathbf{k} = \mathbf{e}_j} (\overline l_j m_j + l_j\overline m_j)\mathbf{H}_{\mathbf{l}}\mathbf{H}_{\mathbf{m}}.
\end{equation}
\tag{4.4}
$$
To obtain equation (4.4), we have replaced in $v_{1,\mathbf{k}}$ and $\mathbf{v}_{2,\mathbf{k}}$ the minuses by pluses, dropped the exponential multipliers, and added some new positive (for positive $\mathbf{H}_{\mathbf{l}}$ and $\mathbf{H}_{\mathbf{m}}$) terms. The main property of system (4.3), (4.4) is such that if, $|\widehat H_{\mathbf{k}}|\leqslant\widehat{\mathbf{H}}_{\mathbf{k}}$ for any $\mathbf{k}\in\mathbb{Z}_\diamond^{2n}$, then, for any $\delta\geqslant 0$, the inequality $|v_{1,\mathbf{k}} + \mathbf{v}_{2,\mathbf{k}}| \leqslant \mathbf{V}_\mathbf{k}$ holds, that is, conditions (a) and (b) from Definition 7.2 hold.
System (4.3), (4.4) has a nilpotent form. Therefore, by Theorem 7.1 we may use the majorant principle: if $\mathbf{H}$ is a solution of (4.3), (4.4), then (3.11) has a solution $\mathcal{H}$ and $\mathcal{H}\ll\mathbf{H}$ for any $\delta\geqslant 0$. System (4.3), (4.4) can be written in a shorter form as
$$
\begin{equation}
\partial_\delta\mathbf{H} = 4\sum_{j=1}^n \partial_{z_j}\mathbf{H}\, \partial_{\overline z_j}\mathbf{H}, \qquad \mathbf{H}|_{\delta=0} = f(\zeta).
\end{equation}
\tag{4.5}
$$
Since the initial condition $\mathbf{H}|_{\delta=0}$ depends on $z$ and $\overline z$ only through the variable $\zeta$, we may look for a solution of (4.5) in the form $\mathbf{H}(z,\overline z,\delta)=F(\zeta,\delta)$. The function $F$ satisfies the equation
$$
\begin{equation*}
\partial_\delta F = 4n (\partial_\zeta F)^2, \qquad F|_{\delta=0} = f(\zeta).
\end{equation*}
\notag
$$
The function $G=\partial_\zeta F$ satisfies the inviscid Burgers equation
$$
\begin{equation}
\partial_\delta G = 8n G \, \partial_\zeta G, \qquad G|_{\delta=0} = \partial_\zeta f(\zeta) = O_2(\zeta).
\end{equation}
\tag{4.6}
$$
By using the method of characteristics, we find that the function $G=G(x,t)$, which solves (4.6), satisfies the equation By Lemma 7.3, we can take $f(\zeta) = h\rho\zeta^3 / (\rho-\zeta)$. Now by Lemma 7.4,
$$
\begin{equation*}
f'(\zeta) \ll \frac{a\zeta^2}{b-\zeta}, \qquad a= 2h\rho, \quad b= \frac{\rho}2.
\end{equation*}
\notag
$$
Putting $\tau=8n\delta$, we obtain from (4.7) This is a quadratic equation with respect to $G$. The solution is
$$
\begin{equation*}
G = \frac{2a\zeta^2}{b-\zeta-2a\tau\zeta + \sqrt{(b-\zeta-2a\tau\zeta)^2 - 4a\tau\zeta^2(1+a\tau)}}.
\end{equation*}
\notag
$$
It is analytic for
$$
\begin{equation*}
|\zeta| < \frac{b}{1+2a\tau + 2\sqrt{a\tau(1+a\tau)}} .
\end{equation*}
\notag
$$
If then This implies that
$$
\begin{equation*}
|\mathcal{H}| \leqslant |F| \leqslant \frac{ab^2}{2(1 + 2a\tau)^3}\quad \text{in the domain } \ \biggl\{|\mathbf{z}| \leqslant \frac{b}{2(1+2a\tau)} \biggr\}.
\end{equation*}
\notag
$$
Theorem 4.2 is proved. Corollary 4.3. Suppose $\widehat H\in\mathcal{A}\cap\mathcal{F}_r$. Then $\phi^\delta_{\xi_*}(\widehat H)\in\mathcal{A}\cap\mathcal{F}_r$ for any $\delta\geqslant 0$. Note that, for a typical $\widehat H\in\mathcal{A}\cap\mathcal{F}_\diamond$, one should expect that $\phi^{+\infty}_{\xi_*}(\widehat H)\in\mathcal{N}_\diamond$ does not belong to $\mathcal{A}$ [10]. § 5. Degenerate normal form5.1. Normalization theoremSuppose the normal form $H_2 + N_\diamond$ is such that $N_\diamond := \lim_{\delta\to +\infty} \phi_{\xi_*}^\delta (\widehat H) \in \mathcal{N}$ vanishes in several first orders in $\mathbf{z}$. In this section, we obtain more precise estimates (in comparison with those from Theorem 4.2) for the analyticity domain of the Hamiltonian function $\mathcal{H}(\,{\cdot}\,,\delta)$ with increasing $\delta$. For any $j\in\mathbb{N}$, we put
$$
\begin{equation}
\Omega_s = \max \biggl\{ \frac1{|\langle\omega,q\rangle|} \colon q\in\mathbb{Z}^n\setminus\mathbb{L}_\omega, \, |q|\leqslant s \biggr\}.
\end{equation}
\tag{5.1}
$$
Definition 5.1. We say that $\{a_j\}_{j\in\mathbb{Z}_+}$, $a_j\geqslant 1$, is a Bruno sequence if $\{a_j\}$ is monotone non-decreasing and Obviously, if $\{a_j\}$ and $\{b_j\}$ are two Bruno sequences, then the product $\{a_j b_j\}$ is also a Bruno sequence. Definition 5.2. We say that the vector $\omega\in\mathbb{R}^n$ satisfies the Bruno condition if $\{a_j\}$, $a_j=\max\{1,\Omega_{2^j+2}\}$ is a Bruno sequence. Theorem 5.1. Suppose $\widehat H\in\mathcal{A}^\rho\cap\mathcal{F}_\diamond$, $\omega$ satisfies the Bruno condition, and $N_\diamond = O_r(\mathbf{z})$. Then, for any $\delta\geqslant 0$, the solution $\mathcal{H}(\mathbf{z},\delta) = \sum \mathcal{H}_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}$ of (3.11) belongs to the space $\mathcal{A}^{g(\delta)}$, where for some constant $C$ depending on $\rho,\|\widehat H\|_\rho,\{\Omega_j\}$, and $n$. More precisely, for some $C_H = C_H(\rho,\|\widehat H\|_\rho,\{\Omega_j\},n)$ and $C_z = C_z(\rho,\|\widehat H\|_\rho,\{\Omega_j\},n)$, In particular, if $r=3$, Theorem 5.1 essentially coincides with Theorem 4.2. If $r\geqslant 4$, (5.2) improves estimate (4.2). The proof of Theorem 5.1 is presented in § 6. 5.2. Resonance of codimension oneSuppose that $\operatorname{rank}\mathbb{L}_\omega = n-1$. Then there exists a unique collection of integers $q_1,\dots,q_n,p$, and $\lambda>0$ such that
$$
\begin{equation*}
p>0, \qquad \operatorname{GCD}(|q_1|,\dots,|q_n|,p) = 1, \quad \text{and}\quad \omega = \frac{\lambda q}{p}.
\end{equation*}
\notag
$$
Hence in (3.12) we have $\omega_{\mathbf{l}',\mathbf{m}'} \geqslant \lambda / p$. If the normal form of $H_2+\widehat H$ equals $H_2 + O_r(\mathbf{z})$, we may use Theorem 5.1 to obtain the following corollary. Corollary 5.1. Suppose $\operatorname{rank}\mathbb{L}_\omega = n-1$, $\widehat H\in\mathcal{A}^\rho\cap\mathcal{F}_\diamond$, and the normal form of $H_2+\widehat H$ is of order $O_r(\mathbf{z})$. Then there exists a canonical change of variables $\mathbf{w}\mapsto\mathbf{z}$ which transforms the Hamiltonian to $H_2(\mathbf{w})+G(\mathbf{w})$, $G=O_r(\mathbf{w})$, where and in the domain the following estimate holds:Estimates (5.5) mean that the non-resonant part $G^*$ may be reduced to an exponentially small quantity of order $e^{-\lambda\delta/p}$. The price is a little size ($\sim \delta^{-1/(r-2)}$) of the domain in which the Hamiltonian function $G$ is defined. To obtain estimates (5.5) in the domain (5.4), we split the function $\mathcal{H}(\mathbf{z},\delta)$, obtained in Theorem 5.1 into the resonant, $\mathcal{H}^0$, and non-resonant, $\mathcal{H}^*$, parts
$$
\begin{equation*}
\mathcal{H}^0 = \sum_{|\mathbf{k}|\geqslant 3,\, \mathbf{k}'\in\mathbb{L}_\omega} \mathcal{H}_\mathbf{k} \mathbf{z}^\mathbf{k}, \qquad \mathcal{H}^* = \sum_{|\mathbf{k}|\geqslant 3,\, \mathbf{k}'\in\mathbb{Z}^n\setminus\mathbb{L}_\omega} \mathcal{H}_\mathbf{k} \mathbf{z}^\mathbf{k} .
\end{equation*}
\notag
$$
The corresponding function $H(\mathbf{z},\delta)$ equals $G^0 + G^*$,
$$
\begin{equation*}
G^0 = \mathcal{H}^0, \qquad G^* = \sum_{|\mathbf{k}|\geqslant 3,\, \mathbf{k}'\in\mathbb{Z}^n\setminus\mathbb{L}_\omega} \mathcal{H}_\mathbf{k} e^{-\omega_{\mathbf{k}'}\delta} \mathbf{z}^\mathbf{k} .
\end{equation*}
\notag
$$
Estimates (5.5) follow from (5.3) and the inequality $\omega_{\mathbf{k}'}\geqslant\lambda/p$ for any $\mathbf{k}'\in\mathbb{Z}^n\setminus\mathbb{L}_\omega$.
§ 6. Proof of Theorem 5.16.1. Inductive stepDefinition. The sequence $\{b_j\}_{j\in\mathbb{N}}$ is said to be sublinear if $\lim_{j\to+\infty} b_j / j \,{=}\, 0$. The sequence $\{b_j\}_{j\in\mathbb{N}}$ is said to be convex if $b_{j-1} - 2b_j + b_{j+1} \geqslant 0$ for all $j\geqslant 2$. Lemma 6.1. Suppose $\widehat H\in\mathcal{A}\cap\mathcal{F}_\diamond$ is such that (1) $\widehat H = \sum_{|\mathbf{k}|\geqslant s} \widehat H_\mathbf{k}\mathbf{z}^\mathbf{k} = O_s(\mathbf{z})$, $|\widehat H_\mathbf{k}| \leqslant c e^{b_{|\mathbf{k}|} + \alpha |\mathbf{k}|}$, $\alpha\geqslant 0$, $s\geqslant 3$, (2) the sequence $\{b_j\}$ is non-increasing, sublinear and convex.6 Then an analytic canonical transformation $\mathbf{w}\mapsto\mathbf{z}=\vartheta(\mathbf{w})$ of coordinates reduces the Hamiltonian $H_2 + \widehat H$ to the form $H_2\circ\vartheta+\widehat H\circ\vartheta = H_2 + G^0 + G$, where
$$
\begin{equation}
G^0 = \sum_{s\leqslant |\mathbf{k}| \leqslant 2s-3,\,\langle\omega,\mathbf{k}'\rangle=0} \widehat H_\mathbf{k} \mathbf{w}^\mathbf{k} \in \mathcal{N}_\diamond,
\end{equation}
\tag{6.1}
$$
$$
\begin{equation}
G = \sum_{|\mathbf{k}|\geqslant 2s-2} G_\mathbf{k}\mathbf{w}^\mathbf{k}, \qquad |G_\mathbf{k}| \leqslant c e^{b_{|\mathbf{k}|} + (\alpha+\varepsilon) |\mathbf{k}|},
\end{equation}
\tag{6.2}
$$
Moreover, for any $0 < \rho\leqslant e^{-\alpha}$ and $\rho'$ satisfying we have $\vartheta\colon D_{\rho'}\to D_\rho$. Proof. We construct the transformation $\vartheta$ by using continuous averaging with the following operator $\xi_s$. Suppose $H_2(\mathbf{z}_\delta) + H(\mathbf{z}_\delta,\delta)$,
$$
\begin{equation*}
H(\mathbf{z},\delta) = \sum_{|\mathbf{k}|\geqslant s} H_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}
\end{equation*}
\notag
$$
is the Hamiltonian function obtained as a result of the coordinate transformation $\mathbf{z}_\delta\mapsto\mathbf{z}$. We define
$$
\begin{equation*}
\xi_s H(\mathbf{z},\delta) = -i \sum_{\mathbf{k}\in\mathcal{Z}_s} \sigma_{\mathbf{k}'} H_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}, \qquad \mathcal{Z}_s = \{\mathbf{k}\in\mathbb{Z}_\diamond^{2n} \colon s\leqslant |\mathbf{k}|\leqslant 2s-3\},
\end{equation*}
\notag
$$
where $\sigma_{\mathbf{k}'}$ is defined by (3.5). We have $H = H^- + H^+ + H^0 + H^*$,
$$
\begin{equation*}
\begin{gathered} \, H^\pm = \sum_{\mathbf{k}\in\mathcal{Z}_s^\pm} H_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}, \qquad H^0 = \sum_{\mathbf{k}\in\mathcal{Z}_s^0} H_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}, \qquad H^* = \sum_{|\mathbf{k}|\geqslant 2s-2} H_\mathbf{k}(\delta) \mathbf{z}^\mathbf{k}, \\ \mathcal{Z}_s^\pm = \{\mathbf{k}\in\mathcal{Z}_s \colon \pm\sigma_{\mathbf{k}'} > 0\}, \qquad \mathcal{Z}_s^0 = \{\mathbf{k}\in\mathcal{Z}_s \colon \sigma_{\mathbf{k}'} = 0\}. \end{gathered}
\end{equation*}
\notag
$$
By definition $\xi_s H = i(H^- - H^+)$, equation (3.4) takes the form Here, we used the fact that, for any two monomials of degree at least $s$, their Poisson bracket has degree at least $2s-2$.
We present system (6.5), (6.6) in a more detailed form
$$
\begin{equation}
\partial_\delta H_\mathbf{k} = - \omega_{\mathbf{k}'} H_\mathbf{k} \quad \text{for } \ \mathbf{k}\in\mathcal{Z}_s,
\end{equation}
\tag{6.7}
$$
$$
\begin{equation}
\partial_\delta H_\mathbf{k} = - i \{H^- - H^+, H^- + H^+ + H^0 + H^*\}_\mathbf{k} \quad \text{for } \ |\mathbf{k}| \geqslant 2s-2,
\end{equation}
\tag{6.8}
$$
Here, $\omega_{\mathbf{k}'}$ is defined by (3.5), and $\{\,,\}_\mathbf{k}$ denotes the Taylor coefficient with the number $\mathbf{k}$ in the expansion of the bracket. We solve the IVP (6.7)–(6.9) on the interval $\delta\in [0,+\infty)$. The required change of variables is obtained with $\delta=+\infty$.
Equations (6.7) can be easily solved: $H_\mathbf{k} = e^{-\omega_{\mathbf{k}'}\delta} \widehat H_\mathbf{k}$ for $\mathbf{k}\in\mathcal{Z}_s$. As a result, we get (6.1). Now we associate with system (6.7), (6.8) a majorant system for the functions
$$
\begin{equation*}
\begin{aligned} \, \mathbf{H}^\pm &= \sum_{\mathbf{k}\in\mathcal{Z}_s^\pm} \mathbf{H}_\mathbf{k} \mathbf{z}^\mathbf{k} \gg H^\pm, \\ \mathbf{H}^0 &= \sum_{\mathbf{k}\in\mathcal{Z}_s^0} \mathbf{H}_\mathbf{k} \mathbf{z}^\mathbf{k} \gg H^0 \equiv \widehat H^0, \\ \mathbf{H}^* &= \sum_{|\mathbf{k}|\geqslant 2s-2} \mathbf{H}_\mathbf{k} \mathbf{z}^\mathbf{k} \gg H^*. \end{aligned}
\end{equation*}
\notag
$$
By condition (1) of Lemma 6.1, we may choose
$$
\begin{equation*}
\begin{aligned} \, \widehat{\mathbf{H}}_\mathbf{k} &= \mathbf{H}_\mathbf{k} |_{\delta=0} = c e^{b_{|\mathbf{k}|} + \alpha |\mathbf{k}|}, \qquad |\mathbf{k}| \geqslant s, \\ \mathbf{H}_\mathbf{k}(\delta) &= e^{-\delta/\Omega_{2s-2}} c e^{b_{|\mathbf{k}|} + \alpha |\mathbf{k}|}, \qquad \mathbf{k}\in\mathcal{Z}_s^- \cup \mathcal{Z}_s^+ , \quad \delta\geqslant 0. \end{aligned}
\end{equation*}
\notag
$$
We take
$$
\begin{equation}
\begin{gathered} \, \partial_\delta\mathbf{H}_\mathbf{k} \geqslant e^{-\delta / \Omega_{2s-2}} \Sigma_1(\mathbf{k}), \qquad |\mathbf{k}| \geqslant 2s-2, \\ \nonumber \Sigma_1(\mathbf{k}) = \sum_{j=1}^n\, \sum_{\mathbf{l}\in\mathcal{Z}_s^-\cup\mathcal{Z}_s^+,\, |\mathbf{m}|\geqslant s,\, \mathbf{l}+\mathbf{m}=\mathbf{k}+\mathbf{e}_j} (\overline l_j m_j + l_j\overline m_j) \widehat{\mathbf{H}}_\mathbf{l} \mathbf{H}_\mathbf{m} \end{gathered}
\end{equation}
\tag{6.10}
$$
as a majorant system for (6.8).
For any $|\mathbf{k}|\geqslant 2s-2$, we put $\mathbf{H}_\mathbf{k} = c\, e^{b_{|\mathbf{k}|} + \alpha |\mathbf{k}|} \mathbf{h}_\mathbf{k}$, $\mathbf{h}_\mathbf{k}(\delta) \geqslant \mathbf{h}_\mathbf{k}(0) = 1$. We use the notation Lemma 6.2. Suppose the sequence $\{b_j\}_{j\in\mathbb{N}}$ is non-increasing and convex. Then, for any $\mathbf{k}\in\mathbb{Z}_+^{2n}$, $|\mathbf{k}|\geqslant 2s-2$, We prove Lemma 6.2 at the end of this section. By Lemma 6.2, the differential inequalities (6.10) follow from (and may be replaced by) the system
$$
\begin{equation*}
\partial_\delta \mathbf{h}_\mathbf{k} = c |\mathbf{k}| e^{-\delta / \Omega_{2s-2}} \Lambda \chi_{|\mathbf{k}|}, \qquad \mathbf{h}_\mathbf{k}(0) = 1.
\end{equation*}
\notag
$$
Since the multiplier $c e^{-\delta / \Omega_{2s-2}} \Lambda$ does not depend on $\mathbf{k}$, we conclude7 that $\chi_{|\mathbf{k}|} = \mathbf{h}_\mathbf{k}$ and we may take
$$
\begin{equation*}
\mathbf{h}_\mathbf{k}(\delta) = \exp\biggl( c |\mathbf{k}| \Lambda \int_0^\delta e^{-\widetilde\delta / \Omega_{2s-2}}\, d\widetilde\delta \biggr).
\end{equation*}
\notag
$$
In particular, $\mathbf{h}_\mathbf{k} = \mathbf{h}_\mathbf{k}(\delta)$ is an increasing function of $\delta\in [0,+\infty)$. Taking $\delta = +\infty$, we obtain
$$
\begin{equation*}
\mathbf{h}_\mathbf{k}(+\infty) = \exp( c \Lambda \Omega_{2s-2} |\mathbf{k}|).
\end{equation*}
\notag
$$
This implies that we may use $\varepsilon$ given by (6.3) in (6.2). Let $\vartheta^{-1}_\delta$ be the $\delta$-shift along solutions of the Hamiltonian system
$$
\begin{equation*}
\dot z_j = i\, \partial_{\overline z} \xi_s H, \quad \dot{\overline z_j} = -i\, \partial_z \xi_s H, \qquad \xi_s H = - i \sum_{\mathbf{k}\in\mathcal{Z}_s} \sigma_{\mathbf{k}'} \widehat H_\mathbf{k} e^{-\omega_{\mathbf{k}'}\delta} \mathbf{z}^\mathbf{k}.
\end{equation*}
\notag
$$
We estimate $\rho'$ such that, for any $\mathbf{z}\in D_{\rho'}$ and $\delta\in [0,+\infty)$, the point $\mathbf{z}(\delta) = \vartheta^{-1}_\delta(\mathbf{z})$ does not leave $D_\rho$. We have $\mathbf{z}(\delta)=(z_1(\delta),\dots,\overline z_n) \in D_\rho$, $\rho e^\alpha \leqslant 1$, and so
$$
\begin{equation}
|\dot z_j(\delta)| \leqslant \sum_{\mathbf{k}\in\mathcal{Z}_s} |\mathbf{k}| c e^{b_{|\mathbf{k}|}+\alpha |\mathbf{k}| - \delta / \Omega_{2s-2}} \rho^{|\mathbf{k}|-1} \leqslant (2s)^{2n} c e^{b_s - \delta/\Omega_{2s-2}} (\rho e^\alpha)^{s-1} e^\alpha,
\end{equation}
\tag{6.12}
$$
where $j$ is any number from the set $\{1,\dots,n\}$. Here, we have used that the sequence $b_s$ does not increase, $s\leqslant |\mathbf{k}|\leqslant 2s$, and the sum contains less than $(2s)^{2n-1}$ terms. Estimate (6.12) implies
$$
\begin{equation*}
|z_j(\delta) - z_j(0)| \leqslant (2s)^{2n} c e^{b_s} \Omega_{2s-2} (\rho e^\alpha)^{s-1} e^\alpha, \qquad j = 1,\dots,n.
\end{equation*}
\notag
$$
Since the sequence $\{b_j\}$ does not increase, we have $b_s - b_{2s-2}\geqslant 0$. Hence
$$
\begin{equation*}
|z_j(\delta) - z_j(0)| \leqslant (2s)^{2n} c e^{2b_s - b_{2s-2}} \Omega_{2s-2} (\rho e^\alpha)^{s-1} e^\alpha = \frac{1}{n s e^\alpha} (\rho e^\alpha)^{s-1} \varepsilon.
\end{equation*}
\notag
$$
The differences $\overline z_j(\delta) - \overline z_j(0)$ can be estimated in the same way. This implies (6.4). Lemma 6.1 is proved. Proof of Lemma 6.2. For any $\mathbf{l},\mathbf{m},\mathbf{k}\in\mathbb{Z}_+^{2n}$, the equation $\mathbf{l}+\mathbf{m}=\mathbf{k}+\mathbf{e}_j$ implies $|\mathbf{l}|+|\mathbf{m}|=|\mathbf{k}|+2$. This implies the estimate
$$
\begin{equation}
\begin{aligned} \, \Sigma_1(\mathbf{k}) &\leqslant e^{\alpha |\mathbf{k}| + 2\alpha} \chi_{|\mathbf{k}|} \Sigma_2(\mathbf{k}), \\ \nonumber \Sigma_2(\mathbf{k}) &= \sum_{j=1}^n \, \sum_{\mathbf{l}\in\mathcal{Z}_s^-\cup\mathcal{Z}_s^+,\, |\mathbf{m}|\geqslant s,\, \mathbf{l}+\mathbf{m}=\mathbf{k}+\mathbf{e}_j} c^2\, |\mathbf{l}|\cdot |\mathbf{m}| e^{b_{|\mathbf{l}|} + b_{|\mathbf{m}|} - b_{|\mathbf{k}|} }. \end{aligned}
\end{equation}
\tag{6.13}
$$
Since in $\Sigma_2$ $|\mathbf{m}|\leqslant |\mathbf{k}|$, we have $\Sigma_2(\mathbf{k})\leqslant c^2 |\mathbf{k}| \Sigma_3(\mathbf{k})$, where
$$
\begin{equation*}
\Sigma_3(\mathbf{k}) = \sum_{j=1}^n S_j(\mathbf{k}), \qquad S_j(\mathbf{k}) = \sum_{\mathbf{l}\in\mathcal{Z}_s^-\cup\mathcal{Z}_s^+,\, |\mathbf{m}|\geqslant s,\, \mathbf{l}+\mathbf{m}=\mathbf{k}+\mathbf{e}_j} |\mathbf{l}|\cdot e^{b_{|\mathbf{l}|} + b_{|\mathbf{m}|} - b_{|\mathbf{k}|}}.
\end{equation*}
\notag
$$
The number of terms in the sum $S_j(\mathbf{k})$ with fixed $|\mathbf{l}|\leqslant 2s-3$ does not exceed
$$
\begin{equation*}
\# \{\mathbf{l}'\in\mathbb{Z}_+^{2n} \colon |\mathbf{l}'| = |\mathbf{l}|\} < (2s)^{2n-1}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
S_j(\mathbf{k}) \leqslant \sum_{p+q=|\mathbf{k}|+2,\, s\leqslant p \leqslant 2s - 3,\, q\geqslant s} (2s)^{2n} e^{b_p + b_q - b_{|\mathbf{k}|}}.
\end{equation*}
\notag
$$
The sequence $\{b_j\}$ is convex. Therefore, by Lemma 7.5, if $|\mathbf{k}| \geqslant 2s-2$, $p+q=|\mathbf{k}|+2$, and $p,q\geqslant s$, we have
$$
\begin{equation*}
b_p + b_q \leqslant b_{|\mathbf{k}|+2-s} + b_s \quad\text{and}\quad b_{|\mathbf{k}|+2-s} - b_{|\mathbf{k}|} \leqslant b_s - b_{2s-2}.
\end{equation*}
\notag
$$
This implies $b_p + b_q - b_{|\mathbf{k}|} \leqslant 2b_s - b_{2s-2}$. Hence
$$
\begin{equation*}
S_j(\mathbf{k}) \leqslant \frac12 (2s)^{2n+1} e^{2b_s-b_{2s}-2} = \frac{\Lambda e^{-2\alpha}}{n}, \qquad \Sigma_3(\mathbf{k}) \leqslant e^{-2\alpha}\Lambda
\end{equation*}
\notag
$$
and we obtain (6.11). Lemma 6.2 is proved. 6.2. Low order normalizationIn this section, assuming that the normal form of $H_2 + \widehat H$ is $H_2+N_\diamond$, $N_\diamond=O_r(\mathbf{z})$, we apply Lemma 6.1 inductively to reduce the Hamiltonian to the form $H_2 + G$, $G=O_r$. Assuming $\widehat H\in\mathcal{A}$, we estimate $G$ in terms of $\widehat H$. As a by-product, we obtain another proof of the Bruno–Rüssmann theorem on convergence of the normalization in the case of zero normal form. Motivated by conditions of Theorem 5.1 we will assume that (1) $\omega\in\mathbb{R}^n$ satisfies the Bruno condition, (2) $\widehat H\in\mathcal{F}_\diamond\cap\mathcal{A}$, (3) the normal form of the Hamiltonian $H_2 + \widehat H$ is $H_2+N_\diamond$ $N_\diamond=O_r(\mathbf{z})$, $r\geqslant 3$. Now by Lemma 7.6, there exists a sublinear convex negative sequence $\{b_s\}$ such that where $\{a_j\}$ is the Bruno sequence
$$
\begin{equation}
a_j = n 2^{2n+j} (2^j+2)^{2n+1} \Omega_{2^j+2}, \qquad j\in\mathbb{N}.
\end{equation}
\tag{6.15}
$$
By Lemma 7.1, there exist $\widehat c_0$ and $\widehat\alpha_0$ such that
$$
\begin{equation*}
\widehat H = \sum_{|\mathbf{k}|\geqslant 3} \widehat H_\mathbf{k} \mathbf{z}^\mathbf{k}, \qquad |\widehat H_\mathbf{k}| \leqslant \widehat c_0 e^{b_{|\mathbf{k}|} + \widehat\alpha_0 |\mathbf{k}|}.
\end{equation*}
\notag
$$
Choosing $\alpha_0\geqslant\widehat\alpha_0$, we have
$$
\begin{equation}
|\widehat H_\mathbf{k}| \leqslant c_0 e^{b_{|\mathbf{k}|}+\alpha_0 |\mathbf{k}|}, \qquad |\mathbf{k}| \geqslant 3,
\end{equation}
\tag{6.16}
$$
where $c_0 = \widehat c_0 e^{3(\widehat\alpha_0 - \alpha_0)}$. This means that choosing $\alpha_0$ sufficiently large, we may assume that
$$
\begin{equation}
c_0 e^{2\alpha_0} \leqslant \frac1{16}, \qquad n e^{2\alpha_0} \geqslant 2.
\end{equation}
\tag{6.17}
$$
Theorem 6.1. Suppose that $\omega$ and $\widehat H$ satisfy conditions (1)–(3) and (6.14)–(6.17). Then an analytic canonical transformation of coordinates reduces the Hamiltonian $H_2 + \widehat H$ to the form $H_2\circ\vartheta + \widehat H\circ\vartheta = H_2 + G$, whereRemark 6.1. The normalization from the proof of Theorem 6.1 deals with a finite number of “small divisors” $\langle\omega,\mathbf{k}'\rangle$. Hence the assumption that $\omega$ satisfies the Bruno condition looks too strong.8 We use this assumption to obtain uniform with respect to $r$ estimates (6.18), (6.19) for $\rho_*$ and $G_\mathbf{k}$. Corollary 6.1 (the Bruno–Rüssmann theorem). Suppose that (a) $\omega\in\mathbb{R}^n$ satisfies the Bruno condition, (b) $\widehat H \in\mathcal{A}\cap\mathcal{F}_\diamond$, (c) the normal form of the Hamiltonian $H_2 + \widehat H$ equals $H_2$. Then an analytic canonical transformation of coordinates $\mathbf{w}\mapsto\mathbf{z}=\vartheta(\mathbf{w})$ reduces the Hamiltonian $H_2+\widehat H$ to the normal form $H_2\circ\vartheta + \widehat H\circ\vartheta = H_2$. To prove Corollary 6.1, it suffices to make $r\to +\infty$ in Theorem 6.1. 6.3. Proof of Theorem 6.1The initial Hamiltonian satisfies estimate (6.16). We apply Lemma 6.1 inductively. Let Then $s$ in Lemma 6.1 takes the values We obtain the sequences $\alpha_0,\alpha_1,\dots,\alpha_N$, $\varepsilon_1,\dots,\varepsilon_N$, $\Lambda_1,\dots,\Lambda_N$, which estimate the Hamiltonian on the $m$th step,
$$
\begin{equation}
\alpha_{m+1} = \alpha_m + \varepsilon_{m+1}, \qquad \varepsilon_m = c_0 \Lambda_m \Omega_{s_m}, \quad \Lambda_m = \frac{n e^{2\alpha_{m-1}}}{2} (2s_m)^{2n+1} e^{2b_{s_m} - b_{s_{m+1}}}.
\end{equation}
\tag{6.20}
$$
By (6.14), (6.15) with $j=m-1$
$$
\begin{equation}
e^{2b_{s_m} - b_{s_{m+1}}} n 2^{m-2} (2 s_m)^{2n+1} \Omega_{s_m} = 1.
\end{equation}
\tag{6.21}
$$
Equation (6.21) combined with (6.20) implies
$$
\begin{equation}
\varepsilon_m = \frac{c_0 n e^{2\alpha_{m-1}}}{2} (2s_m)^{2n+1} e^{2b_{s_m} - b_{s_{m+1}}} \Omega_{s_m} = \frac{c_0 e^{2\alpha_{m-1}}}{2^{m-1}} = \frac{2c_0 e^{2\alpha_0 + 2(\varepsilon_0+\dots+\varepsilon_{m-1})}}{2^m}.
\end{equation}
\tag{6.22}
$$
Now we prove by induction that we $\varepsilon_m\leqslant 2^{-m-2}$ for $c_0 e^{2\alpha_0}\leqslant 1/16$ (see (6.17)). Indeed, for $m=0$, we have the estimate $\varepsilon_0 = 2c_0^2 e^{2\alpha_0} \leqslant 1/8$ by (6.22). Suppose this inequality is valid for $m\leqslant m_0$. Then
$$
\begin{equation*}
\varepsilon_{m_0+1} \leqslant \frac18 \frac{e^{2^{-2}+\dots+2^{m-3}}}{2^{m_0+1}} < \frac{e^{1/2}}{2^{m_0+2}} < 2^{-m_0-3}.
\end{equation*}
\notag
$$
Putting $\beta=\alpha_N$, we obtain $\beta - \alpha_0 \leqslant \sum_{m=1}^\infty \varepsilon_m \leqslant 1/2$. This implies (6.19). To estimate $\rho_*$, we define the sequence $\vartheta_1,\dots,\vartheta_N$, where $\vartheta_m \colon D_{\rho_{m-1}}\to D_{\rho_m}$ is the map from Lemma 6.1 on the $m$th step. We also define
$$
\begin{equation}
\rho_0,\dots,\rho_N, \qquad \rho_0 = e^{-\alpha_0}, \quad \rho_N = \rho_*.
\end{equation}
\tag{6.23}
$$
This implies that the map $\vartheta=\vartheta_1\circ\dots\circ\vartheta_N \colon D_{\rho_N}\to D_{\rho_0}$ is well-defined. Inequality (6.4) on $m$th step reads
$$
\begin{equation}
rho_{m+1} \leqslant \rho_m - \frac{\varepsilon_{m+1}}{n s e^\alpha_m}\, (\rho_m e^{\alpha_m})^{s_{m+1}-1}.
\end{equation}
\tag{6.24}
$$
Now we show that $\rho_m$, $m\geqslant 1$, determined by the equation $\rho_m e^{\alpha_m}=1$, satisfies (6.24). Indeed, $\rho_0 e^{\alpha_0}=1$ by (6.23). Assuming $\rho_m e^{\alpha_m}=1$ for some $m\geqslant0$, we have
$$
\begin{equation*}
\biggl( \rho_m - \frac{\varepsilon_{m+1}}{ns e^{\alpha_m}} (\rho_m e^{\alpha_m})^{s_{m+1}} \biggr) = e^{\varepsilon_{m+1}} - \frac{\varepsilon_{m+1}}{ns e^{\alpha_m}} e^{\alpha_{m+1}} = e^{\varepsilon_{m+1}} \biggl( 1 - \frac{\varepsilon_{m+1}}{ns} \biggr).
\end{equation*}
\notag
$$
The last quantity exceeds 1 because $\varepsilon_m\leqslant\varepsilon_0\leqslant 1/8$. Hence $\rho_{m+1}=e^{-\alpha_{m+1}}$ satisfies (6.24). Therefore, $\rho_* =\rho_N = \rho_0 e^{-\sum_1^N \varepsilon_m} \geqslant \rho_0 e^{-1/2}$. Theorem 6.1 is proved. 6.4. Proof of Theorem 5.1Let $\{b_s\}$ be a sublinear convex negative sequence satisfying (6.14) with Bruno sequence (6.15). By Theorem 6.1 there exists an analytic canonical transformation of coordinates, which reduces the Hamiltonian $H_2 + \widehat H$ to the form $H_2 + G$, where the function $G$ satisfies (6.19). Since $b_{|\mathbf{k}|}$ are negative, we have
$$
\begin{equation*}
|G_\mathbf{k}| \leqslant c_0 e^{\beta |\mathbf{k}|}, \qquad |\mathbf{k}| \geqslant r.
\end{equation*}
\notag
$$
We now start another continuous averaging procedure with the operator $\xi=\xi_*$. We repeat the proof of Theorem 4.2 until equation (4.7). In our case $f'=O_{r-1}(\mathbf{z})$. By Lemma 7.3,
$$
\begin{equation*}
f'(\zeta) \ll \frac{a\zeta^{r-1}}{b-\zeta}, \qquad a= a(c_0,\beta), \quad b = b(c_0,\beta).
\end{equation*}
\notag
$$
Putting $\tau=8n\delta$, we obtain
$$
\begin{equation}
G = \frac{a (\zeta + \tau G)^{r-1}}{b - (\zeta+\tau G)}.
\end{equation}
\tag{6.25}
$$
To estimate the function $G=G(\zeta)$, we have to use a quantitative version of the implicit function theorem (Lemma 6.3 below). First, we introduce the function $\widetilde G(\zeta) = \zeta + \tau G(\zeta)$. It satisfies the equation
$$
\begin{equation}
\zeta = \widetilde G + \varphi(\widetilde G), \qquad \varphi(\widetilde G) = -\frac{a\tau\widetilde G^{r-1}}{b - \widetilde G},
\end{equation}
\tag{6.26}
$$
which is a consequence of (6.25). Lemma 6.3 (see [3]). Let $x=y+\varphi(y)$ in the complex ball $\{y\in\mathbb{C} \colon |y|\leqslant 6\rho\}$, where the function $\varphi$ is analytic and $|\varphi|\leqslant\rho/2$. Then there is a unique analytic function $\psi$ defined for $|x|\leqslant\rho$ such that Applying Lemma 6.3 to equation (6.26) with $x=\zeta$ and $y=\widetilde G$, we take
$$
\begin{equation*}
\rho = \min\biggl\{ \frac{b}{12}, \frac16 \biggl(\frac b{24 a\tau}\biggr)^{1/(r-2)} \biggr\}.
\end{equation*}
\notag
$$
In the ball $|\widetilde G| \leqslant 6\rho$, we have
$$
\begin{equation*}
|\varphi| \leqslant \frac{a\tau (6\rho)^{r-1}}{b - 6\rho} \leqslant \frac{2a\tau (6\rho)^{r-1}}{b} \leqslant \frac \rho 2.
\end{equation*}
\notag
$$
This implies that $\widetilde G = \zeta + \tau G = \zeta + \psi(\zeta)$, where $|\psi| \leqslant \rho/2$ for $|\zeta| \leqslant \rho$. If $\tau = 8n\delta$ is sufficiently large, then
$$
\begin{equation*}
\rho = \frac16 \biggl(\frac b{24 a\tau}\biggr)^{1/(r-2)} \geqslant \frac{c_z}{1 + \delta^{1/(r-2)}}
\end{equation*}
\notag
$$
for some constant $c_z$. Hence in the domain $\{|\zeta|\leqslant \rho\}$
$$
\begin{equation*}
|G| = \biggl|\frac{\widetilde G - \zeta}{\tau}\biggr| \leqslant \frac{\rho}{2\tau} \leqslant \frac{C}{1 + \delta^{1 + 1/(r-2)}}
\end{equation*}
\notag
$$
for some constant $C$. The function $\int_0^\zeta G(\widehat\zeta)\, d\widehat\zeta$ is a majorant for $|\mathcal{H}|$. This implies $|\mathcal{H}| \leqslant |\rho G| \leqslant C_H / (1 +\delta^{1+2/(r-2)})$. Theorem 5.1 is proved. Generically $\omega$ does not admit resonances of order less than $4$ (this includes the case of non-resonant $\omega$). Then $r = 4$ and diameter of the analyticity domain for $\mathcal{H}$ is of order $\delta^{-1/2}$. This estimate is better than $\delta^{-1}$ declared in [18]. § 7. Technical part7.1. MajorantsLemma 7.1. Suppose $F = \sum_{\mathbf{k}\in\mathbb{Z}_+^{2n}} F_\mathbf{k} \mathbf{z}^\mathbf{k} \in\mathcal{A}^\rho$, and the sequence $\{b_j\}_{j\in\mathbb{Z}_+}$ is sublinear. Then, for any $\alpha > - \ln\rho$, there exists $c>0$ such that Proof. By Lemma 2.1, we have $|F_\mathbf{k}|\leqslant c_F \rho^{-|\mathbf{k}|}$, $c_F = \|F\|_\rho$. Hence in (7.1) we have to choose $c$ such that For any $\alpha > - \ln\rho$, the function $q\mapsto e^{-b_q - (\alpha + \ln\rho) q}$, $q\in\mathbb{Z}_+$, is bounded because $b_q$ is sublinear. This proves the lemma. For any $F,\mathbf{F}\in\mathcal{F}$ we say that $F\ll\mathbf{F}$ if and only if $|F_\mathbf{k}| \leqslant \mathbf{F}_\mathbf{k}$, $\mathbf{k}\in\mathbb{Z}_+^{2n}$, for their Taylor coefficients. In this case, we say that $\mathbf{F}$ is a majorant for $F$. Lemma 7.2. Suppose $F\ll\mathbf{F}$ and $\widehat F\ll\widehat{\mathbf{F}}$. Then: (1) $F+\widehat F\ll\mathbf{F}+\widehat{\mathbf{F}}$, $F\widehat F\ll \mathbf{F}\widehat{\mathbf{F}}$, $\partial_{z_s} F\ll\partial_{z_s}\mathbf{F}$, and $\partial_{\overline z_s} F\ll\partial_{\overline z_s}\mathbf{F}$, $s=1,\dots,n$; (2) if $F$ and $\mathbf{F}$ depend on the parameter $\delta\in [\delta_1,\delta_2]$, then We skip an obvious proof. Lemma 7.3. Suppose $F\in\mathcal{A}^\rho$, $F = \sum_{|\mathbf{k}|\geqslant s} F_\mathbf{k} \mathbf{z}^\mathbf{k} = O_s(\mathbf{z})$, and $|F_\mathbf{k}| \leqslant a\rho^{s - |\mathbf{k}|}$. Then Proof. Note that $\sum_{|\mathbf{k}|=j} \mathbf{z}^\mathbf{k} \ll \zeta^j$ for any $j\in\mathbb{Z}_+$. Then proving the lemma. Proof. Since
$$
\begin{equation*}
\partial_\zeta \frac{\rho\zeta^3}{\rho - \zeta} = \frac{3\rho\zeta^2}{\rho - \zeta} + \frac{\rho\zeta^3}{(\rho - \zeta)^2},
\end{equation*}
\notag
$$
the lemma follows from the two estimates
$$
\begin{equation*}
\begin{gathered} \, \frac{3\rho\zeta^2}{\rho - \zeta} \ll \frac{3\rho\zeta^2}{2(\rho/2 - \zeta)}, \\ \frac{\rho\zeta}{(\rho - \zeta)^2} = \sum_{k=0}^\infty k \biggl(\frac\zeta\rho\biggr)^k = \sum_{k=0}^\infty \frac{k}{2^k} \biggl(\frac\zeta{\rho/2}\biggr)^k \ll \sum_{k=0}^\infty \frac12 \biggl(\frac\zeta{\rho/2}\biggr)^k = \frac{\rho}{2(\rho/2 - \zeta)^2}. \end{gathered}
\end{equation*}
\notag
$$
7.2. Majorant principleWe use the majorant method to obtain estimates for solutions of initial value problems (IVP) in $\mathcal{F}$. As an example, consider the IVP
$$
\begin{equation}
\partial_\delta F = \Phi(F,\delta), \qquad F|_{\delta=0} = \widehat F.
\end{equation}
\tag{7.4}
$$
Here, $F\in\mathcal{F}$ depends on the parameter $\delta$, and $\Phi$ is a map from $\mathcal{F}\times\mathbb{R}_+$ to $\mathcal{F}$. Definition dfn:pr. The IVP (7.4) is said to be power regular if, for any $\widehat F\in\mathcal{F}$, equation (7.4) has a unique solution $F = F(\mathbf{z},\delta)\in\mathcal{F}$ for all $\delta>0$. We associate with (7.4) the so-called majorant system
$$
\begin{equation}
\partial_\delta \mathbf{F} = \Psi(\mathbf{F},\delta), \qquad \mathbf{F}|_{\delta=0} = \widehat{\mathbf{F}}.
\end{equation}
\tag{7.5}
$$
We put $\Phi_\mathbf{k} = p_\mathbf{k}\circ\Phi$ and $\Psi_\mathbf{k} = p_\mathbf{k}\circ\Psi$. Definition 7.2. The IVP (7.5) is said to be a majorant IVP for (7.4) if: (a) $\widehat F\ll \widehat{\mathbf{F}}$; (b) for any $F\ll\mathbf{F}$, $\mathbf{k}\in\mathbb{Z}_+^{2n}$, and $\delta \geqslant 0$, we have $\Phi_\mathbf{k}(F,\delta) \ll \Psi_\mathbf{k}(\mathbf{F},\delta)$. Majorant principle. Suppose the IVP (7.4) is power regular. Suppose also that there exists a solution $\mathbf{F} = \mathbf{F}(\,{\cdot}\,,\delta)\in\mathcal{A}$ of (7.5) on the interval $\delta\in [0,\delta_0]$. Then (7.4) has a unique analytic solution $F$ on $[0,\delta_0]$ and $F(\,{\cdot}\,,\delta) \ll \mathbf{F}(\,{\cdot}\,,\delta)$. Remark 7.1. Definitions dfn:pr and 7.2 as well as the majorant principle obviously extend to systems of equations, where $F,\widehat F\in\mathcal{F}^m$ and $\Phi\colon \mathcal{F}^m\times\mathbb{R}_+\to\mathcal{F}$. Remark 7.2. One may replace the first equation (7.5) by the inequality $\partial_\delta \mathbf{F} \gg \Psi(\mathbf{F},\delta)$. The majorant principle presented here differs from the majorant argument used since Cauchy times. Traditionally, the evolution variable (in our case $\delta$) is regarded complex as well and Taylor expansions in it are used. In our approach, this variable is a real parameter in both exact solution and a majorant. Due to this we are able to obtain majorant estimates for solutions of (7.4) on large (even infinite) intervals of $\delta$. Theorem 7.1. Suppose both systems (7.4) and (7.5) have nilpotent structure (see Definition 4.1). Then the majorant principle holds true. We expect that majorant principle is valid in a much wider generality. But in this paper we are only interested in the case of systems having nilpotent form. Proof of Theorem 7.1. Let $\mathbf{k}^0$ be an index with minimal possible degree $|\mathbf{k}^0|$. For example, in system (3.11), $|\mathbf{k}^0|=3$. Nilpotent form of (7.4) implies that Hence $F_{\mathbf{k}^0}(\delta)\ll\mathbf{F}_{\mathbf{k}^0}(\delta)$ for $\delta\geqslant 0$.
We proceed by induction on $|\mathbf{k}|$. Suppose $F_{\mathbf{k}}(\delta)\ll\mathbf{F}_{\mathbf{k}}(\delta)$, $\delta\geqslant 0$, provided $|\mathbf{k}|< K$. For any $\mathbf{k}$ such that $|\mathbf{k}|=K$ we have by induction assumption and Definition 7.2, (b),
$$
\begin{equation*}
\partial_\delta (\mathbf{F}_\mathbf{k} - F_\mathbf{k}) = \Psi_\mathbf{k}(\mathbf{F}(\,{\cdot}\,,\delta),\delta) - \Phi_\mathbf{k}(\mathbf{F}(\,{\cdot}\,,\delta),\delta) \gg 0.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\mathbf{F}_\mathbf{k}(\delta) = \widehat{\mathbf{F}}_\mathbf{k}(\delta) + \int_0^\delta \Psi_\mathbf{k}(\mathbf{F}(\,{\cdot}\,,\lambda),\lambda)\, d\lambda \gg F_\mathbf{k}(\delta) = \widehat F_\mathbf{k}(\delta) + \int_0^\delta \Phi_\mathbf{k}(F(\,{\cdot}\,,\lambda),\lambda)\, d\lambda.
\end{equation*}
\notag
$$
Here, we used that arguments of $\Psi_\mathbf{k}$ and $\Phi_\mathbf{k}$ are known by the induction assumption.
This majorant inequality makes sense if the left-hand side is defined, that is, for any $\delta\in [0,\delta_0]$. The theorem is proved. 7.3. On convex sequencesLemma 7.5. Suppose the sequence $\{b_j\}_{j\in\mathbb{N}}$ is convex. Then: (1) for any $1\leqslant m < k < l$ (2) for any $1\leqslant m < k \leqslant l$ Proof. (1) First, consider the case $l-k=1$. If $k-m=1$ then (7.6) coincides with definition of convexity. We use induction on $k-m$. Suppose (7.6) holds for $k-m=q$. Then inequality (7.6) is obtained if we add the inequalities which hold by the induction assumption.
The case $l-k>1$ follows from the case $l-k=1$ also by induction. (2) Equation (7.6) implies the inequalities
$$
\begin{equation*}
\begin{aligned} \, (l-k) b_{k-m} + (k-m-l) b_k + m b_l &\geqslant 0, \\ m b_k + (k-l-m) b_l + (l-k) b_{l+m} &\geqslant 0 \end{aligned}
\end{equation*}
\notag
$$
from which (7.7) follows. Lemma is proved. Lemma 7.6. (1) For any convex sublinear sequence $\{b_s\}$, the sequence $\{a_j\}$, satisfying (6.14) is Bruno. (2) For any Bruno sequence $\{a_j\}$, there exists a sublinear convex negative non-increasing sequence $\{b_s\}$ satisfying (6.14). Proof. (1) Suppose $\{b_s\}$ is convex and sublinear, and $\{a_j\}$ satisfies (6.14). First, note that
$$
\begin{equation*}
\ln a_j - \ln a_{j-1} = - b_{2^{j+1}+2} + 3b_{2^j+2} - 2b_{2^{j-1}+2}.
\end{equation*}
\notag
$$
The latter expression is non-negative by convexity of $\{b_s\}$ (see Lemma 7.5). Hence, $a_j$ is non-decreasing.
Equation (6.14) implies
$$
\begin{equation*}
\frac12 \sum_{j=0}^{J-1} 2^{-j} \ln a_j = \sum_{j=0}^{J-1} \bigl( 2^{-j-1} b_{2^{j+1}+2} - 2^{-j} b_{2^j+2} \bigr) = 2^{-J} b_{2^J+2} - b_3.
\end{equation*}
\notag
$$
Since $\{b_j\}$ is sublinear, $2^{-J} b_{2^J+2} \to 0$ as $J\to\infty$. Therefore, $\{a_j\}$ is a Bruno sequence.
(2) Suppose $\{a_j\}$ is a Bruno sequence. We compute $\{b_{2^j+2}\}$ from the equation
$$
\begin{equation*}
\frac{b_{2^J+2}}{2^J} = b_3 + \frac12 \sum_{j=1}^{J-1} \frac{\ln a_j}{2^j}
\end{equation*}
\notag
$$
which implies (6.14). To have sublinearity, we need $\lim_{J\to +\infty} 2^{-J} b_{2^J+2} = 0$. Hence, we take $b_3 = - \frac12 \sum_{j=1}^\infty 2^{-j} \ln a_j$. Such a choice of $b_3$ implies that $b_{2^J+2} < 0$, $J=0,1,\dots$ .
The equation
$$
\begin{equation*}
b_{2^J+2} = - 2^{J-1} \sum_{j=J}^\infty \frac{\ln a_j}{2^j} = - \frac12 \sum_{j=0}^\infty \frac{\ln a_{J+j}}{2^j}
\end{equation*}
\notag
$$
implies
$$
\begin{equation*}
b_{2^{J+1}+2} - b_{2^J+2} = \frac12 \ln a_J - \frac12 \sum_{j=1}^\infty \frac{\ln a_{J+j}}{2^j} \leqslant \frac12 \ln a_J \biggl( 1 - \sum_{j=1}^\infty 2^{-j} \biggr) = 0.
\end{equation*}
\notag
$$
Hence the sequence $\{b^{2^j+2}\}$ does not increase.
For any integer $s\in (2^j+2,2^{j+1}+2)$, we define $b_s$ by linear interpolation
$$
\begin{equation*}
b_s = \frac{s - 2^j - 2}{2^j}\, b_{2^j+2} + \frac{2^{j+1} + 2 - s}{2^j}\, b_{2^{j+1}+2}.
\end{equation*}
\notag
$$
Such $\{b_s\}$ is obviously sublinear.
We check the convexity of $\{b_s\}$ first, on the subsequence $\{b_{2^j+2}\}_{j\in\mathbb{Z}_+}$. We have to prove the inequality
$$
\begin{equation}
2^J b_{2^{J-1}+2} - (2^J + 2^{J-1}) b_{2^J+2} + 2^{J-1} b_{2^{J+1}+2} \geqslant 0.
\end{equation}
\tag{7.8}
$$
By (6.14), the left-hand side of (7.8) equals $2^{J-1}(\ln a_J - \ln a_{J-1})$. For any $J\geqslant 1$, this expression is non-negative because $\{a_j\}$ is non-decreasing.
If $b_s$, $s\ne 2^j+2$, are determined by linear interpolation, the convexity, negativity, and monotonicity remain true for all the sequence $\{b_s\}_{s\in\mathbb{N}}$. Lemma 7.6 is proved.
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