# Convergence of the Birkhoff normal form sometimes
implies convergence of a normalizing transformation

Rafael de la Llave Supported in part by NSF DMS 1800241 School of Mathematics,
Georgia Institute of Technology  Maria Saprykina Supported in part by the
Swedish Research Council, VR 2015-04012 Dept. of Mathematics, KTH Royal
Institute of Technoligy

###### Abstract

Consider an analytic Hamiltonian system near its analytic invariant torus
$\mathcal{T}_{0}$ carrying zero frequency. We assume that the Birkhoff normal
form of the Hamiltonian at $\mathcal{T}_{0}$ is convergent and has a
particular form: it is an analytic function of its non-degenerate quadratic
part. We prove that in this case there is an analytic canonical
transformation—not just a formal power series—bringing the Hamiltonian into
its Birkhoff normal form.

##  1 Introduction

The goal of this paper is to study the convergence of the transformations of
an analytic Hamiltonian system in a neighborhood of an invariant torus to the
Birkhoff normal form. Here we assume that the frequency vector at the
invariant torus is very resonant, hence already at the formal level, the
existence of the Birkhoff normal form has obstructions. The main result,
Theorem 1 below, will show that if the obstructions for the formal equivalence
between the system and its Birkhoff normal form vanish and the normal form is
convergent and has a particular form, then the system is analytically
equivalent to its normal form. Hence, this result can be considered as a part
of the rigidity program: identifying obstructions for a weak form of
equivalence whose vanishing implies a stronger form of equivalence.

###  1.1 Classical theory of normal forms: existence and uniqueness.

Consider an analytic function

$\displaystyle H(I,\theta)=\left<{\lambda}_{0},I\right>+\mathcal{O}^{2}(I),$
(1.1)

where $\theta\in{\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$,
$I\in({\mathbb{R}}^{d},0)$, $\left<\cdot,\cdot\right>$ denotes the usual
scalar product in ${\mathbb{R}}^{d}$, and $\lambda_{0}\in{\mathbb{R}}^{d}$ is
a constant vector called the frequency vector. The Hamiltonian system
associated to it is $\dot{I}=\partial_{\theta}H(I,\theta),\
\dot{\theta}=-\partial_{I}H(I,\theta)$. Note that we are assuming the standard
symplectic form. In particular, the set
$\mathcal{T}_{0}:=\\{0\\}\times{\mathbb{T}}^{d}$ is an invariant torus of this
system. We say that $H(I,\theta)$ has a Birkhoff normal form (BNF) $N(I)$ in a
neighborhood of $\mathcal{T}_{0}$ if $N(I)$ is a formal power series, and
there exists a formal symplectic transformation $\Psi(I,\theta)$, tangent to
the identity

$\Psi(I,\theta)=(I+\mathcal{O}^{2}(I),\theta+\mathcal{O}(I))$

such that

$H\circ\Phi(I,\theta)=N(I)$

in the sense of formal power series. Any canonical coordinate change
$\Phi(I,\theta)$ as above is called a normalizing transformation. The
following fundamental result is called the Birkhoff normal form [SM71, MHO].
For $H(I,\theta)$ as above, assume that $\lambda_{0}$ satisfies a Diophantine
condition: there exist constants $(C,\tau)$ such that for all
$k\in{\mathbb{Z}}^{d}\setminus\\{0\\}$ we have

$\displaystyle\left|\langle{\lambda}_{0},k\rangle\right|\geq C|k|^{-\tau}.$
(1.2)

Then $H(I,\theta)$ has a (formal) Birkhoff normal form. Moreover, if a normal
form exists and $\lambda_{0}$ is rationally independent, then the Birkhoff
normal form is unique (up to trivial changes relabelling the actions). Note
that the normalizing transformations are not unique, since composing
$\Phi(I,\theta)$ with any transformation that preserves $I$ gives a
normalizing transformation.

Birkhoff normal form is an important tool in the study of Hamiltonian systems.
Already the assumption of existence and nondegeneracy of the normal form has
strong dynamical consequences (see, e.g., [EFK15] Th.C). The importance of the
BNF becomes even stronger if the normal form is convergent, and even more so
if there exists an analytic normalizing transformation.

The standard way of constructing BNF, which we will review in more detail
later, is to proceed iteratively, devising transformations that normalize
$H(I,\theta)$ up to the coefficients of order $I^{n}$. The normalization step
involves solving differential equations with analytic conditions. The
Diophantine conditions (1.2) can be somewhat weakened to subexponential growth
($\lim_{N\to\infty}\frac{1}{N}\log\sup_{|k|\leq
N}\left|\langle{\lambda}_{0},k\rangle\right|^{-1}=0$).

If ${\lambda}_{0}$ is resonant, one cannot guarantee the existence of the
Birkhoff normal form even at the level of formal power series, since there may
be some terms in the formal power series of $H$ that cannot be eliminated by a
canonical transformation. On the other hand, there are, of course, systems
(e.g the BNF itself, or changes of variables from it) for which one can
construct a BNF even in the resonant case. Then one speaks of the Birkhoff-
Gustavson normal form [Gu66].

Analogous definitions and statements hold true for symplectic maps in a
neighborhood of a fixed point. Even if the formal elimination procedures are
very similar, the analysis is very different. Handy references for the
classical theory of Birkhoff normal forms are [SM71, MHO, Mu, EFK13, EFK15].

###  1.2 Generic divergence both of the Birkhoff Normal Form and the
normalizing transformation.

The BNF and the normalizing transformations are constructed as formal power
series. The following natural questions are of great importance: the first one
is whether the BNF converges for Hamiltonians in a certain class. The
second—whether there is a convergent normalizing transformation.

Concerning the first question, R. Perez-Marco [PM] proved the following
dichotomy: for any given nonresonant quadratic part, either the BNF is
generically divergent or it always converges. The original proof was done in
the setting of Hamiltonian systems having a non resonant elliptic fixed point.
The extension of this result to the case of the torus, that is not completely
straightforward, has been worked out by R. Krikorian, see Theorem 1.1 in
[Kri].

Up to very recently it was unclear which of the possibilities is actually
realized. A large progress has been made by R. Krikorian [Kri], who proved
that there exists a real analytic symplectic diffeomorphism $f$ of a two-
dimensional annulus such
$f({\mathbb{T}}\times\\{0\\})=({\mathbb{T}}\times\\{0\\})$,
$f(\theta,0)=(\theta+\omega_{0},0)$ with $\omega_{0}$ Diophantine and having a
non-degenerate divergent Birkhoff normal form. Combined with the
aforementioned result of Perez-Marco, this implies that Birkhoff Normal Form
of an analytic Hamiltonian is “in general” divergent.

Concerning the normalizing transformations, H. Poincaré proved that they are
divergent for a generic Hamiltonian. C. L. Siegel proved the same statement in
a neighborhood of an elliptic fixed point (in fact, for a larger class of
Hamiltonians than just generic, [Si54]). This is implied by showing that the
orbit structure of the map in any neighborhood is very different from that of
the Birkhoff normal form (which is integrable). Analogous results for
symplectic maps near an elliptic fixed point appear in [Rü59]. Very different
arguments showing divergence of normalizing transformations for generic
systems appear in [Ze73] and for some concrete polynomial mappings in [Mo60].

###  1.3 Convergence of the transformations under the Diophantine conditions
for some particularly simple BNF

There are classes of Hamiltonians for which we can guarantee the convergence
of the normalizing transformation. The following influential rigidity result
was proved independently by A. D. Bruno [Br71] and H. Rüssmann [Rü67]. Note
that the main assumption is that the (in principle only formal) BNF is of a
particular kind.

Consider an analytic Hamiltonian $H(I,\theta)$, whose frequency
${\lambda}_{0}$ satisfies a Diophantine condition (1.2). Assume moreover that
the Birkhoff normal form $N(I)$ of $H(I,\theta)$ is a formal function $B$ of
one single variable $\Lambda_{0}:=\langle{\lambda},I\rangle$, i.e.,

$N(I)=B(\Lambda_{0}(I)).$

Then there exists an analytic normalizing transformation, and the BNF is, in
fact, analytic.

We remark that Bruno proves the above result under a weaker condition on
${\lambda}_{0}$ than (1.2). For analogous statements in the case of invariant
tori see [Br89]. Other modifications can be found in [Rü02, Rü04]. This result
has been recently generalised to a much more general context by Eliasson,
Fayad and Krikorian [EFK13, EFK15]. We stress that in all these works
mentioned above, ${\lambda}_{0}$ is assumed to be non-zero and the crucial
assumption is that ${\lambda}_{0}$ satisfies a Diophantine-type condition and
that the BNF is of a very simple form.

###  1.4 “Sometimes” convergence of the BNF implies convergence of a
normalizing transformation.

Our main result is close in spirit to the above works, but it does not rely on
a Diophantine condition. In fact, we consider a special class of
diffeomorphisms such that the frequency ${\lambda}_{0}$ is zero. Thus, the BNF
is degenerate in the previous sense. But within this class of Hamiltonians we
just use a standard non-degeneracy assumption on the quadratic part. Namely,
we prove the following.

######  Theorem 1.

Assume the following:

  * $(A_{1})$

$H(I,\theta)$ has a formal Birkhoff normal form $N(I)$ that starts with
quadratic terms in $I$, i.e there exists a formal symplectic change of
variables $\Psi(I,{\theta})$, tangent to the identity, i.e.
$\Psi(I,\theta)=(I+\mathcal{O}^{2}(I),\phi+\mathcal{O}(I))$, such that

$H\circ\Psi(I,{\theta})=N(I)=N_{0}(I)+\mathcal{O}^{3}(I)$

in the sense of power series.

  * $(A_{2})$

$N_{0}(I)=I^{tr}\Omega I$ (for some symmetric $\Omega$) is non-degenerate:
$\det{\Omega}\neq 0$.

  * $(A_{3})$

$N(I)=B(N_{0}(I))=N_{0}+\sum_{j=2}^{\infty}b_{j}(N_{0}(I))^{j}$ where $B$ is
an analytic function.

Then there exists an invertible analytic symplectic transformation

$\Phi(I,\theta)=(I+\mathcal{O}^{2}(I),\phi+\mathcal{O}(I))$

such that

$H\circ\Phi(I,{\theta})=N(I).$ (1.3)

Note that we start from a resonant torus, so that the existence of a BNF of
the form we assume, requires vanishing of (formal) obstructions. Hence, our
main result can be reformulated as saying that the formal assumptions imply
convergence of the normalizing transformation.

Similar rigidity statements have appeared in other contexts. In [Po92, Ch. 5],
H.Poincaré studied the formal power series of canonical transformations, which
send a family of Hamiltonian systems into a family of integrable systems (in
the sense of power series). In [Po92] it was shown that these formal power
series do not exist unless there are some conditions (which are not met in the
three body problem for arbitrary masses). The non-existence of formal power
series, a fortiori implies the non-existence of analytic families of analytic
transformations integrating the three body problem.

The paper [Ll] proved a converse to the result in [Po92]: if the system
satisfies a very specific and generic non-degeneracy condition, then,
existence of a formal power series that integrates the family of
transformations in the sense of power series implies existence of a convergent
one.

Assumption $A_{3}$ is there for technical purposes, see Sec. 3.3. Note that it
is trivial for $d=1$. This assumption reminds of that of Rüssmann in [Rü67,
Rü02, Rü04].

The assumption that the Birkhoff normal form is a function of $N_{0}$ has been
discussed in [Ga] under the name of relative integrability. Two Hamiltonian
dynamical systems are relatively integrable when one of them can be obtained
from the other by a symplectic change of coordinates and a reparameterization
of the time which only depends on the total energy. That is, the orbit
structures of the two systems in an energy surface are equivalent up to a
change of scale of time. The paper [Ga] includes several arguments for why the
notion of relative integrability is natural when discussing formal
equivalence. In the present paper, however, the focus lies on the notion of
equivalence under a symplectic change of variables. We show that, for a
certain class of systems, equivalence in the sense of formal power series
implies equivalence in the sense of analytic canonical changes of variables.
Hence, our main result can be understood as a rigidity result. The class of
systems for which this rigidity result holds can be succinctly described as
the set of systems that relatively integrable with respect to the main term.

In the context of formal equivalence implying analytically convergent
equivalence, it is natural to formulate:

Conjecture. Assume that an analytic Hamiltonian $H(I,\theta)$ as in (1.1) has
a convergent BNF that satisfies the non-degeneracy assumption that the
frequency map is a local diffeomormphism. Then there is a convergent
normalizing transformation.

Note that the problems studied in [Rü67] and [Br71] do not satisfy the
hypothesis of the conjecture, even though they satisfy the conclusions.

In the other direction one can construct examples [S] of analytic maps near a
hyperbolic fixed point such that the Birkhoff normal form is quadratic (in the
above notations, $N=\Lambda_{0}$) with a non-resonant set of eigenvalues, and
any normalizing transformation to the normal form diverges. In these examples,
the eigenvalues form carefully chosen Liouville vectors. That is, the paper
[S] shows that, depending on the Diophantine conditions, quadratic normal
forms may be rigid or not. The models in [S] do not satisfy the hypothesis of
the conjecture above.

###  1.5 Overview of the proof.

The standard method of obtaining the Birkhoff Normal form is an iterative
procedure in which we construct the transformations order by order: at the
$n$-th step of the procedure one computes the $n$-th order terms in the Taylor
expansions, assuming that all the terms of lower orders are computed. It would
appear natural to follow this scheme and try to estimate the transformations
at each step of the recursive procedure. Unfortunately, this seems technically
unfeasible. One of the main complications in any possible proof of convergence
of the transformations is that even if the BNF is unique, the formal
transformations $\Phi_{N}$ are very far from unique (Since the BNF depeds only
on the actions, the $\Phi_{N}$ can be composed with any canonical
transformation which moves the angles but preserves the actions. So, an
essential ingredient of any proof of convergence should be a especification of
how to choose the normalizing transformations.

In this paper we use a quadratically convergent method in which we double the
number of known coefficients at each step. Roughly – see more details in the
next paragraphs – we will show that if the formal obstructions vanish we can
choose a sequence of canonical transformations that proceed to converge
quadratically: doubling the order of the BNF at every step of the
construction. More importantly, there is a specific choice of the
transformation that satisfies very explicit bounds. The bounds on the new
transformation in terms of the remainder turn out to involve a loss of
derivatives. Therefore we need to implement a Nash-Moser scheme to estimating
the important objects in a sequence of domains which decrease slowly.

Here is a short overview of the proof; all the necessary notations are
introduced in the next section. At the $n$-th step of the iterative procedure
we will start with a Hamiltonian of the form

$H_{n}(I,{\theta})=N_{n}(I)+\widetilde{R_{n}}(I,{\theta}),$

where $N_{n}(I)$ is a polynomial in $I$ of degree $m_{n}=2^{n}+1$, and the
remainder term $\widetilde{R_{n}}$ is small in the following sense: for a
certain domain-dependent norm, introduced in Sec. 2.1.1, for a certain small
${\delta}_{n}$ (we assume ${\delta}_{n}\to 0$ with $n\to\infty$) and
${\kappa}>0$ the remainder term satisfies
$|\widetilde{R_{n}}|_{\rho_{n},\rho_{n}}\leq\delta_{n}^{\kappa}$.

At this step we construct a symplectic change of coordinates $\Phi_{n}$, such
that

$H_{n}\circ\Phi_{n}(I,{\theta})=N_{n+1}(I)+\widetilde{R_{n+1}}(I,{\theta}),$

where $N_{n+1}$ has degree $m_{n+1}=2m_{n}-1$, and
$|\widetilde{R_{n+1}}|_{\rho_{n+1},\rho_{n+1}}\leq\delta_{n+1}^{\kappa}=2^{-{\kappa}}\delta_{n}^{\kappa}$.

We construct $\Phi_{n}$ as a time one map of a the flow of a Hamiltonian
vector field $F_{n}$. The main ingredient consists in constructing and
estimating the norm of $F_{n}$ (and thus $\Phi_{n}$), which is found as a
solution of a certain homological equation (see (3.1) and in a simplified form
(4.1)). In general, this equation may not have even a formal solution unless
some constraints are met. However, the assumption of Theorem 1 implies that
this equation does have a formal solution. The key observation in this paper
is the following: if this homological equation has a formal solution, then it
also has an analytic solution with tame estimates for it (in the sense of
Nash-Moser theory). This statement is the contents of Lemma 6. We note that
the tame estimates use an argument different from the matching of powers.

The procedure can be repeated, because the main assumption used to show the
existence of solutions of the Newton equation is that there is a formal
solution to all orders. This assumption is clearly preserved if we make any
analytic change of variables. Once we know that the Newton procedure can be
repeated infinitely often, the convergence is more or less standard.

##  2 Notations and a step of induction.

###  2.1 Notations.

####  2.1.1 Norms and majorants.

Let ${\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ be a $d$-dimensional
torus, and for ${\sigma}>0$ consider its complex extension
${\mathbb{T}}^{d}_{\sigma}=\left({\mathbb{R}}^{d}+(-{\sigma},{\sigma})\sqrt{-1}\right)/{\mathbb{Z}}^{d}$.
Let ${\mathbb{D}}^{d}_{\rho}=\\{I\in{\mathbb{C}}^{d}:|I|<\rho\\}$ be a complex
disc, and define the ”d-dimensional annulus”

$\mathbb{A}_{\rho,{\sigma}}:={\mathbb{D}}_{\rho}^{d}\times{\mathbb{T}}^{d}_{\sigma}.$

Let $\mathcal{O}(\mathbb{A}_{\rho,{\sigma}})$ be the set of functions
holomorphic in $\mathbb{A}_{\rho,{\sigma}}$ that are real symmetric, i.e.,
such that $\overline{f(\bar{I},\bar{\theta})}=f(I,{\theta})$ (where the bar
stands for the complex conjugate). We use supremum norms over
$\mathbb{A}_{\rho,{\sigma}}$, denoted by $\|f\|_{\rho,{\sigma}}$. In the same
way we define the set $\mathcal{O}({\mathbb{D}}_{\rho})$ with the
corresponding norm $\|f\|_{\rho}$ being the sup-norms over the disc
${\mathbb{D}}^{d}_{\rho}$.

For a function $f\in\mathcal{O}(\mathbb{A}_{\rho,{\sigma}})$ consider its
Taylor-Fourier representation in the powers of $I$:
$f(I,{\theta})=\sum_{j\in{\mathbb{N}}^{d}}\sum_{k\in{\mathbb{Z}}^{d}}f_{j,k}e^{2\pi
i\langle k,\theta\rangle}I^{j}$. Consider a majorant for $f$ of the form

$\widehat{f}(I)=\sum_{j\in{\mathbb{N}}^{d}}\sum_{k\in{\mathbb{Z}}^{d}}|f_{j,k}|I^{j}e^{2\pi|k|{\sigma}}$

We denote by $|f|_{\rho,{\sigma}}$ the norm of the corresponding majorant
$\widehat{f}(I)$:

$|f|_{\rho,{\sigma}}=\|\widehat{f}\|_{\rho,{\sigma}}.$

Clearly, $\|f\|_{\rho,{\sigma}}\leq|f|_{\rho,{\sigma}}$. Analogous notation
$|f|_{\rho}$ corresponds to the norm $\|f\|_{\rho}$ above.

In what follows we will mostly have ${\sigma}=\rho$.

####  2.1.2 Important constants for the iterative procedure.

  * $\square$

Let $\rho_{0}=\mathop{\mbox{min}}\\{1,\rho\\}$,

  * $\square$

The order of polynomials involved in the $n$-th step of the iterative
procedure is

$m_{n}=2^{n}+1.$

  * $\square$

The norm of the rest term $\widetilde{R_{n}}$ at the $n$-th step will be
estimated as $|\widetilde{R_{n}}|_{\rho_{n}}\leq\delta_{n}^{\kappa}$. Let

$\displaystyle\kappa=d+6,$ $\displaystyle b=2^{-(\kappa+3)}$
$\displaystyle\delta_{0}=\rho_{0}b2^{-3}=\rho_{0}2^{-(\kappa+6)}$
$\displaystyle\delta_{n+1}=2^{-1}\delta_{n}.$

  * $\square$

Finally, let

$q_{n}=(2b)^{2^{-(n+1)}},$

and

$\rho_{n+1}=(\rho_{n}-3{\delta}_{n})q_{n}.$

####  2.1.3 Polynomials.

In the iterative procedure we will work with polynomials in $I$ whose
coefficients depend on ${\theta}$.

  * $\square$

Let

$N_{0}(I)=I^{tr}\Omega I$ (2.1)

where $\Omega$ is a symmetric non-degenerate matrix: $\det{\Omega}\neq 0$.

  * $\square$

An expression $M=f({\theta})I^{k}$ (where $k$ is a multi-index) is called a
monomial.

  * $\square$

We will say that a monomial $M_{k,l}=I^{k}e^{2\pi i\langle l,{\theta}\rangle}$
is resonant if it satisfies $\\{N_{0},M\\}=0$.

  * $\square$

$R^{[j]}(I,{\theta})$ stands for a homogeneous polynomial in $I$ of degree $j$
with coefficients depending on ${\theta}$:

$R^{[j]}(I,{\theta})=\sum_{|k|=j}r_{k}({\theta})I^{k}.$

  * $\square$

We also use notation $R^{[m,n]}$ to denote the range of degrees in $I$:

$R^{[m,n]}(I,{\theta})=\sum_{j=m}^{n}R^{[j]}(I,{\theta}),\quad R^{[\geq
m]}(I,{\theta})=\sum_{j=m}^{\infty}R^{[j]}(I,{\theta}).$

Let $m_{n}$ be as above. The following functions will be of special
importance.

  * $\square$

The normal form $N(I)$ is assumed to have the form

$N(I)=B(N_{0}(I))=N_{0}(I)+\sum_{j=2}^{\infty}b_{j}(N_{0}(I))^{j}.$ (2.2)

Denote

$\displaystyle N_{n}=N^{[2,m_{n}]}=\left(B(N_{0})\right)^{[2,m_{n}]};$ (2.3)

in particular, since $m_{0}=2$, $N_{0}=N_{0}^{[2,m_{0}]}=N_{0}^{[2]}$ is
quadratic.

  * $\square$

The rest term at the $n$-th inductive step is $\widetilde{R_{n}}(I,\theta)$:

$\displaystyle\widetilde{R_{n}}=\widetilde{R_{n}}^{[>m_{n}]}.$ (2.4)

  * $\square$

We will also need polynomials in $I$ with $\theta$-dependent coefficients:
$R_{n}(I,\theta)$ and $F_{n}(I,\theta)$ of the following degrees:

$\displaystyle R_{n}=R_{n}^{[m_{n}+1,m_{n+1}]},\quad
F_{n}=F_{n}^{[m_{n},m_{n+1}-1]}.$ (2.5)

###  2.2 Base of induction: an equivalent problem.

######  Lemma 1.

Suppose that

$H(I,{\theta})=N_{0}(I)+\widetilde{R_{0}}(I,\theta)\in\mathcal{O}(\mathbb{A}_{\rho,{\sigma}}),$

where $|\widetilde{R_{0}}|_{\rho,{\sigma}}\leq\delta$, and there exists a
formal (resp., analytic) symplectic transformation

$\Psi(I,\theta)=\left(\phi(I,\theta),\,\psi(I,\theta)\right)=(I+\mathcal{O}^{2}(I),\theta+\mathcal{O}(I))$

such that

$H\circ\Psi(I,{\theta})=N(I)=N_{0}(I)+\sum_{j=2}^{\infty}b_{j}(N_{0}(I))^{j}.$

Then for any $a>0$ there exists a Hamiltonian $\widehat{H}(I,{\theta})$ and a
formal (resp., analytic) symplectic transformation
$\widehat{\Psi}(I,\theta)=(I+\mathcal{O}^{2}(I),\theta+\mathcal{O}(I))$ such
that

$\widehat{H}\circ\widehat{\Psi}(I,{\theta})=N_{0}(I)+\widehat{R_{0}}(I,\theta)\in\mathcal{O}(\mathbb{A}_{\frac{1}{a}\rho,{\sigma}}),$

where $|\widehat{R_{0}}|_{\frac{1}{a}\rho,{\sigma}}\leq a\delta$, and

$N(I)=N_{0}(I)+\sum_{j=2}^{\infty}b_{j}a^{2(j-1)}(N_{0}(I))^{j}.$

Proof. Define $\widehat{H}(I,{\theta})=\frac{1}{a^{2}}H(aI,{\theta})$, and
$\widehat{\Psi}(I,\theta)=\left(\frac{1}{a}\phi(aI,\theta),\,\psi(aI,\theta)\right)$.
It can be verified directly that $\widehat{\Psi}$ is symplectic and tangent to
the identity. Moreover,

$\widehat{H}\circ\widehat{\Psi}(I,\theta)=\frac{1}{a^{2}}H(\phi(aI,\theta),\psi(aI,\theta))=N_{0}(I)+\sum_{j=2}^{\infty}b_{j}a^{2(j-1)}(N_{0}(I))^{j}.$

$\Box$

###  2.3 Induction step.

While the base of induction is given by formula (2.12), the step of the
iterative procedure is provided by the following proposition.

######  Proposition 1.

For a fixed $n>0$, let $m_{n}$, $\rho_{n}$ and ${\delta}_{n}$ be as in Sec.
2.1.2 above. Suppose that $H_{n}(I,\theta)$ is formally conjugated to the BNF
of the form (2.2):

$N(I)=N_{0}(I)+\sum_{j=2}^{\infty}b_{j}(N_{0}(I))^{j},$

and the normal form satisfies:

$|N^{[m_{n}+j]}|_{\rho_{n}}<{\delta}_{n}^{\kappa+1},\quad j=0,\dots m_{n},$
(2.6)

and denoting $g_{2j}(I)=jb_{j}(N_{0}(I))^{j-1}$, we assume

$|g_{j}|_{\rho_{n}}\leq\frac{1}{4^{j}},\quad j=1,\dots,m_{n};$ (2.7)

Suppose that

$H_{n}(I,\theta)=N_{n}(I)+\widetilde{R_{n}}(I,\theta),$

where $N_{n}(I)=\left(B(N_{0}(I))\right)^{[2,m_{n}]}$ and
$\widetilde{R_{n}}=\widetilde{R_{n}}^{[>m_{n}]}$ satisfies

$|\widetilde{R_{n}}|_{\rho_{n},\rho_{n}}\leq\delta_{n}^{\kappa}.$

Then there exists a symplectic change of coordinates
$\Phi_{n}:(I^{\prime},\theta^{\prime})\mapsto(I,\theta)$,

$\Phi_{n}(I^{\prime},\theta^{\prime})=(U^{(n)}(I^{\prime},\theta^{\prime}),V^{(n)}(I^{\prime},\theta^{\prime})),$

given by a Hamiltonian $F_{n}=F_{n}^{[m_{n},m_{n+1}-1]}$ such that

$\displaystyle
H_{n+1}(I^{\prime},\theta^{\prime}):=H_{n}\circ\Phi_{n}(I^{\prime},\theta^{\prime})=N_{n+1}(I^{\prime})+\widetilde{R_{n+1}}(I^{\prime},\theta^{\prime}),$
(2.8)

where $N_{n+1}(I^{\prime})=N^{[2,m_{n+1}]}(I^{\prime})$,
$\widetilde{R_{n+1}}(I^{\prime},\theta^{\prime})=\widetilde{R_{n+1}}^{[>m_{n+1}]}(I^{\prime},\theta^{\prime})$,
and

$\displaystyle|\widetilde{R_{n+1}}|_{\rho_{n+1},\rho_{n+1}}\leq{\delta}_{n+1}^{\kappa}.$
(2.9)

Moreover,
$\Phi_{n}(I^{\prime},\theta^{\prime})=(U^{(n)}(I^{\prime},\theta^{\prime}),V^{(n)}(I^{\prime},{\theta}^{\prime}))$
satisfies

$\displaystyle\sum_{j=1}^{d}\|U_{j}^{(n)}(I^{\prime},\theta^{\prime})-I^{\prime}_{j}\|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}+\|V_{j}^{(n)}(I^{\prime},\theta^{\prime})-\theta^{\prime}_{j}\|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}<{\delta}_{n},$
(2.10)

and the inverse map,
$\Phi_{n}^{-1}(I,\theta):=(U^{(-n)}(I,\theta),V^{(-n)}(I,\theta))$, satisfies

$\displaystyle\sum_{j=1}^{d}\|U_{j}^{(-n)}(I,\theta)-I_{j}\|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}+\|V_{j}^{(-n)}(I,\theta)-\theta_{j}\|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}<{\delta}_{n}.$
(2.11)

The proof of this proposition constitutes the main technical tool of this
paper. It implies Theorem 1 in a standard way. See, e.g., [Rü67], pp 61-63).
For convenience, we give a proof below.

###  2.4 Proof of Theorem 1.

Lemma 1 permits us to assume without loss of generality that for the given
Hamiltonian
$H_{0}(I,{\theta}):=H(I,{\theta})=N_{0}(I)+\widetilde{R_{0}}(I,{\theta}),$

$|\widetilde{R_{0}}|_{\rho_{0},\rho_{0}}\leq\delta_{0}^{\kappa}.$ (2.12)

Since the function $B$ is analytic, the same lemma permits us to assume that
(2.6) and (2.7) hold for each $n$.

The step of induction is provided by Proposition 1. Since $H_{n}$ is formally
reducible to the normal form $N$, the same can be said about $H_{n+1}$.

Repetition of this process leads to a sequence of transformations

$T_{n}=\Phi_{0}\circ\Phi_{1}\circ\dots\circ\Phi_{n-1}.$

Let us show that $T_{n}$ converges to the desired coordinate change
$\Phi=T_{\infty}$, analytic in the polydisc
$\mathbb{A}_{\rho_{\infty},\rho_{\infty}}$, where
$\rho_{0}b<\rho_{\infty}<\rho_{0}$. Indeed, with the notations of Sec. 2.1.2,

$3\sum_{k=0}^{\infty}{\delta}_{k}\leq 3\cdot 2{\delta}_{0}<3\cdot
2\rho_{0}b2^{-3}<\rho_{0}b.$

Then for any $n$ we have

$\rho_{n+1}=q_{n}(\rho_{n}-3{\delta})\geq\rho_{0}\prod_{j=0}^{n}q_{j}-3\sum_{j=0}^{n}{\delta}_{n}\geq\rho_{0}\prod_{j=0}^{\infty}q_{j}-3\sum_{j=0}^{\infty}{\delta}_{n}\geq\rho_{0}2b-3\cdot
2{\delta}_{0}>b\rho_{0}$

It is left to prove that $T_{n}$ converges of to an analytic function
$T_{\infty}$, satisfying (1.3). Denote the variables, involved in the $n$-th
step of the induction by $w_{n-1}=(I,\theta)$ and
$w_{n}=(I^{\prime},\theta^{\prime})$, where

$w_{n}=\Phi_{n-1}^{-1}w_{n-1}.$

In these notations,

$w_{0}=\Phi_{0}\circ\Phi_{1}\circ\dots\circ\Phi_{n-1}w_{n}=T_{n}w_{n}.$

Now, for $w_{n}=(I^{\prime},\theta^{\prime})$ we have

$H\circ
T_{n}(I^{\prime},\theta^{\prime})=N_{n}(I^{\prime})+\widetilde{R_{n}}(I^{\prime},\theta^{\prime}).$

Since $(\Phi_{n}(I^{\prime},\theta^{\prime})-(I^{\prime},\theta^{\prime}))$
starts with the terms of degree $2^{n}$ in $I^{\prime}$, for each $j$ the
expansion of
$(T_{n}(I^{\prime},\theta^{\prime})-T_{n+j}(I^{\prime},\theta^{\prime}))$
starts with the terms of degree $2^{n}$ in $I^{\prime}$. This implies that the
sequence of maps $T_{n}$ formally converges, when $n\to\infty$, to a formal
map $T_{\infty}$ such that (1.3) holds:

$H\circ T_{\infty}(I^{\prime},\theta^{\prime})=N(I^{\prime}).$

We still need to show that $T_{\infty}$ is analytic. It is more convenient to
prove that the maps

$T_{n}^{-1}:=\Phi_{n-1}^{-1}\circ\dots\circ\Phi_{1}^{-1}\circ\Phi_{0}^{-1}$

converge to an analytic map $T_{\infty}^{-1}$.

By Proposition 1, the map

$w_{n+1}=\Phi_{n}^{-1}w_{n}$

is analytic in $\mathbb{A}_{\rho_{0}b/2,\rho_{0}b/2}$, and for all $n$ we
have:

$|\Phi_{n}^{-1}w_{n}-w_{n}|_{\rho_{0}b/2,\rho_{0}b/2}\leq{\delta}_{n},$

since $\rho_{n}-3{\delta}\geq\rho_{n+1}>\rho_{0}b$ for all $n$. Therefore, the
map $T_{n}^{-1}$ such that

$w_{n}=T_{n}^{-1}w_{0}$

is analytic in $\mathbb{A}_{\rho_{0}b/4,\rho_{0}b/4}$, and for such $w_{0}$ we
have

$|T_{n}^{-1}w_{0}|\leq\sum_{j=0}^{n-1}|T_{j}^{-1}(w_{j})-w_{j}|+|w_{0}|\leq\sum_{j=0}^{\infty}{\delta}_{j}+\rho_{0}b/4\leq\rho_{0}b/2.$

Estimate

$|T_{n+m}^{-1}(w_{0})-T_{n}^{-1}(w_{0})|_{\rho_{0}b/4,\rho_{0}b/4}\leq\sum_{j=n}^{n+m-1}|T_{j}^{-1}(w_{j})-w_{j})|_{\rho_{0}b/4,\rho_{0}b/4}\leq\sum_{j=n}^{\infty}{\delta}_{j}=2^{1-n}{\delta}_{0}$

implies the convergence of the sequence of maps $T_{n}^{-1}$ to an analytic
map $T_{\infty}^{-1}$ in $\mathbb{A}_{\rho_{0}b/4,\rho_{0}b/4}$. Since the
formal inverse of $T_{\infty}^{-1}$ is the series $T_{\infty}$, the latter
also defines an analytic function, providing the desired coordinate change. We
set $\Phi=T_{\infty}$ in the notations of Theorem 1. $\Box$

##  3 Formal analysis.

Here we start the proof of Proposition 1 by the formal analysis of the
iterative procedure.

###  3.1 Iterative Procedure.

Given $H_{n}$ as in Proposition 1, we will construct $\Phi_{n}$ as the time
one map of the flow of a Hamiltonian $F_{n}$, i.e., $\Phi_{n}=X_{F_{n}}^{1}$
where $X_{F_{n}}^{t}$ is the flow defined by

$\dot{I}=F_{\theta}(I,{\theta}),\quad\dot{\theta}=-F_{I}(I,{\theta}).$

In this case, $\Phi_{n}$ is automatically symplectic.

Notice that the normalising transformation $\Phi_{n}$, as well as the
corresponding generating function $F_{n}$, is not unique (one can compose with
rotations in the angles which preserve the actions, for example). Clearly, the
transformation that converges has to be very carefully chosen.

In the following Lemma 2 we show that if a (formal) normalizing transformation
exists, then there exists (another) normalizing transformation of a special
kind. Namely, such that the corresponding generating function is a polynomial
(in the sense of section 2.1.3), $F_{n}=F_{n}^{[m_{n},m_{n+1}-1]}$ and free
from resonant monomials (see notations in Sec. 2.1.3).

The idea of the proof is that we can always move the formal normalizing
transformation by composing with some transformations that do not change the
normal form. Therefore, we can ensure that the normalizing transformations
belong to a space which is transversal to the space spanned by resonant
monomials. Note that in the proof of Lemma 2 we use crucially the fact that
the normal form is a function of $N_{0}$ so that the resonant terms are the
same at all orders.

There are some analogies between Lemma 2 and Proposition 2.6 in [Ll], but that
result is significantly less delicate since there is an extra parameter that
controls the smallness. In our case, the variable $I$ controls both the
smallness and the distance to the origin at the same time.

Let $\\{\cdot,\cdot\\}$ denote the standard Poisson bracket. Recall that for a
differentiable function $G$ it holds:

$\frac{d}{dt}G\circ X_{F}^{t}=\\{G,F\\}\circ X_{F}^{t}.$

######  Lemma 2.

Suppose that for $H(I,{\theta})$ there exist $N_{2m}(I)=N_{0}+B(N_{0})$ with
$B(X)=\sum_{j=2}^{m}b_{j}X^{j}$, $R(I,{\theta})=R^{[>2m]}(I,{\theta})$ and
$G(I,{\theta})=\mathcal{O}^{2}(I)$ such that $\Psi:=X_{G}^{1}$ satisfies

$H\circ\Psi(I,{\theta})=N_{2m}(I)+R(I,{\theta}).$

  1. 1.

Then there exists ${\tilde{G}}(I,{\theta})$, that is free from resonant
monomials of order $<2m$, such that $\tilde{\Psi}:=X_{\tilde{G}}^{1}$
normalises $H$ to the same normal form, i.e., for some
${\tilde{R}}(I,{\theta})=(\tilde{R})^{[>2m]}(I,{\theta})$ we have:

$H\circ{\tilde{\Psi}}(I,{\theta})=N_{2m}(I)+{\tilde{R}}(I,{\theta}).$

  2. 2.

If, an addition to the previous assumption, we have that the original
$H(I,{\theta})$ has the form

$H(I,{\theta})=N_{m}(I)+R^{[>m]}(I,{\theta}),$

where $N_{m}=N_{m}^{[2,\dots,m]}$, then there exists a polynomial
$F=F^{[m,2m-2]}$, that is free from resonant monomials, such that
$\Phi:=X_{F}^{1}$ normalises $H$ to the same normal form, i.e., for some
${\accentset{\approx}{R}}(I,{\theta})={\accentset{\approx}{R}}^{[>2m]}(I,{\theta})$
we have:

$H\circ\Phi(I,{\theta})=N_{2m}(I)+{\accentset{\approx}{R}}(I,{\theta}).$

Proof: (1). All the calculations below are made in the sense of formal Taylor-
Fourier expressions. Suppose that $K(I,{\theta})$ is such that
$\\{N_{0},K\\}=0$. Notice that in this case
$\\{N_{2m},K\\}=B^{\prime}(N_{0})\\{N_{0},K\\}=0$. Use $K(I,{\theta})$ as a
Hamiltonian to define $k(I,{\theta}):=X_{K}^{1}$. Then by Taylor formula we
have:

$\displaystyle H\circ\Psi\circ k$ $\displaystyle=(N_{2m}+R)\circ
k=(N_{2m}+R)\circ X_{K}^{t}\Big{|}_{t=1}=N_{2m}+R+\\{(N_{2m}+R),K\\}$
$\displaystyle+\frac{1}{2}\\{\\{(N_{2m}+R),K\\},K\\}+\dots=N_{2m}+R_{1},$

where $R_{1}(I,{\theta})=R_{1}^{[>2m]}(I,{\theta})$.

It is a classical fact that the composition $\Psi\circ k$ in the sense of
formal power series is the time-one map of another Hamiltonian given by the
Cambell-Baker-Dynkin formula [Dragt, Appendix C], [LlMM, Appendix]; here we
denote it by CBD formula. Note that in these references the usual notation for
the Hamiltonian vector field defined by $G$ is ${\mathcal{L}}_{G}$, and
$\exp({\mathcal{L}}_{G})$ stands for its time one map. In the present paper
the same map is denoted by $X_{G}^{1}$. Now, suppose that $\Psi=X_{G}^{1}$ and
$k=X_{K}^{1}$. CBD formula implies that the composition of these maps
satisfies:

$\begin{split}&\tilde{\Psi}:=\Psi\circ
k=X_{\tilde{G}}^{1},\quad\text{where}\\\
&{\tilde{G}}=G+K+\frac{1}{2}\\{G,K\\}+\frac{1}{12}\\{G,\\{G,K\\}\\}-\frac{1}{12}\\{K,\\{K,G\\}\\}+\cdots\end{split}$

The last sum is to be understood in the sense of formal power series in $I$.

To prove Lemma 2, we use CBD formula, and choose $K$ recursively (order by
order in $I$) so that ${\tilde{G}}$ has no resonant terms up to order $2m$. At
each step of the recursion we choose $(-K(I,\theta))$ to be equal to the
lowest order resonant term of $G$, and set ${\tilde{G}}$ to be the new $G$. As
we saw above, the map $\tilde{\Psi}=\Psi\circ K$, used as a normalization map,
brings $H$ to the same normal form as $\Psi$ did. But its generating
Hamiltonian ${\tilde{G}}$ has no lower order resonant monomials. Iterating
this procedure, we get a normalization with the desired property.

(2). Since we can normalise $H=N_{m}+R^{[>m]}$ to $N_{2m}$ with the help of
the generating function $G=\mathcal{O}^{2}(I)$, then, by (1), we can also
achieve the normalization using the transformation $\tilde{\Psi}$ generated by
a resonance-free Hamiltonian $\tilde{G}$. Note that
$\tilde{G}=\mathcal{O}^{2}(I)$.

By the Taylor formula for power series, we have:

$\displaystyle H\circ\tilde{\Psi}$
$\displaystyle=(N_{m}+R^{[>m]})\circ{\tilde{\Psi}}=(N_{m}+R^{[>m]})\circ
X_{\tilde{G}}^{t}\Big{|}_{t=1}=N_{m}+R^{[>m]}$
$\displaystyle+\\{(N_{m}+R^{[>m]}),\tilde{G}\\}+\frac{1}{2}\\{\\{(N_{m}+R^{[>m]}),\tilde{G}\\},\tilde{G}\\}+\dots=N_{2m}+R_{1}.$

Since $\tilde{G}$ is resonance-free, any monomial $P$ in $\tilde{G}$ gives a
non-zero impact $\\{N_{0},P\\}$ to the sum above, whose order in $I$ is
strictly larger than the order of $P$. By comparing the orders of the
coefficients in $I$ we see that the lowest possible order of a monomial in
$\\{N_{0},\tilde{G}\\}$ is the same as that in $R^{[>m]}$, and hence
$\tilde{G}=\tilde{G}^{[\geq m]}$. Finally notice that the reduced generating
function $F:=\tilde{G}^{[m,2m-2]}$ produces the same normal form.

$\Box$

The following lemma introduces the notations used in the proof of the Main
Theorem. Here we use the results of Lemma 2 to relate the conjugating function
to the solutions of the homological equation (3.1) below.

######  Lemma 3.

Adopt the notations for the degrees of polynomials from Sec. 2.1.3 (in
particular, $N_{n}=N^{[2,m_{n}]}$ as in 2.3, and
$R_{n}=R_{n}^{[m_{n}+1,m_{n+1}]}$). Let $B(X)=\sum_{j=1}^{\infty}b_{j}X^{j}$.
Suppose that $H_{n}$ has the form

$H_{n}=N_{n}+\widetilde{R_{n}}=N_{n}+R_{n}+\widetilde{R_{n}}^{[>m_{n+1}]}.$

where $N_{n}(I)=N_{0}+B(N_{0})^{[4,m_{n}]}$.

Suppose that there exists $G(I,{\theta})=\mathcal{O}^{2}(I)$ such that
$\Psi:=X_{G}^{1}$ satisfies

$H\circ\Psi(I,{\theta})=N_{m+1}(I)+R(I,{\theta}).$

Then there exists a polynomial (in $I$) $F_{n}=F_{n}^{[m_{n},m_{n+1}-1]}$ with
the following properties: the time one map $\Phi_{n}:=X_{F_{n}}^{1}$ satisfies

$H_{n+1}:=H_{n}\circ\Phi_{n}=N_{n+1}+\widetilde{R_{n+1}},$

and $F_{n}$ satisfies

$\\{N_{n},F_{n}\\}^{[m_{n}+1,m_{n+1}]}+R_{n}+N_{n}-N_{n+1}=0,$ (3.1)

and

$\widetilde{R_{n+1}}:=A_{n}+B_{n}+C_{n},$

where

$A_{n}:=\widetilde{R_{n}}^{[>m_{n+1}]}\circ\Phi_{n},\quad
B_{n}:=\int_{0}^{1}\\{(1-t)\\{N_{n},F_{n}\\}+R_{n},F_{n}\\}\circ
X_{F_{n}}^{t}dt,$ (3.2) $C_{n}=(\\{N_{n},F_{n}\\})^{[>m_{n+1}]}.$ (3.3)

Notice that the expressions for $A_{n}$, $B_{n}$, $C_{n}$ start with terms of
order $m_{n+1}+1$, and hence,
$\widetilde{R_{n+1}}=\widetilde{R_{n+1}}^{[>m_{n+1}]}$, as needed.

Proof. Let $m=m_{n}=2^{n}+1$. Then $m_{n+1}=2m-1$. With the notations for the
degrees of polynomials from Sec. 2.1.3, Lemma 2 implies that there exists a
polynomial $F_{n}=F_{n}^{[m_{n},m_{n+1}-1]}$ such that
$\Phi_{n}:=X_{F_{n}}^{1}$ satisfies
$H_{n}\circ\Phi_{n}=N_{n+1}+\widetilde{R_{n+1}}$. By the Taylor formula we
have:

$\displaystyle\begin{aligned}
H_{n}\circ\Phi_{n}=&(N_{n}+R_{n}+\widetilde{R_{n}}^{[>m_{n+1}]})\circ
X_{F_{n}}^{t}\Big{|}_{t=1}=N_{n}+\\{N_{n},F_{n}\\}+R_{n}+\\\
&\int_{0}^{1}\\{(1-t)\\{N_{n},F_{n}\\}+R_{n},F_{n}\\}\circ
X_{F_{n}}^{t}\,dt+\widetilde{R_{n}}^{[>m_{n+1}]}\circ\Phi_{n}\\\
=&N_{n+1}+\widetilde{R_{n+1}}.\end{aligned}$ (3.4)

Notice that by extracting all the terms of orders $m_{n}+1,\dots,m_{n+1}$ from
the equation above, one gets the cohomological equation (3.1).

$\Box$

###  3.2 Homological equation order by order.

Here we rewrite equation (3.1) as a (finite) set of equations for each degree
of $I$. Equations corresponding to degrees $m_{n}+1,\dots,m_{n+1}$ will
formally determine $F_{n}$ (they are written out explicitly in (3.5)). The
rest of equations define $C_{n}$ (which is a part of the new remainder term).
Equating coefficients with the same homogeneous degree in $I$ in both sides of
(3.4) we obtain for the degrees from $m_{n}+1$ to $m_{n+1}$ the following
recursive formula (we write $m$ instead of $m_{n}$ for typographic reasons):

$\displaystyle\begin{aligned} &\\{N_{0},F^{[m]}\\}+R^{[m+1]}=N^{[m+1]},\\\
&\\{N_{0},F^{[m+1]}\\}+\\{N^{[3]},F^{[m]}\\}+R^{[m+2]}=N^{[m+2]},\\\
&\\{N_{0},F^{[m+2]}\\}+\\{N^{[4]},F^{[m]}\\}+\\{N^{[3]},F^{[m+1]}\\}+R^{[m+3]}=N^{[m+3]},\\\
&\dots\\\
&\\{N_{0},F^{[2m-2]}\\}+\sum_{j=0}^{m-3}\\{N^{[m-j]},F^{[m+j]}\\}+R^{[2m-1]}=N^{[2m-1]}.\end{aligned}$
(3.5)

Recall that $2m_{n}-1=m_{n+1}$, see Sec. 2.1.2. From the formal solvability we
know that each of these equations has a formal solution $F_{n}^{[m+j]}$. Of
course, such a solution is not unique. We will make the solution unique by
prescribing the condition

$\int_{{\mathbb{T}}^{d}}F_{n}^{[m+j]}(I,{\theta})=0.$

As we will see, this normalization will allow us to get the estimates needed
for the proof of the convergence. The sum of the terms of orders
$m_{n+1}+1,\dots,m_{n+1}+m_{n}-2$ (i.e., $2m_{n},\dots,3m_{n}-3$) that appear
in equation (4.1) is denoted by $C_{n}$. In the notation $m=m_{n}$, we have:
$C_{n}=C_{n}^{[2m,3m-3]}$. The terms of the uniform degree satisfy

$\displaystyle\begin{aligned}
&C_{n}^{[2m]}=\\{N_{[3]},F^{[2m-2]}\\}+\\{N_{[4]},F^{[2m-3]}\\}+\dots+\\{N_{[m]},F^{[m+1]}\\},\\\
&C_{n}^{[2m+1]}=\\{N_{[4]},F^{[2m-2]}\\}+\\{N_{[5]},F^{[2m-3]}\\}+\dots+\\{N_{[m]},F^{[m+2]}\\}\\\
&\dots\\\ &C_{n}^{[3m-3]}=\\{N_{[m]},F^{[2m-2]}\\}.\end{aligned}$ (3.6)

This can be written more compactly as

$\displaystyle
C_{n}=\sum_{k=1}^{m-2}\\{F^{[2m-1-k]},\sum_{j=k+2}^{m}N^{[k+j]}\\}.$ (3.7)

This should be viewed as a definition of the remainder term $C_{n}$.

###  3.3 An important simplification.

In the case when the normal form is an analytic function of $N_{0}(I)$ as in
(2.2), we have an important simplification. Denote

$\displaystyle g_{2j}(I):=jb_{j}(N_{0}(I))^{j-1}\text{ and }g_{2j+1}(I)\equiv
0.$ (3.8)

Then for $j\in{\mathbb{N}}$ we have:

$\displaystyle\begin{aligned}
&\\{N^{[2j]},F\\}=\\{b_{j}(N_{0})^{j},F\\}=jb_{j}(N_{0})^{j-1}\\{N_{0},F\\}=g_{2j}(I)\\{N_{0},F\\}\\\
&\\{N^{[2j+1]},F\\}=g_{2j+1}(I)\\{N_{0},F\\}\equiv 0.\end{aligned}$ (3.9)

We formulate this as a lemma:

######  Lemma 4.

If the normal form is an analytic function of $N_{0}(I)$ as in (2.2), then
equation (3.5) is equivalent to

$\displaystyle\begin{aligned} &\\{N_{0},F^{[m]}\\}+R^{[m+1]}=N^{[m+1]},\\\
&\\{N_{0},F^{[m+1]}\\}+g_{3}(I)\\{N_{0},F^{[m]}\\}+R^{[m+2]}=N^{[m+2]},\\\
&\\{N_{0},F^{[m+2]}\\}+g_{4}(I)\\{N_{0},F^{[m]}\\}+g_{3}(I)\\{N_{0},F^{[m+1]}\\}+R^{[m+3]}=N^{[m+3]},\\\
&\dots\\\
&\\{N_{0},F^{[2m-2]}\\}+\sum_{j=0}^{m-3}g_{m-j}(I)\\{N_{0},F^{[m+j]}\\}+R^{[2m-1]}=N^{[2m-1]}.\end{aligned}$
(3.10)

and

$\displaystyle
C_{n}=\sum_{k=1}^{m-2}\left(\\{F^{[2m-1-k]},\,N_{0}\\}\cdot\sum_{j=k+2}^{m}g_{j}\right).$
(3.11)

###  3.4 Homological equations in majorants.

Here we study a simple recursive formula and estimate its terms. Later it will
provide an important estimate of $|\\{N_{0},F^{j}\\}|_{\rho_{n},\rho_{n}}$.
Here is the idea: suppose that in the Lemma above for some ${\epsilon}>0$, for
all $j=0,\dots,m$ we have:

$P_{j}:=|R^{[m+j]}|_{\rho_{n},\rho_{n}}+|N^{[m+j]}|_{\rho_{n},\rho_{n}}\leq{\epsilon},\quad|g_{j}|_{\rho_{n}}\leq
1/4^{j}.$

Define $S_{j}$ by the relations (3.12) below. Then, by Lemma 4, for all
$j=0,\dots,m$ we have

$|\\{N_{0},F^{j}\\}|_{\rho_{n},\rho_{n}}\leq S_{j}.$

######  Lemma 5.

Given ${\epsilon}>0$, suppose that for all $j=1,\dots,m-1$ the numbers $P_{j}$
satisfy

$0<P_{j}\leq{\epsilon}.$

Let $S_{j}$ be defined recursively by equations

$\displaystyle S_{1}=P_{1},$ (3.12) $\displaystyle
S_{2}=P_{2}+\frac{1}{4}S_{1},$ $\displaystyle
S_{3}=P_{3}+\frac{1}{4}S_{2}+\frac{1}{4^{2}}S_{1}$ $\displaystyle
S_{4}=P_{4}+\frac{1}{4}S_{3}+\frac{1}{4^{2}}S_{2}+\frac{1}{4^{3}}S_{1}$
$\displaystyle\dots$ $\displaystyle
S_{m-1}=P_{m-1}+\sum_{j=1}^{m-1}\frac{1}{4^{j}}S_{m-1-j}.$

Then for each $j$ we have

$S_{j}\leq 2{\epsilon},\quad j=1,\dots,m-1.$

Proof. By the formula for $S^{[j]}$ above,

$S_{j}\leq
P_{j}+\frac{1}{4}S_{j-1}+\frac{1}{4}(S_{j-1}-P_{j-1})=P_{j}+2\frac{1}{4}S_{j-1}\leq
P_{j}+S_{j-1}/2.$

This implies

$S_{j}\leq\sum_{k=0}^{j-1}2^{-k}P_{j-k}\leq{\epsilon}\sum_{k=0}^{j-1}2^{-k}<2{\epsilon}.$

$\Box$

##  4 Formal solution provides analytic with estimates.

In this section we study a homological equation (4.1) below with an analytic
right-hand side $Q(I,{\theta})$. Assuming that it has a formal solution, we
will find an analytic one, and estimate it in terms of the right hand side.
Similar procedures appear in [Ll].

######  Lemma 6.

Let $N_{0}(I)=I^{tr}{\Omega}I$ where ${\Omega}$ is a symmetric matrix with
$\det{\Omega}\neq 0$, and let $Q(I,{\theta})$ be analytic in an annulus
$\mathbb{A}_{\rho,\sigma}$ for some $\rho$, $\sigma>0$. Suppose that the
following equation has a formal solution $\widetilde{F}(I,{\theta})$:

$\displaystyle\\{N_{0},\widetilde{F}\\}=Q.$ (4.1)

Then equation (4.1) has an analytic solution $F(I,{\theta})$, defined in
$\mathbb{A}_{\rho,\sigma}$, and for any $0<{\delta}<\rho$, $0<\gamma<\sigma$
we have:

$|F|_{\rho-{\delta},\sigma-\gamma}\leq
c(d,{\Omega})\frac{1}{{\delta}\gamma^{d}}|Q|_{\rho,\sigma},$

where $c(d,{\Omega})$ is a constant only depending on $d$ and ${\Omega}$.

Moreover, if $Q(I,{\theta})$ is a homogeneous polynomial in $I$ with
coefficients depending on ${\theta}$, then so is $F(I,{\theta})$.

Proof.  Expanding $F$ formally into a Fourier series:
$F=\sum_{k\in{\mathbb{Z}}^{d}}\widehat{F}_{k}(I)e^{2\pi i\langle
k,{\theta}\rangle}$, we get:

$\\{N_{0},F\\}=\sum_{j=1}^{d}F_{{\theta}_{j}}(N_{0})_{I_{j}}=2\pi
i\sum_{k\in{\mathbb{Z}}^{d}}\langle
k,2{\Omega}I\rangle\widehat{F}_{k}(I)e^{2\pi i\langle k,{\theta}\rangle}.$

Recall that ${\Omega}$ is symmetric, so $\langle
k,{\Omega}I\rangle=\langle{\Omega}k,I\rangle$. Expressing
$Q=\sum_{k\in{\mathbb{Z}}^{d}}\widehat{Q}_{k}(I)e^{2\pi i\langle
k,{\theta}\rangle}$, we can rewrite equation (4.1) as a series of equations
indexed by $k$:

$\displaystyle\widehat{Q}_{k}(I)=4\pi
i\langle{\Omega}k,I\rangle\widehat{F}_{k}(I).$ (4.2)

If $\langle k,{\Omega}I\rangle\neq 0$, we can express
$\widehat{F}_{k}=\widehat{Q}_{k}(I)/(4\pi i\langle{\Omega}k,I\rangle)$.

Since we have assumed existence of a formal solution of the homological
equation (4.1) (and hence, a solution of (4.2) for each $k$), we have:

$\langle{\Omega}k,I\rangle=0\Rightarrow\widehat{Q}_{k}(I)=0.$

Hence, for $\langle{\Omega}k,I\rangle=0$, the equation is satisfied for any
value of $\widehat{F}_{k}(I)$. We define $\widehat{F}_{k}$ at these points by
continuity. A way to do it is the following. Differentiate equation (4.2) in
the direction of ${\Omega}k$:

$\langle{\Omega}k,\nabla\widehat{Q}_{k}(I)\rangle=4\pi
i\left(|{\Omega}k|^{2}\widehat{F}_{k}(I)+\langle{\Omega}k,I\rangle\langle{\Omega}k,\nabla\widehat{F}_{k}(I)\rangle\right),$

where for a vector $v\in{\mathbb{R}}^{d}$ we denote
$|v|^{2}=\sum_{j=1}^{d}v_{j}^{2}$. For $\langle{\Omega}k,I\rangle=0$, define
$\widehat{F}_{k}(I)=\langle{\Omega}k,\nabla\widehat{Q}_{k}(I)\rangle/(4\pi
i|{\Omega}k|^{2})$. Summing up, we have defined a continuous function
$\widehat{F}_{k}(I)$ by

$\widehat{F}_{k}(I)=\frac{1}{4\pi
i}\begin{cases}\langle{\Omega}k,I\rangle^{-1}\widehat{Q}_{k}(I),\quad\langle{\Omega}k,I\rangle\neq
0,\\\
\frac{1}{|{\Omega}k|^{2}}\langle{\Omega}k,\nabla\widehat{Q}_{k}(I)\rangle,\ \
\langle{\Omega}k,I\rangle=0.\end{cases}$

Moreover, since $\widehat{F}_{k}(I)$ is analytic in
${\mathbb{D}}_{\rho}\setminus\\{\langle{\Omega}k,I\rangle=0\\}$ and bounded in
${\mathbb{D}}_{\rho}$, it is analytic in ${\mathbb{D}}_{\rho}$. Notice that if
in equation (4.2) $\widehat{Q}_{k}(I)$ is a homogeneous polynomial in $I$,
then so is $\widehat{F}_{k}(I)$.

Now let us estimate the norm of the solution. Fix $0<{\delta}<\rho/2$,
$0<\gamma<\sigma$. For each fixed $k\in{\mathbb{Z}}^{d}$, we will estimate the
corresponding $\widehat{F}_{k}(I)$ in two steps: first ‘$\delta/2$-close” to
the resonant plane $\langle{\Omega}k,I\rangle$, and then in the rest of
${\mathbb{D}}_{\rho-{\delta}}$.

For the first step, let
$\Pi_{\delta}=\\{\langle{\Omega}k,I\rangle=0\\}\cap{\mathbb{D}}_{\rho-{\delta}}$
be the part of the resonant plane falling into ${\mathbb{D}}_{\rho-{\delta}}$.
Notice that the orthogonal complement to this plane is formed by the vectors
$\alpha e^{2\pi i\phi}{\Omega}k$, $\alpha\geq 0$, $\phi\in[0,1)$. Let

$\Delta=\left\\{I=\alpha\frac{{\Omega}k}{|{\Omega}k|}e^{2\pi
i\phi}\,\Big{|}\,\alpha<{\delta}/2,\,\phi\in[0,1)\right\\}$

be the complex disk of radius ${\delta}/2$ centered at zero and orthogonal to
$\Pi_{\delta}$. Note that the restrictions of $\widehat{Q}_{k}(I)$ and
$\widehat{F}_{k}(I)$ to this disc are analytic. Consider the
${{\delta}}/2$-neighbourhood $O_{\delta}$ of $\Pi_{\delta}$:
$O_{\delta}=\bigcup_{I_{0}\in\Pi_{\delta}}(I_{0}+\Delta)$. Then and
$O_{\delta}\subset{\mathbb{D}}_{\rho-{\delta}}$.

For each fixed $I\in O_{\delta}$ there exists $I_{0}\in\Pi_{\delta}$ such that
$I\in I_{0}+\Delta$. We can estimate $|\widehat{F}_{k}(I)|$ by the maximum
modulus principle on the disk $I_{0}+\Delta$. Namely, for $I$ lying on the
boundary of this disk we have:
$|\langle{\Omega}k,I\rangle|=|\langle{\Omega}k,I_{0}\rangle+\langle{\Omega}k,{\delta}{\Omega}k/(2|{\Omega}k|)\rangle|=|{\Omega}k|{\delta}/2$.
Hence, for such $I$ we have

$|\widehat{F}_{k}(I)|\leq\frac{2|\widehat{Q}_{k}|_{\rho}}{4\pi{\delta}|{\Omega}k|}<\frac{|\widehat{Q}_{k}|_{\rho}}{{\delta}|{\Omega}k|}.$

As the second step in this estimate, consider
$I\in{\mathbb{D}}_{\rho-{\delta}}\setminus O_{\delta}$. Here
$|\langle{\Omega}k,I\rangle|\geq|{\Omega}k|{\delta}/2$, so
$|\widehat{F}_{k}(I)|$ satisfies the same estimate as above.

By Cauchy estimates, we have:

$|\widehat{Q}_{k}|_{\rho}\leq|Q|_{\rho,\sigma}e^{-|k|\sigma}.$

Since det$\,{\Omega}\neq 0$, there exists a constant $c({\Omega})$ such that
$|{\Omega}k|\geq|k|/c({\Omega})$ for all $k$. Then

$\displaystyle|\widehat{F}_{k}|_{\rho-{\delta}}\leq\frac{1}{{\delta}|{\Omega}k|}|\widehat{Q}_{k}|_{\rho}\leq
c({\Omega})\frac{e^{-\sigma|k|}}{{\delta}|k|}|Q|_{\rho,\sigma}.$

Finally, for small ${\delta}$ and $\gamma$ we have:

$\displaystyle|F|_{\rho-{\delta},\sigma-\gamma}\leq$
$\displaystyle\sum_{k\in{\mathbb{Z}}^{d}\setminus\\{0\\}}e^{(\sigma-\gamma)|k|}|\widehat{F}_{k}|_{\rho-{\delta}}\leq\frac{c({\Omega})}{{\delta}}\sum_{k\in{\mathbb{Z}}^{d}\setminus\\{0\\}}\frac{e^{-\gamma|k|}}{|k|}|Q|_{\rho,\sigma}$
$\displaystyle\leq$
$\displaystyle\frac{c(d,{\Omega})}{{\delta}\gamma^{d}}|Q|_{\rho,\sigma},$

where $c(d,{\Omega})$ is a constant only depending on $d$ and ${\Omega}$. The
estimates above are very wasteful, but they are enough for our purposes.
$\Box$

##  5 Proof of Proposition 1.

Here we summarize the preparatory work to complete the proof of Proposition 1.
Let us return to the original problem. For a fixed $n$, let the necessary
constants be as in Sec.2.1.2,
$|\widetilde{R_{n}}|_{\rho_{n}}\leq{\delta}_{n}^{\kappa}$, and let
$g_{2j}(I)=j\,b_{j}\,(N_{0}(I))^{j-1}$ as in (3.8).

###  5.1 Estimate of $|\\{N_{0},F_{n}\\}|_{\rho_{n},\rho_{n}}$ and
$|C_{n}|_{\rho_{n},\rho_{n}}$.

For $j=1,\dots,m_{n}-1$ denote

$P_{j}:=|N^{[m_{n}+j]}|_{\rho_{0}}+|R^{[m_{n}+j]}|_{\rho_{n}}.$

By the choice of $\rho_{0}$, see Sec. 2.1.2, for all $j=1,\dots,m_{n}-1$ we
have:

$|g_{j}(I)|_{\rho_{0}}\leq
4^{-j},\quad|N^{[m_{n}+j]}|_{\rho_{0}}\leq{\delta}_{n}^{\kappa}.$

Since for $j=1,\dots,m_{n}-1$ we have
$|R^{[m_{n}+j]}|_{\rho_{n}}\leq|\widetilde{R_{n}}|_{\rho_{n}}\leq{\delta}_{n}^{\kappa}$,
and so for these values of $j$ we get

$P_{j}\leq 2{\delta}_{n}^{\kappa}.$

let $S_{j}$ be defined by (3.12). By Lemma 5, for $j=1,\dots m-1$ we have
$S_{j}\leq 2{\epsilon}$. Equations (3.10) imply that for $j=1,\dots m-1$ we
have

$\displaystyle|\\{N_{0},F_{n}^{[m+j-1]}\\}|_{\rho_{n},\rho_{n}}\leq S_{j}\leq
2{\epsilon}=4{\delta}_{n}^{\kappa}.$ (5.1)

By linearity,

$|\\{N_{0},F_{n}\\}|_{\rho_{n},\rho_{n}}\leq\sum_{j=1}^{m_{n}-1}|\\{N_{0},F_{n}^{[m_{n}+j-1]}\\}|_{\rho_{n},\rho_{n}}\leq
4m_{n}{\delta}_{n}^{\kappa}\leq 4{\delta}_{n}^{\kappa-1}.$

The latter estimate follows from the definition of $m_{n}$ and ${\delta}_{n}$,
see Section 2.1.2.

Moreover, by (3.11),

$\displaystyle|C_{n}|_{\rho_{n}}$
$\displaystyle=\sum_{k=1}^{m-2}\left(S_{m-k}\,\sum_{j=k+2}^{m}G_{j}\right)\leq\sum_{k=1}^{m-2}\left(S_{m-k}\,\sum_{j=k+2}^{\infty}4^{-j}\right)$
$\displaystyle\leq\frac{1}{3}\sum_{k=1}^{m-2}4^{-(k+1)}S_{m-k}\leq\frac{1}{2}{\epsilon}={\delta}_{n}^{\kappa}.$

Hence,

$\displaystyle|C_{n}|_{\rho_{n}}\leq{\delta}_{n}^{\kappa}.$ (5.2)

###  5.2 Estimates for $F_{n}$.

Consider equation (5.1). Lemma 6 with $\rho=\sigma=\rho_{n}$,
${\delta}=\gamma={\delta}_{n}$ and $|Q|_{\rho,\sigma}\leq
4{\delta}_{n}^{\kappa}$, implies:

$|F_{n}^{[m+j-1]}|_{\rho_{n}-{\delta}_{n},\rho_{n}-{\delta}_{n}}\leq
4c(d,{\Omega}){\delta}_{n}^{\kappa-d-1}.$

Since $F_{n}=F_{n}^{[m_{n},m_{n}+j-1]}$ where $m_{n}\leq{\delta}_{n}^{-1}$, we
get:

$|F_{n}|_{\rho_{n}-{\delta}_{n},\rho_{n}-{\delta}_{n}}\leq\sum_{j=1}^{m_{n}-1}|F_{n}^{[m+j-1]}|_{\rho_{n}-{\delta}_{n},\rho_{n}-{\delta}_{n}}\leq
m_{n}\,4c(d,{\Omega}){\delta}_{n}^{\kappa-d-1}\leq{\delta}_{n}^{\kappa-d-3}\leq{\delta}_{n}^{3}.$
(5.3)

The latter estimate follows from the definition of $\kappa$, see Section
2.1.2.

###  5.3 Estimates for $\Phi_{n}$.

Here we prove that with $F_{n}$ as above, estimates (2.10) and (2.11) hold
true. Indeed, the coordinate change $\Phi_{n}=X_{F_{n}}^{1}$ is the time one
map of the flow $X_{F_{n}}^{t}$ defined by the equations

$\dot{I}=\partial_{\theta}F_{n}(I,{\theta}),\quad\dot{\theta}=-\partial_{I}F_{n}(I,{\theta}).$

By (5.3) and Cauchy estimates we get

$|\partial_{I}F_{n}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-{\delta}_{n}}\leq{\delta}_{n}^{2},\quad|\partial_{\theta}F_{n}|_{\rho_{n}-{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{2}.$
(5.4)

Then for any $t\leq 1$:

$|X_{F_{n}}^{t}(I,{\theta})-(I,{\theta})|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}\leq
t\,{\delta}_{n}^{-1}|F_{n}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{2}.$
$X_{F_{n}}^{t}:\mathbb{A}_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}\mapsto\mathbb{A}_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}$
(5.5)

In particular, since $\Phi_{n}=X_{F_{n}}^{1}$, we get the desired formulas
(2.10) and (2.11).

###  5.4 Estimate of the new remainder $\widetilde{R_{n+1}}$.

######  Lemma 7.

For $F_{n}$ constructed above, estimate (2.9) holds:

$|\widetilde{R_{n+1}}|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}<4{\delta}_{n}^{\kappa}.$

Proof. By Lemma 3,

$\widetilde{R_{n+1}}=A_{n}+B_{n}+C_{n},$

where $A_{n}$, $B_{n}$ and $C_{n}$ are defined by (3.2) and (3.3).

Estimate of $A_{n}$:  Using (5.5), we get:

$|\widetilde{R_{n}}^{[>m_{n+1}]}\circ\Phi_{n}|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}\leq|\widetilde{R_{n}}|_{\rho_{n}-2\rho_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{\kappa}.$

Estimate of $C_{n}$: We showed in section 5.1 that

$|C_{n}|_{\rho_{n},\rho_{n}}\leq{\delta}_{n}^{\kappa}.$

Estimate of $B_{n}$: By (5.4),
$|\partial_{I}F_{n}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-{\delta}_{n}}\leq{\delta}_{n}^{2}$
and
$|\partial_{\theta}F_{n}|_{\rho_{n}-{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{2}.$
By (2.9)

$|R_{n}|_{\rho_{n},\rho_{n}}\leq|\widetilde{R_{n}}|_{\rho_{n},\rho_{n}}\leq{\delta}_{n}^{\kappa}.$

This implies, using Cauchy estimates, that

$|\\{R_{n},F_{n}\\}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{\kappa}.$

Notice that, by formulas (3.1) and (3.3), we have
$\\{N_{n},F_{n}\\}=R_{n}+N_{n}-N_{n-1}+C_{n}$.

By (2.6),

$|N_{n}-N_{n-1}|_{\rho_{0},\rho_{0}}=\sum_{j=1}^{m_{n}}N^{[m_{n}+j]}\leq
m_{n}{\delta}_{n}^{\kappa+1}\leq{\delta}_{n}^{\kappa}.$

and therefore

$|\\{N_{n},F_{n}\\}|_{\rho_{n},\rho_{n}}=|R_{n}|_{\rho_{n},\rho_{n}}+|N_{n}-N_{n-1}|_{\rho_{n},\rho_{n}}+|C_{n}|_{\rho_{n},\rho_{n}}\leq
3{\delta}_{n}^{\kappa}.$

Combining the above estimates, we get

$|\\{\\{N_{n},F_{n}\\},F_{n}\\}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq{\delta}_{n}^{\kappa},$

Since, by (5.5), for any $t\leq 1$ we have
$X_{F_{n}}^{t}:\mathbb{A}_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}\mapsto\mathbb{A}_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}$,
we obtain

$|\\{\\{N_{n},F_{n}\\}+R_{n},F_{n}\\}\circ
X_{F_{n}}^{t}|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}\leq|\\{\\{N_{n},F_{n}\\}+R_{n},F_{n}\\}|_{\rho_{n}-2{\delta}_{n},\rho_{n}-2{\delta}_{n}}\leq
2{\delta}_{n}^{\kappa}.$

$\Box$

Here we get the desired estimate for the remainder term. We have proved above
that

$|\widetilde{R_{n+1}}|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}<4{\delta}_{n}^{{\kappa}}$

Recall that $\widetilde{R_{n+1}}=\widetilde{R_{n+1}}^{[>m_{n+1}]}$. By Lemma 8
proved below, this implies the desired estimate

$|\widetilde{R_{n+1}}|_{\rho_{n+1},\rho_{n+1}}<{\delta}_{n+1}^{{\kappa}}$

This finishes the proof of Proposition 1, and hence Theorem 1 (as explained in
the introduction). $\Box$

######  Lemma 8.

Suppose that the constants ${\kappa}$, $b$, ${\delta}_{n}$, $q_{n}$,
$\rho_{n}$ are defined in Section 2.1.2, and an analytic function
$G(I,\theta)$ satisfies $G=G^{[>m_{n+1}]}$, and

$|G|_{\rho_{n}-3{\delta}_{n},\rho_{n}-3{\delta}_{n}}<4{\delta}_{n}^{{\kappa}}.$

Then

$|G|_{\rho_{n+1},\rho_{n+1}}<{\delta}_{n+1}^{{\kappa}}.$

Proof.  By the definition of ${\kappa}$ in Section 2.1.2 we have:
$q_{n}^{m_{n+1}+1}=q_{n}^{2^{n+1}+2}<q_{n}^{2^{n+1}}=2b=2^{-{\kappa}-2}$. Also
recall that ${\delta}_{n+1}=2^{-1}{\delta}_{n}$.

Since $G$ starts with terms of degree $m_{n+1}=2^{n+1}+2$, we have:

$|G|_{q_{n}(\rho_{n}-3{\delta}_{n}),q_{n}(\rho_{n}-3{\delta}_{n})}<q_{n}^{2^{n+1}+2}\,4{\delta}_{n}^{{\kappa}}\leq
2^{-{\kappa}-2}\,4{\delta}_{n}^{{\kappa}}\leq{\delta}_{n+1}^{{\kappa}}.$

$\Box$

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E-mail addresses:

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