Content-Type: multipart/mixed; boundary="-------------0003170823538" This is a multi-part message in MIME format. ---------------0003170823538 Content-Type: text/plain; name="00-119.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-119.comments" in press Bullettin Societe Mathematique de France ---------------0003170823538 Content-Type: text/plain; name="00-119.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-119.keywords" Siegel's center problem,small divisors, Gevrey classes ---------------0003170823538 Content-Type: application/x-tex; name="carletti.marmi.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="carletti.marmi.tex" %&LaTeX \documentclass{amsart} \usepackage{amsmath,amsfonts} \usepackage{amscd,amsthm} \theoremstyle{plain} %% This is the default \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{notation}[theorem]{Notation} \newtheorem{Proposition 3.9}{prop39} \renewcommand{\thenotation}{} \def\limind{\mathop{\oalign{lim\cr\hidewidth$\longrightarrow$\hidewidth\cr}}} \def\limproj{\mathop{\oalign{lim\cr\hidewidth$\longleftarrow$\hidewidth\cr}}} \numberwithin{equation}{section} \DeclareMathOperator{\Val}{Val \!} \begin{document} \vspace{-30em} IN PRESS: BULLETIN SOCIETE MATHEMATIQUE DE FRANCE \vspace{+3em} \title[Linearization of germs of diffeomorphisms] {Linearization of analytic and non--analytic germs of diffeomorphisms of $({\mathbb C},0)$} \author{Timoteo Carletti, Stefano Marmi} \date{\today} \address[Timoteo Carletti, Stefano Marmi]{Dipartimento di Matematica ``Ulisse Dini'', Viale Morgagni 67/A, 50134 Florence, Italy} \email[Timoteo Carletti]{carletti@udini.math.unifi.it} \email[Stefano Marmi]{marmi@udini.math.unifi.it} %\subjclass{Primary 05C38, 15A15; Secondary 05A15, 15A18} \keywords{Siegel's center problem, small divisors, Gevrey classes} \begin{abstract} We study Siegel's center problem on the linearization of germs of diffeomorphisms in one variable. In addition of the classical problems of formal and analytic linearization, we give sufficient conditions for the linearization to belong to some algebras of ultradifferentiable germs closed under composition and derivation, including Gevrey classes. In the analytic case we give a positive answer to a question of J.-C. Yoccoz on the optimality of the estimates obtained by the classical majorant series method. In the ultradifferentiable case we prove that the Brjuno condition is sufficient for the linearization to belong to the same class of the germ. If one allows the linearization to be less regular than the germ one finds new arithmetical conditions, weaker than the Brjuno condition. We briefly discuss the optimality of our results. \end{abstract} \maketitle \section{introduction} \indent In this paper we study the Siegel center problem \cite{Herman}. Consider two subalgebras $A_{1}\subset A_{2}$ of $z\mathbb{C}\left[\left[ z\right]\right]$ closed with respect to the composition of formal series. For example $z\mathbb{C}\left[\left[ z\right]\right]$, $z\mathbb{C}\{ z \}$ (the usual analytic case) or Gevrey--$s$ classes, $s>0$ (i.e. series $F(z)=\sum_{n\ge 0}f_{n}z^n$ such that there exist $c_{1}, c_{2}>0$ such that $|f_{n}|\le c_{1}c_{2}^n(n!)^s$ for all $n\ge 0$). Let $F\in A_{1}$ being such that $F^{\prime}\left( 0 \right)=\lambda\in \mathbb{C}^{*}$. We say that $F$ is linearizable in $A_{2}$ if there exists $H\in A_{2}$ tangent to the identity and such that % \begin{equation} F \circ H = H \circ R_\lambda \label{homological} \end{equation} % where $R_\lambda \left( z \right)= \lambda z$. When $|\lambda|\not= 1$, the Poincar\'e-Konigs linearization theorem assures that $F$ is linearizable in $A_{2}$. When $|\lambda|=1$, $\lambda = e^{2\pi i \omega}$, the problem is much more difficult, especially if one looks for {\it necessary and sufficient} conditions on $\lambda$ which assure that {\it all} $F\in A_{1}$ {\it with the same $\lambda$ are linearizable in $A_{2}$}. The only trivial case is $A_{2}= z\mathbb{C}\left[\left[ z\right]\right]$ (formal linearization) for which one only needs to assume that $\lambda$ is not a root of unity, i.e. $\omega\in \mathbb{R}\setminus\mathbb{Q}$. In the analytic case $A_{1}=A_{2}=z\mathbb{C}\{z\}$ let $S_{\lambda}$ denote the space of analytic germs $F\in z\mathbb{C}\{z\}$ analytic and injective in the unit disk $\mathbb{D}$ and such that $DF(0)=\lambda$ (note that any $F\in z\mathbb{C}\{z\}$ tangent to $R_{\lambda}$ may be assumed to belong to $S_{\lambda}$ provided that the variable $z$ is suitably rescaled). Let $R(F)$ denote the radius of convergence of the unique tangent to the identity linearization $H$ associated to $F$. J.-C. Yoccoz \cite{Yoccoz} proved that the {\it Brjuno condition} (see Appendix \ref{appfrazioni}) is necessary and sufficient for having $R(F)>0$ for all $F\in S_{\lambda}$. More precisely Yoccoz proved the following estimate: assume that $\lambda = e^{2\pi i \omega}$ is a Brjuno number. There exists a universal constant $C>0$ (independent of $\lambda$) such that % \begin{equation*} \label{Yoccozestimate} |\log R(\omega ) +B(\omega )|\le C \end{equation*} % where $R(\omega )= \inf_{F\in S_{\lambda}} R(F)$ and $B$ is the Brjuno function (\ref{Brjunofunction}). Thus $\log R(\omega )\ge -B(\omega ) - C$. \indent Brjuno's proof \cite{Brjuno} gives an estimate of the form % \begin{equation*} \label{Brjunoestimate} \log r(\omega )\ge -C'B(\omega )-C'' \end{equation*} % where one can choose $C'=2$ \cite{Herman}. Yoccoz's proof is based on a geometric renormalization argument and Yoccoz himself asked whether or not was possible to obtain $C'=1$ by direct manipulation of the power series expansion of the linearization $H$ as in Brjuno's proof (\cite{Yoccoz}, Remarque 2.7.1, p. 21). Using an arithmetical lemma due to Davie \cite{Davie} (Appendix \ref{appDavie}) we give a positive answer (Theorem \ref{Yoccozlower}) to Yoccoz's question. \indent We then consider the more general ultradifferentiable case $A_{1}\subset A_{2}\not= z{\mathbb C}\{z\}$. If one requires $A_{2}=A_{1}$, i.e. the linearization $H$ to be as regular as the given germ $F$, once again the Brjuno condition is sufficient. Our methods do not allow us to conclude that the Brjuno condition is also necessary, a statement which is in general false as we show in section 2.3 where we exhibit a Gevrey--like class for which the sufficient condition coincides with the optimal arithmetical condition for the associated linear problem. Nevertheless it is quite interesting to notice that given any algebra of formal power series which is closed under composition (as it should if one whishes to study conjugacy problems) and derivation a germ in the algebra is linearizable {\it in the same algebra} if the Brjuno condition is satisfied. If the linearization is allowed to be less regular than the given germ (i.e. $A_{1}$ is a proper subset of $A_{2}$) one finds a new arithmetical condition, weaker than the Brjuno condition. This condition is also optimal if the small divisors are replaced with their absolute values as we show in section 2.4. We discuss two examples, including Gevrey--$s$ classes.\footnote{We refer the reader interested in small divisors and Gevrey--$s$ classes to \cite{Lochak, GramchevYoshino1, GramchevYoshino2}.} \indent {\it Acknwoledgements.} We are grateful to J.--C. Yoccoz for a very stimulating discussion concerning Gevrey classes and small divisor problems. \section{the Siegel center problem} Our first step will be the formal solution of equation (\ref{homological}) assuming only that $F\in z\mathbb{C}[[z]]$. Since $F\in z\mathbb{C}\left[ \left[ z \right] \right]$ is assumed to be tangent to $R_{\lambda}$ then $F(z)= \sum_{n\ge 1}f_{n}z^n$ with $f_{1}=\lambda$. Analogously since $H\in z\mathbb{C}\left[ \left[ z \right] \right]$ is tangent to the identity $H(z)=\sum_{n=1}^\infty h_{n}z^n$ with $h_{1}=1$. If $\lambda$ is not a root of unity equation (\ref{homological}) has a unique solution $H\in z^{}\mathbb{C}\left[ \left[ z \right] \right]$ tangent to the identity: the power series coefficients satisfy the recurrence relation % \begin{equation} \label{recursive} h_{1}= 1 \; , \; h_{n} = \frac {1}{\lambda^n -\lambda} \sum_{m=2}^n f_{m}\sum_{n_{1}+\ldots +n_{m}= n\, , \, n_{i}\ge 1 } h_{n_{1}}\ldots h_{n_{m}}\; . \end{equation} In \cite{Teo} it is shown how to generalize the classical Lagrange inversion formula to non--analytic inversion problems on the field of formal power series so as to obtain an explicit non--recursive formula for the power series coefficients of $H$. \subsection{The analytic case: a direct proof of Yoccoz's lower bound} Let $S_{\lambda}$ denote the space of germs $F\in z\mathbb{C}\{z\}$ analytic and injective in the unit disk $\mathbb{D}=\{z\in \mathbb{C}\, , \, |z|<1\}$ such that $DF(0)=\lambda$ and assume that $\lambda = e^{2\pi i\omega}$ with $\omega\in \mathbb{R}\setminus\mathbb{Q}$. With the topology of uniform convergence on compact subsets of $\mathbb{D}$, $S_{\lambda}$ is a compact space. Let $H_{F}\in z\mathbb{C}[[z]]$ denote the unique tangent to the identity formal linearization associated to $F$, i.e. the unique formal solution of (\ref{homological}). Its power series coefficients are given by (\ref{recursive}). Let $R(F)$ denote the radius of convergence of $H_{F}$. Following Yoccoz (\cite{Yoccoz}, p. 20) we define % \begin{equation*} R(\omega ) = \inf_{F\in S_{\lambda}} R(F)\; . \end{equation*} % We will prove the following \begin{theorem} {\bf Yoccoz's lower bound}. \label{Yoccozlower} % \begin{equation} \log R(\omega) \ge -B(\omega ) -C \end{equation} % where $C$ is a universal constant (independent of $\omega$) and $B$ is the Brjuno function (\ref{Brjunofunction}). \end{theorem} Our method of proof of Theorem \ref{Yoccozlower} will be to apply an arithmetical lemma due to Davie (see Appendix B) to estimate the small divisors contribution to (\ref{recursive}). This is actually a variation of the classical majorant series method as used in \cite{Siegel, Brjuno}. \proof Let $s\left( z\right) =\sum_{n\geq 1}s_nz^n$ be the unique solution analytic at $z=0$ of the equation $s\left( z\right) =z+\sigma \left( s\left( z\right) \right)$, where $\sigma (z) = \frac{z^{2}(2-z)}{(1-z)^{2}}= \sum_{n\ge 2}nz^n$. The coefficients satisfy % \begin{equation} \label{recursive-s} s_{1}= 1 \; , \; s_{n} = \sum_{m=2}^n m\sum_{n_{1}+\ldots +n_{m}= n\, , \, n_{i}\ge 1 } s_{n_{1}}\ldots s_{n_{m}}\; . \end{equation} % Clearly there exist two positive constants $\gamma_{1},\gamma_{2}$ such that % \begin{equation} \label{sestimate} |s_n| \leq \gamma_{1}\gamma_{2}^{n}\; . \end{equation} % From the recurrence relation (\ref{recursive}) and Bieberbach--De Branges's bound $|f_{n}|\le n $ for all $n\ge 2$ we obtain % \begin{equation} \label{hest1} |h_{n}| \le \frac {1}{|\lambda^n -\lambda|} \sum_{m=2}^n m\sum_{n_{1}+\ldots +n_{m}= n\, , \, n_{i}\ge 1 } |h_{n_{1}}|\ldots |h_{n_{m}}|\; . \end{equation} % We now deduce by induction on $n$ that $|h_{n}|\le s_{n}e^{K(n-1)}$ for $n\ge 1$, where $K$ is defined in Appendix B. If we assume this holds for all $n'0$ (Davie's lemma, Appendix B). \endproof \subsection{The ultradifferentiable case} A classical result of Borel says that the map $J_{\mathbb{R}}\, : \, \mathcal{C}^\infty ([-1,1],\mathbb{R}) \rightarrow \mathbb{R}[[x]]$ which associates to $f$ its Taylor series at $0$ is surjective. On the other hand, $\mathbb{C}\{z\} = \limind_{r>0}\mathcal{O}(\mathbb{D}_{r})$, where $\mathbb{D}_{r}=\{z\in \mathbb{C}\, , \, |z|0$; \item{1.} There exists $C_{1}>0$ such that $M_{n+1}\le C_{1}^{n+1}M_{n}$ for all $n\ge 1$; \item{2.} The sequence $(M_{n})_{n\ge 1}$ is logarithmically convex; \item{3.} $M_{n}M_{m}\le M_{m+n-1}$ for all $m,n\ge 1$. \begin{definition} \label{ultradifferentiable} Let $f= \sum_{n\ge 1}f_{n}z^n\in z\mathbb{C} \left[ \left[ z \right] \right]$; $f$ belongs to the algebra $z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})}$ if there exist two positive constants $c_{1},c_{2}$ such that % \begin{equation} \label{coeffs} |f_{n}| \le c_{1}c_{2}^nM_{n}\;\; \hbox{for all}\; n\ge 1\; . \end{equation} % \end{definition} The role of the above assumptions on the sequence $(M_{n})_{n\ge 1}$ is the following: 0. assures that $z\mathbb{C}\{ z \}\subset z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})}$; 1. implies that $z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})}$ is stable for derivation. Condition 2. means that $\log M_{n}$ is convex, i.e. that the sequence $(M_{n+1}/M_{n})$ is increasing; it implies that $z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})_{n\ge 1}}$ is an algebra, i.e. stable by multiplication. Condition 3. implies that this algebra is {\it closed for composition}: if $f,g\in z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})_{n\ge 1}}$ then $f\circ g \in z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})_{n\ge 1}}$. This is a very natural assumption since we will study a {\it conjugacy} problem. Let $s>0$. A very important example of ultradifferentiable algebra is given by the algebra of {\it Gevrey}--$s$ series which is obtained chosing $M_{n}= (n!)^s$. It is easy to check that the assumptions 0.--3. are verified. But also more rapidly growing sequences may be considered such as $M_{n} = n^{a n^b}$ with $a >0$ and $10$ and $10 $ and $1 <\beta 0$. \endproof {\it Problem.} Are the arithmetical conditions stated in Theorem \ref{theultradifferentiable} optimal? In particular is it true that given any algebra $A= z\mathbb{C}\left[ \left[ z \right] \right]_{(M_{n})}$ and $F\in A$ then $H\in A$ {\it if and only if} $\omega$ is a Brjuno number? We believe that this problem deserves further investigations and that some surprising results may be found. In the next two sections we will give some preliminary results. \subsection{A Gevrey--like class where the linear and non linear problem have the same sufficient arithemtical condition} Let $\mathbb{C} \left[ \left[z \right] \right]_s$ denote the algebra of Gevrey--$s$ complex formal power series, $s>0$. If $s^{\prime}>s>0$ then $z\mathbb{C} \left[ \left[z \right] \right]_s \subset z\mathbb{C} \left[ \left[z \right] \right]_{s^\prime}$; let % \begin{equation*} A_{s} = \bigcap_{s^\prime > s} z\mathbb{C} \left[ \left[z \right] \right]_{s^\prime}\; . \end{equation*} % Clearly $A_{s}$ is an algebra stable w.r.t. derivative and composition. This algebra can be equivalently characterized requiring that given $f \left( z \right) = \sum_{n \geq 1} f_n z^n \in z\mathbb{C} \left[ \left[z \right] \right]$ one has % \begin{equation} \label{conditionAs} \limsup_{n \rightarrow \infty} \frac{\log \lvert f_n \rvert}{n \log n} \leq s \end{equation} % Consider Euler's derivative (see \cite{Du}, section 4) % \begin{equation} (\delta_{\lambda}f)(z) = \sum_{n=2}^\infty (\lambda^n-\lambda)f_{n}z^n \; , \end{equation} % with $\lambda = e^{2\pi i \omega}$. It acts linearly on $zA_{s}$ and it is a linear automorphism of $zA_{s}$ if and only if % \begin{equation} \label{lineararithmetic} \lim_{k \rightarrow \infty}\frac{\log q_{k+1}}{q_k \log q_k}=0 \end{equation} % where, as usual, $\left( q_k \right)_{k \in \mathbb{N}}$ is the sequence of the denominators of the convergents of $\omega$. This fact can be easily checked by applying the law of the best approximation (Lemma \ref{bestapproximation}, Appendix ~\ref{appfrazioni}) and the charaterization (\ref{conditionAs}) to \begin{equation*} h(z) = (\delta_{\lambda}^{-1}f)(z) = \sum_{n \geq 2} \frac{f_n}{\lambda^n -\lambda}z^n\; . \end{equation*} % Note that the arithmetical condition $\log q_{k+1}={\it o} \left(q_k \log q_k \right)$ is much weaker than Brjuno's condition. We now consider the Siegel problem associated to a germ $F\in A_{s}$. Applying the third statement of Theorem \ref{theultradifferentiable} with $N_n=\left( n! \right)^{s+\eta}$ and $M_n=\left( n! \right)^{s+\epsilon}$ for any positive fixed $\epsilon > \eta>0$ one finds that if the following arithmetical condition is satisfied % \begin{equation} \label{nonlineararithmetic} \lim_{k \rightarrow \infty}\frac{1}{\log q_{k}} \sum_{i=0}^{k} \frac{\log q_{i+1}}{q_i}=0 \end{equation} % then the linearization $H_{F}$ also belongs to $A_{s}$.\footnote{In Theorem \ref{theultradifferentiable} we proved that a sufficient condition with this choice of $M_n$ and $N_n$ is % \begin{equation*} \limsup_{n\rightarrow +\infty} \left( \sum_{i=0}^{k(n)} \frac{\log q_{i+1}}{q_{i}} - \frac{\epsilon - \eta}{n} \log \left( n! \right) \right) \leq C < +\infty \end{equation*} % which can be rewritten as % \begin{equation*} \limsup_{n\rightarrow +\infty} \left( \sum_{i=0}^{k(n)} \frac{\log q_{i+1}}{q_{i}} - \left( \epsilon - \eta \right) \log q_{k\left( n \right)} -C \right) =0 \end{equation*} % from which (\ref{nonlineararithmetic}) is just obtained dividing by $\log q_{k\left( n \right)}$.} The equivalence of (\ref{nonlineararithmetic}) and (\ref{lineararithmetic}) is the object of the following \begin{lemma} \label{equivalency} Let $\left( q_l \right)_{l \geq 0}$ be the sequence of denominators of the convergents of $\omega \in \mathbb{R} \setminus \mathbb{Q}$. The following statements are all equivalent: % \begin{enumerate} \item $\lim_{n \rightarrow \infty} \frac{1}{\log n}\sum_{l=0}^{k\left( n \right)}\frac{\log q_{l+1}}{q_l}=0$ \item $\sum_{l=0}^{k\left( n \right)}\frac{\log q_{l+1}}{q_l}={\it o}\left( \log q_{k} \right)$ \item $\log q_{k+1}={\it o}\left( q_k \log q_k \right)$ \end{enumerate} % \end{lemma} \proof 1. $\Longrightarrow$ 2. is trivial (choose $n=q_{k\left( n \right)}$). \indent 2. $\Longrightarrow$ 3. Writing for short $k$ istead of $k \left( n \right)$ % \begin{align*} \frac{1}{\log q_k}\sum_{l=0}^{k} \frac{\log q_{l+1}}{q_l} &= \frac{\log q_{k+1}}{q_k \log q_{k}}+\frac{1}{\log q_k}\sum_{l=0}^{k-1} \frac{\log q_{l+1}}{q_l} \\ &=\frac{\log q_{k+1}}{q_k \log q_{k}}+ \frac{{\it o}\left( \log q_{k-1} \right)}{\log q_{k}} \end{align*} % Since $\lim_{k\rightarrow \infty} \frac{{\it o}\left( \log q_{k-1} \right)}{\log q_{k}} = 0$ we get 3. \indent 3. $\Longrightarrow$ 1. First of all note that since $q_{k\left( n \right)} \leq n$ 2. trivially implies 1. Thus it is enough to show that 3. $\Longrightarrow$2. $\log q_{k+1}={\it o}\left( q_k \log q_k \right)$ means: % \begin{equation*} \forall \epsilon >0 \; \; \exists \Hat{n} \left( \epsilon \right) \text{ such that } \forall l > \Hat{ n }\left( \epsilon \right) \quad \frac{\log q_{l+1}}{q_l \log q_l}<\epsilon \end{equation*} % \indent If $\log q_{l+1} k\left( \epsilon \right)>\hat{n}(\epsilon )$, for some positive constant $C_2$. \indent Putting these estimates together we can bound (\ref{threeterms}) with: % \begin{equation*} \frac{1}{\log q_k}\sum_{l=0}^k \frac{\log q_{l+1}}{q_l} \leq \epsilon + \epsilon C_1+\epsilon C_2 \end{equation*} % for all $\epsilon >0$ and for all $k > k\left( \epsilon \right)$, thus $\sum_{l=0}^k\frac{\log_{l+1}}{q_l}={\it o}\left( \log q_k \right)$ \endproof \subsection{Divergence of the modified linearization power series when the artihmetical conditions of Theorem \ref{theultradifferentiable} are not satisfied} In Theorem \ref{theultradifferentiable} we proved that if $F \in z\mathbb{C}\left\{ z \right\}$ and $\omega$ verifies condition (\ref{BrjunoM}) then the linearization $H\in z\mathbb{C}\left[ \left[ z \right] \right]_{\left( M_n \right)}$. The power series coefficients $h_{n}$ of $H$ are given by (\ref{recursive}). Let us define the sequence of strictly positive real numbers $(\Tilde{h}_n)_{n\ge 0}$ as follows: % \begin{equation} \Tilde{h}_0=1\; , \;\; \Tilde{h}_n =\frac{1}{|\lambda^n-1|} \sum_{m=2}^{n+1}|f_{m}| \sum_{n_{1}+\ldots +n_{m}= n+1-m\, , \, n_{i}\ge 0 } \Tilde{h}_{n_{1}}\ldots \Tilde{h}_{n_{m}}\; . \label{htilde} \end{equation} % Clearly $|h_{n}|\le \Tilde{h}_{n+1}$. Let $\Tilde{H} $ denote the formal power series associated to the sequence $(\Tilde{h}_n)_{n\ge 0}$ % \begin{equation} \Tilde{H}(z) = \sum_{m=1}^{\infty}\Tilde{h}_{n-1}z^n \label{Htilde} \end{equation} % Following closely \cite{Yoccoz}, Appendice 2, in this section we will prove that if condition (\ref{BrjunoM}) is violated then $\Tilde{H}$ doesn't belong to $z\mathbb{C}\left[ \left[ z \right] \right] _{\left( M_n \right)}$. Note that since it is not restrictive to assume that $|f_{2}|\ge 1$ one has % \begin{equation} \label{lowerh} \Tilde{h}_{n}> \sum_{k=0}^{n-1}\Tilde{h}_{k}\Tilde{h}_{n-1-k} \ge \Tilde{h}_{n-1}\; , \end{equation} \indent % thus the sequence $(\Tilde{h}_{n})_{n\ge 0}$ is strictly increasing. Let $\omega$ be an irrational number which violates (\ref{BrjunoM}) and let $U=\{ q_j:q_{j+1} \geq \left( q_j +1 \right)^2 \}$ where $\left( q_j \right)_{j \geq 1}$ are the denominators of the convergents of $x$. Since $\inf_n \frac{1}{n}\log M_n=c>-\infty$ we have: % \begin{equation*} \sum_{q_j \not\in U,j=0}^{k\left( n \right)}\frac{\log q_{j+1}}{q_j}-\frac{\log M_n}{n} \leq \sum_{q_j \not\in U,j=0}^{k\left( n \right)}\frac{2\log \left( q_j +1 \right)}{q_j}-c= \tilde{c} < +\infty \end{equation*} % where $k\left( n \right)$ is defined by: $q_{k\left( n \right)} \leq n < q_{k\left( n \right)+1}$. On the other hand $\limsup_{n \rightarrow \infty} \left( \sum_{j=0}^{k\left( n \right)}\frac{\log q_{j+1}}{q_j} -\frac{\log M_n}{n} \right)= ~\infty$ thus % \begin{equation} \limsup_{n \rightarrow \infty} \left( \sum_{q_j \in U:j=0}^{k\left( n \right)}\frac{\log q_{j+1}}{q_j} -\frac{\log M_n}{n} \right)= \infty \end{equation} % this implies that $U$ is not empty. From now on the elements of $U$ will be denoted by: $q_0^\prime < q_1^\prime < \ldots$. \indent Let $n_i= \lfloor \frac{q_{i+1}^\prime}{q_{i}^\prime+1} \rfloor$. \begin{lemma} The subsequence $\left( \Tilde{h}_{q_{i}^\prime} \right)_{i \geq 0}$ verifies: % \begin{equation} \label{subsequenceestimate} \Tilde{h}_{q_{i+1}^\prime} \geq \frac{1}{ |\lambda^{q_{i+1}^\prime}-1| } \Tilde{h}_{q_{i}^\prime}^{n_i} \; . \end{equation} % \end{lemma} \proof From the definition (\ref{htilde}) and the assumption $|f_{2}|\ge 1$ it follows that % \begin{equation*} \Tilde{h}_{2s-1}\ge \frac{|f_{2}|}{|\lambda^{2s-1}-1|}\Tilde{h}_{s-1}^{2} \ge \frac{\Tilde{h}_{s-1}^{2}}{2} \end{equation*} % thus for all $i\ge 2$ and $s\ge 1$ one has % \begin{equation} \label{geometric-h} \Tilde{h}_{2s-1}\ge \frac{\Tilde{h}_{s-1}^{i}}{2}\; . \end{equation} % \indent Choosing $s=q_{i}^\prime +1$, $i=n_{i}$ this leads to the desired estimate: % \begin{equation*} \Tilde{h}_{q_{i+1}^\prime}\ge \frac{2|f_{2}|}{|\lambda^{q_{i+1}^\prime}-1|} \Tilde{h}_{q_{i+1}^\prime-1}\ge \frac{2|f_{2}|}{|\lambda^{q_{i+1}^\prime}-1|} \Tilde{h}_{n_{i}(q_{i}^\prime +1)-1}\ge \frac{\Tilde{h}_{q_{i}^\prime}^{n_i}}{|\lambda^{q_{i+1}^\prime}-1|}\; . \end{equation*} % \endproof \indent By means of the previous lemma we can now prove that $\limsup_{n \rightarrow \infty}\frac{1}{n}\log \frac{\Tilde{h}_n}{M_n}=+\infty$. Let $\alpha_{i}=n_{i}\frac{q_{i}^\prime}{q_{i+1}^\prime}$. Then $1 \geq \alpha_{i}\geq \left( 1-\frac{1}{q_i^\prime +1}\right)^2$, which assures that $\prod_{i \geq 0}\alpha_{i} =c$ for some finite constant $c$ (depending on $\omega$). Then from (\ref{subsequenceestimate}) we get: % \begin{equation*} \frac{1}{q_{i+1}^\prime}\log \frac{\Tilde{h}_{q_{i+1}^\prime}}{M_{q_{i+1}^\prime}}\geq c \left[ \sum_{j=1}^{i+1}-\frac{\log |\lambda^{q_{j}^\prime}-1|}{q_j^\prime} -\frac{1}{q^\prime_{i+1}}\log M_{q^\prime_{i+1}}\right]+c_4 \end{equation*} % which diverges as $i\rightarrow \infty$. \appendix \section{continued fractions and Brjuno's numbers} \label{appfrazioni} Here we summarize briefly some basic notions on continued fraction development and we define the Brjuno numbers. \indent For a real number $\omega$, we note $\lfloor \omega \rfloor$ its integer part and $\{ \omega\}=\omega - \lfloor \omega \rfloor$ its fractional part. We define the Gauss' continued fraction algorithm: % \begin{itemize} \item $a_0=\lfloor \omega \rfloor$ and $\omega_0=\{ \omega\}$ \item for all $n \geq 1$: $a_n=\lfloor \frac{1}{\omega_{n-1}} \rfloor$ and $\omega_n=\{ \frac{1}{\omega_{n-1}} \}$ \end{itemize} % namely the following representation of $\omega$: % \begin{equation*} \omega=a_0+\omega_0=a_0+\frac{1}{a_1+\omega_1}=\ldots \end{equation*} % For short we use the notation $\omega=\left[ a_0,a_1,\ldots,a_n,\ldots\right]$. \indent It is well known that to every expression $\left[ a_0,a_1,\ldots,a_n, \ldots \right]$ there corresponds a unique irrational number. Let us define the sequences $\left( p_n \right)_{n\in \mathbb{N}}$ and $\left( q_n \right)_{n\in \mathbb{N}}$ as follows: % \begin{eqnarray*} q_{-2}=1\text{, }q_{-1}=0\text{, }q_n=a_n q_{n-1}+q_{n-2}\\ p_{-2}=0\text{, }p_{-1}=1\text{, }p_n=a_n p_{n-1}+p_{n-2} \end{eqnarray*} % It is easy to show that: $\frac{p_n}{q_n}=\left[ a_0,a_1,\ldots,a_n\right]$. \indent For any given $\omega \in \mathbb{R} \setminus \mathbb{Q}$ the sequence $\left( \frac{p_n}{q_n} \right)_{n\in \mathbb{N}}$ satisfies % \begin{equation} \label{growthqn} q_{n}\ge \left(\frac{\sqrt{5}+1}{ 2}\right)^{n-1}\; , \; \; n \ge 1 \end{equation} % thus % \begin{equation} \label{universalconstants} \sum_{k\geq 0}\frac 1{q_k}\leq \frac{\sqrt{5}+5}{2}\;\;\;\text{ and }\;\;\;\sum_{k\geq 0}\frac{\log q_k}{q_k}\leq \frac 1e\frac{2^{\frac 54}}{2^{\frac 34}-1}\; , \end{equation} % and it has the following important properties: % \begin{lemma} for all $n \geq 1$ then: $\frac{1}{q_n+q_{n+1}} \leq \lvert q_n \omega - p_n \rvert < \frac{1}{q_{n+1}}$. \end{lemma} % \begin{lemma} \label{lemmanumb} If for some integer $r$ and $s$, $\mid \omega -\frac {r}{s} \mid \leq \frac {1}{2 s^{2}}$ then $\frac {r}{s} = \frac {p_k}{q_k}$ for some integer $k$. \end{lemma} % \begin{lemma} \label{bestapproximation} The law of best approximation: if $1\le q\le q_{k}$, $(p,q)\not= (p_{n},q_{n})$ and $n\ge 1$ then $|qx-p|>|q_{n}x-p_{n}|$. Moreover if $(p,q)\not= (p_{n-1},q_{n-1})$ then $|qx-p|>|q_{n-1}x-p_{n-1}|$. \end{lemma} % For a proof of these standard lemmas we refer to \cite{Hardy}. \indent The growth rate of $\left( q_n \right)_{n\in \mathbb{N}}$ describes how rapidly $\omega$ can be approximated by rational numbers. For example $\omega$ is a diophantine number \cite{Siegel} if and only if there exist two constants $c >0$ and $\tau \ge 1$ such that $q_{n+1}\le c q_{n}^\tau$ for all $n\ge 0$. \indent To every $\omega \in \mathbb{R} \setminus \mathbb{Q}$ we associate, using its convergents, an arithmetical function: % \begin{equation} \label{Brjunofunction} B\left( \omega \right)= \sum_{n \geq 0} \frac{\log q_{n+1}}{q_n} \end{equation} % We say that $\omega$ is a {\it Brjuno number} or that it satisfies the {\it Brjuno condition} if $B\left( \omega \right)<+\infty$. The Brjuno condition gives a limitation to the growth rate of $\left( q_n \right)_{n\in \mathbb{N}}$. It was originally introduced by A.D.Brjuno \cite{Brjuno}. The Brjuno condition is weaker than the Diophantine condition: for example if $a_{n+1} \le c e^{a_{n}}$ for some positive constant $c$ and for all $n\ge 0$ then $\omega = [a_{0}, a_{1}, \ldots , a_{n}, \ldots ]$ is a Brjuno number but is not a diophantine number. \section{Davie's lemma} \label{appDavie} In this appendix we summarize the result of \cite{Davie} that we use, in particular Lemma \ref{lemmaDavie}. Let $\omega \in \mathbb{R} \setminus \mathbb{Q}$ and $\left\{ q_n \right\}_{n \in \mathbb{N}}$ the partial denominators of the continued fraction for $\omega$ in the Gauss' development. \begin{definition} Let $A_k = \left\{ n \geq 0 \mid \| n \omega \| \leq \frac{1}{8q_k} \right\}$, $E_k=\max \left( q_k , q_{k+1}/4 \right)$ and $\eta_k = q_k / E_k$. Let $A_k^{*}$ be the set of non negative integers $j$ such that either $j \in A_k$ or for some $j_1$ and $j_2$ in $A_k$, with $j_2-j_1 < E_k$, one has $j_1 < j < j_2$ and $q_k$ divides $j-j_1$. For any non negative integer $n$ define: % \begin{equation*} l \left( n \right) = \max \left\{ \left( 1+\eta_k \right) \frac{n}{q_k}-2 , \left( m_n \eta_k+n \right) \frac{1}{q_k}-1 \right\} \end{equation*} % where $m_n = \max \{ j \mid 0 \leq j \leq n , j \in A_k^{*} \}$. We then define the function $h_k \left( n \right)$ % \begin{equation*} h_k \left( n \right)= \begin{cases} \frac{m_n+\eta_k n}{q_k}-1& \text{if $m_n+q_k \in A_k^{*}$} \\ l \left( n \right)& \text{if $m_n+q_k \not\in A_k^{*}$} \end{cases} \end{equation*} % \end{definition} The function $h_k \left( n \right)$ has some properties collected in the following proposition \begin{proposition} The function $h_k \left( n \right)$ verifies; % \begin{enumerate} \item $\frac{\left( 1+\eta_k \right)n}{q_k}-2 \leq h_k \left( n \right) \leq \frac{\left( 1+\eta_k \right)n}{q_k}-1$ for all $n$. \item If $n>0$ and $n \in A_k^{*}$ then $h_k \left( n \right) \geq h_k \left( n -1 \right)+1$. \item $h_k \left( n \right) \geq h_k \left( n-1 \right)$ for all $n>0$. \item $h_k \left( n+q_k \right) \geq h_k \left( n \right) +1$ for all $n$. \end{enumerate} % \end{proposition} \indent Now we set $g_k \left( n \right)= \max \left( h_k \left( n \right), \lfloor \frac{n}{q_k} \rfloor \right)$ and we state the following proposition \begin{proposition} \label{gpropos} The function $g_k $ is non negative and verifies: % \begin{enumerate} \item $g_k \left( 0 \right)=0$ \item $g_k \left( n \right) \leq \frac{\left( 1+\eta_k \right)n}{q_k}$ for all $n$ \item $g_k \left( n_1 \right) + g_k \left( n_2 \right) \leq g_k \left( n_1 +n_2 \right)$ for all $n_1$ and $n_2$ \item if $n \in A_k$ and $n>0$ then $g_k \left( n \right) \geq g_k \left( n-1 \right)+1$ \end{enumerate} % \end{proposition} The proof of these propositions can be found in \cite{Davie}. Let $k(n)$ be defined by the condition $q_{k(n)}\le n 0$ such that % \begin{equation*} K(n)\le n\left(\sum_{k=0}^{k(n)}\frac{\log q_{k+1}}{q_{k}}+\gamma_{3}\right)\; ; \end{equation*} % \item $K(n_{1})+K(n_{2})\le K(n_{1}+n_{2})$ for all $n_{1}$ and $n_{2}$; \item $-\log |\lambda^n -1| \le K(n)-K(n-1)$. \end{enumerate} % \end{lemma} The proof is a straightforward application of Proposition \ref{gpropos}. \begin{thebibliography}{XXX} \bibitem[Br]{Brjuno} A.D.Brjuno: Analitycal form of differential equation, Trans. Moscow Math. Soc. $\mathbf{25}$, $131-288$ $\left( 1971 \right)$ \bibitem[Ca]{Teo} T. Carletti: The Lagrange inversion formula on non--archimedean fields, preprint (1999) \bibitem[Da]{Davie} A.M.Davie: The critical function for the semistandard map. Nonlinearity $\mathbf{7}$, $21 - 37$ $\left( 1990 \right)$ \bibitem[Du]{Du}D.Duverney: $U$-D\'erivation, Annales de la Facult\'e des Sciences de Toulouse, vol II, $\mathbf{3}$ $\left( 1993 \right)$ \bibitem[GY1]{GramchevYoshino1} T. Gramchev and M. Yoshino: WKB Analysis to Global Solvability and Hypoellipticity, Publ. Res. Inst. Math. Sci. Kyoto Univ. $\mathbf{31}$, $443-464$, $(1995)$ \bibitem[GY2]{GramchevYoshino2} T. Gramchev and M. Yoshino: Rapidly convergent iteration method for simultaneous normal forms of commuting maps, preprint, $(1997)$ \bibitem[He]{Herman} M.R.Herman: Recent Results and Some Open Questions on Siegel's Linearization Theorem of Germs of Complex Analytic Diffeomorphisms of ${\mathbb C}^n$ near a Fixed Point, Proc. VIII Int. Conf. Math. Phys. Mebkhout Seneor Eds. World Scientific, $138-184$, $(1986)$ \bibitem[HW]{Hardy} G.H.Hardy and E.M.Wright: An introduction to the theory of numbers, $5^{th}$ edition Oxford Univ. Press \bibitem[Lo]{Lochak} P. Lochak: Canonical perturbation theory via simultaneous approximation, Russ. Math. Surv. $\mathbf{47}$, $57-133$, $(1992)$ \bibitem[MMY]{MMY} S.Marmi, P.Moussa and J.-C. Yoccoz: The Brjuno functions and their regularity properties, Communications in Mathematical Physics $\mathbf{186}$, $265-293$, $(1997)$ \bibitem[Si]{Siegel} C.L.Siegel: Iteration of analytic functions, Annals of Mathematics $\mathbf{43}$ $\left(1942\right)$ $807-812$ \bibitem[Yo]{Yoccoz} J.-C.Yoccoz: Th\'eor\`eme de Siegel, polyn\^omes quadratiques et nombres de Brjuno, Ast\'erisque $\mathbf{231}$, $3 - 88 \left( 1995 \right)$ \end{thebibliography} \end{document} ---------------0003170823538--