%&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} \renewcommand{\thenotation}{} \newcommand{\R}{\mathop{\mathbb{R}}} \newcommand{\Q}{\mathop{\mathbb{Q}}} \newcommand{\Z}{\mathop{\mathbb{Z}}} \newcommand{\N}{\mathop{\mathbb{N}}} \newcommand{\C}{\mathop{\mathbb{C}}} \newcommand{\D}{\mathop{\mathbb{D}_{#1}}} \newcommand{\diag}{\mathop{\it{diag}(#1)}} \numberwithin{equation}{section} \begin{document} \title[Nekhoroshev--like estimate for non--linearizable analytic germs.]{Exponentially long time stability for non--linearizable analytic germs of $(\C^n,0)$.} \author{Timoteo Carletti} \date{\today} \address[Timoteo Carletti]{Dipartimento di Matematica "U. Dini", viale Morgagni 67/A, 50134 Firenze, Italy} \email[Timoteo Carletti]{carletti@math.unifi.it} \keywords{Siegel center problem, Gevrey class, Bruno condition, effective stability, Nekoroshev like estimates} % %subj class %32A05 Power series, series of functions %37F50 Small divisors, rotation domains and linearization; Fatou and Julia sets %34E05 Asymptotic expansions % \begin{abstract} We study the Siegel--Schr\"oder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey--$s$, $s>0$ category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey--$s$ formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin for the analytic germ. \end{abstract} \maketitle \section{Introduction} In this paper we consider the {\em Siegel--Schr\"oder center problem}~\cite{Herman,CarlettiMarmi,Carletti} in some class of ultradifferentiable germs of $(\C^n,0)$, $n \geq 1$; let us consider two classes of formal power series $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \mathbb{C}^n \left[ \left[ z_1,\dots ,z_n \right] \right]$, closed w.r.t. to derivation and composition, let $F\in \mathcal{A}_1$ and call $DF(0)=A\in GL(n,\C)$, we say that $F$ is {\em linearizable in $\mathcal{A}_2$} if there exists $H\in \mathcal{A}_2$, normalized with $DH(0)=\mathbb{I}$, which solves~\footnote{Here $F\circ H$ means the composition of $F$ and $H$; in the following we will denote the composition of $F$ $n$-times with itself, by $F^n$ instead of $F^{\circ n}$.}: \begin{equation} \label{eq:linearization} F \circ H (z)= H \circ R_{A}(z) \, , \end{equation} where $R_{A}(z)=Az$. In the following we will assume $A$ to be diagonal with eigenvalues of unit modulus $\lambda_1,\dots,\lambda_n$, thus $A=\diag{\lambda_1,\dots,\lambda_n}$. \indent If both $\mathcal{A}_1$ and $\mathcal{A}_2$ coincide with the ring of formal power series then we have formal linearization if and only if $A$ is {\em non--resonant}, namely for all $\alpha \in \N^n$ such that $|\alpha|=\sum_{1\leq i \leq n}\alpha_i\geq 2$, and for all $j\in \{ 1,\dots,n \}$ then $\lambda^{\alpha}-\lambda_j \neq 0$ (where we used the standard notation $\lambda^{\alpha}=\lambda_1^{\alpha_1} \dots \lambda_n^{\alpha_n}$). When $F$ is a germ of analytic diffeomorphisms defined in a neighborhood of the origin and we want to solve~\eqref{eq:linearization} in the same class of analytic germs, we have to consider several cases. If $A$ is the {\em Poincar\'e domain}, namely $\sup_{1\leq j\leq n} |\lambda_j| <1$ or $\sup_{1\leq j\leq n} |\lambda^{-1}_j| <1$, then Koenigs~\cite{Koenigs} and Poincar\'e~\cite{Poincare1} proved that every analytic germ $F \in Dif\!f(\C^n,0)$ such that $F(0)=0$ and $DF(0)=A$, is analytically linearizable. When $A$ is not in the Poincar\'e domain, we say that it is in the {\em Siegel domain}; the question is harder and some additional arithmetical conditions on $(\lambda_j)_j$ are needed (see~\cite{Herman} \S 17 page 158). Let $p\in \N$, $p\geq 2$ and let us define for non--resonant $\lambda_1 ,\dots , \lambda_n$: \begin{equation} \label{eq:smalldiv} \Omega(p)= \min_{1\leq j\leq n} \inf_{\substack{\alpha \in \N^n \\ 0<|\alpha|0$and$\tau\geq 0$such that for all$\beta\in \N^n\setminus \{ 0 \}$we have$\Omega(|\beta|)\geq \gamma |\beta|^{-\tau}$. Siegel~\cite{Siegel} in 1942 for the$n=1$case and then Sternberg~\cite{Sternberg} and Gray~\cite{Gray} in the general case proved that if$A$verifies a Diophantine condition then the linearization problem has an analytic solution. Bruno~\cite{Bruno} weakened the arithmetical condition by asking the convergence of the series$\sum_{k} \frac{\log \Omega^{-1}(2^{k+1})}{2^k}$. We remark that in the one dimensional case the Bruno condition~\footnote{In this case let$\omega \in (0,1)\setminus \Q$such that$\lambda = e^{2\pi i \omega}$and let$(q_n)_n$be the denominators of the convergents~\cite{HardyWright} to$\omega$, then the Bruno condition is equivalent to the convergence of the series$\sum_{k\geq 0}\frac{\log q_{k+1}}{q_k}$.} is optimal, as proved by Yoccoz~\cite{Yoccoz}. In~\cite{CarlettiMarmi} authors studied the Siegel--Schr\"oder center problem in the case of general algebras of ultradifferentiable germs of$(\C,0)$, including the Gevrey case. In~\cite{Carletti} the multidimensional case is considered: if$\mathcal{A}_1=\mathcal{A}_2$and$A$verifies a Bruno condition, then every$F\in \mathcal{A}_1$with$F(0)=0$and$DF(0)=A$is linearizable in$\mathcal{A}_2$, whereas if$\mathcal{A}_1$is properly contained in$\mathcal{A}_2$new conditions weaker than Bruno are sufficient to ensure linearizability in$\mathcal{A}_2$. \indent In this paper we consider in detail the case where$\mathcal{A}_1$is the ring of germs of analytic diffeomorphisms at the origin of$n\geq 1$complex variables, and$\mathcal{A}_2$is the algebra of {\em Gevrey}--$s$,$s>0$, formal power series: the {\em Gevrey--$s$linearization of analytic germs}. Let$\Hat F=\sum f_{\alpha} z^{\alpha}$,$(f_{\alpha})_{\alpha \in \N^n} \subset \C^n$be a formal power series, then we say that it is {\em Gevrey--$s$}~\cite{Balser,Ramis1},$s>0$, if there exist two positive constants$C_1,C_2$such that: % \begin{equation} \label{eq:gevreydefvect} |f_{\alpha}| \leq C_1 C_2^{-s|\alpha|} |\alpha|!^s \quad \forall \alpha \in \mathbb{N}^n \, . \end{equation} % We denote the class of all formal vector valued power series Gevrey--$s$by$\mathcal{C}_s$. It is closed w.r.t. derivation and composition. \indent In the Gevrey--$s$case the arithmetical condition introduced in~\cite{CarlettiMarmi,Carletti} will be called {\em Bruno}--$s$condition,$s>0$: for short$A\in \mathcal{B}_s$if there exists an increasing sequence of positive integer$(p_k)_k$such that: \begin{equation} \label{eq:brunosndim} \limsup_{|\alpha|\rightarrow +\infty}\left( 2\sum_{m=0}^{\kappa(\alpha)} \frac{\log \Omega^{-1}(p_{m+1})}{p_m}-s\log |\alpha|\right)< +\infty \, , \end{equation} where$\kappa(\alpha)$is defined by$p_{\kappa(\alpha)}\leq |\alpha| < p_{\kappa(\alpha)+1}$. When$n=1$the Bruno--$s$condition can be slightly weakened (see~\cite{CarlettiMarmi}); let$\omega \in (0,1)\setminus \Q$and$\lambda=e^{2\pi i \omega}$, then the Bruno--$s$,$s>0$, condition reads: \begin{equation} \label{eq:brunos1dim} \limsup_{n\rightarrow +\infty} \left( \sum_{j=0}^{k(n)}\frac{\log q_{j+1}}{q_{j}} - s\log n \right) <+\infty \, , \end{equation} where$k(n)$is defined by$q_{k(n)}\leq n < q_{k(n)+1}$. We remark that in both cases the new conditions are weaker than Bruno condition, which is recovered when$s=0$. When$n=1$we prove that the set$\mathcal{B}_s$is$PSL(2,\Z)$--invariant (see remark~\ref{rem:invariance}). The main result of~\cite{Carletti} in the case of Gevrey--$s$classes reads: % \begin{theorem}[Gevrey--$s$linearization] \label{thm:gevreylin} Let$\lambda_1,\dots,\lambda_n$be complex numbers of unit modulus and$A=\diag{\lambda_1,\dots,\lambda_n}$; let$D_1 = \{ z \in \C^n : |z_i|<1 \, , 1\leq i\leq n \}$be the isotropic polydisk of radius$1$and let$F:D_1\rightarrow \C^n$be an analytic function, such that$F(z)=Az+f(z)$, with$f(0)=Df(0)=0$. If$A$is non--resonant and verifies a Bruno--$s$,$s>0$, condition~\eqref{eq:brunosndim} (or~\eqref{eq:brunos1dim} when$n=1$), then there exists a formal Gevrey--$s$linearization$\Hat{H}$which solves~\eqref{eq:linearization}. \end{theorem} % The aim of this paper is to show that the Gevrey character of the formal linearization can give information concerning the dynamics of the analytic germ. Let$F(z)=Az+f(z)$be a germ of analytic diffeomorphism verifying the hypothesis of Theorem~\ref{thm:gevreylin}, assume moreover$F$not to be analytically linearizable. We will show that even if there is not {\em Siegel disk}, where the dynamics of$F$is conjugate to the dynamics of its linear part, we have an open neighborhood of the origin which behaves as a Siegel disk'' under the iterates of$F$for finite but long time, which results exponentially long: the {\em effective stability}~\cite{GDFGS} of the fixed point. In the case of analytic linearization,$|H_i^{-1}(z)|$,$i=1, \dots, n$, (which is well defined sufficiently close to the origin because$H$is tangent to the identity) is {\em constant along the orbits}, namely it is a {\em first integral} % in fact let$z_0$be close enough to belong %to the Siegel disk of$F$and let us denote, for any positive integer$m$, %$z_m=F^m(z_0)$, then from~\eqref{eq:linearization} it follows %$H^{-1}(z_m)=H^{-1}\circ F^m(z_0)=R_{A^m}\circ H^{-1}(z_0)$, namely for any %$i=1,\dots, n$, we have$|H_i^{-1}(z_m)|=|H_i^{-1}(z_0)|$, %being$A^m=\diag{\lambda_1^m,\dots,\lambda_n^m}$and$|\lambda_i^m|=1$for all$i$. %Because$F(z)=H\circ R_{A} \circ H^{-1}(z)$in the analytically %linearizable case, and$|F^m(z_0)|$is bounded for all$m$and sufficiently small$|z_0|$. We will prove that any non--zero$z_0$belonging to a polydisk of sufficiently small radius$r>0$, can be iterate a number of times$K=\mathcal{O}(exp \{ r^{-1/s} \} )$, being$s>0$the Gevrey exponent of the formal linearization, and we can find an {\em almost first integral}: a function which varies by a quantity of order$r$under$m\leq K$iterations, which implies that$F^m(z_0)$is well defined and bounded for$m\leq K$. More precisely we prove the following \begin{theorem} \label{thm:maintheorem} Let$\lambda_1,\dots,\lambda_n$be complex numbers of unit modulus and$A=\diag{\lambda_1,\dots,\lambda_n}$; let$F:D_1\rightarrow \C^n$be an analytic and univalent function, such that$F(z)=Az+f(z)$, with$f(0)=Df(0)=0$. If$A$is non--resonant and verifies a Bruno--$s$,$s>0$, condition~\eqref{eq:brunosndim} (or~\eqref{eq:brunos1dim} when$n=1$), then for all sufficiently small$0< r_{**} <1$, there exist positive constants$A_{**},B_{**},C_{**}$such that for all$0<|z_0|0$. \begin{remark}[Invariance of$\mathcal{B}_s$,$n=1$under the action of$PSL(2,\Z)$] \label{rem:invariance} The continued fraction development~\cite{HardyWright,MMY} of an irrational number$\omega$gives us the sequences:$(a_k)_{k\geq 0}$and$(\omega_k)_{k\geq 0}$. Then we introduce$(\beta_{k})_{k\geq -1}$defined by$\beta_{-1}=1$and for all integer$k\geq 0$:$\beta_{k}=\prod_{j=0}^k \omega_k$, which verifies :$1/2<\beta_kq_{k+1}<1$, where$q_k$'s are the denominators of the continued fraction development of$\omega$. We can then prove that condition Bruno--$s$~\eqref{eq:brunos1dim} is equivalent to the following one: \begin{equation} \label{eq:brunosbeta} \limsup_{k \rightarrow +\infty}\left( \sum_{j=0}^k \beta_{j-1} \log \omega_j^{-1} + s \log \beta_{k-1} \right) < +\infty \, . \end{equation} Let us consider the generators of$PSL(2,\Z)$:$T\omega = \omega +1$and$S\omega = 1/\omega$. For any irrational$\omega$,$T$acts trivially being$\beta_k(T\omega)=\beta_k(\omega)$for all$k$, whereas for irrational$\omega\in (0,1)$we have$\beta_k(\omega)=\omega \beta_{k-1}(S\omega)$. Then the invariance of condition~\eqref{eq:brunosbeta} under the action of$GL(2,\Z)$is obtained verifying the invariance under the action of$S$. \end{remark} Let us consider a slightly stronger version of the Bruno--$s$condition:$\omega \in (0,1)\setminus \Q$belongs to$\Tilde{\mathcal{B}}_s$if: \begin{equation} \label{eq:newbruno1} \lim_{n\rightarrow +\infty}\left( \sum_{l=0}^{k} \frac{\log q_{l+1}}{q_l}-s \log q_k \right) < +\infty \, , \end{equation} where$(q_n)_n$are the convergents to$\omega$. Let us introduce two other arithmetical conditions; let us denote by$\mathcal{B}_s^{\prime}$the set of irrational numbers whose convergents verify: \begin{equation} \label{eq:brunoprimes} \lim_{k\rightarrow +\infty} \frac{\log q_{k+1}}{q_k\log q_k}= s \, . \end{equation} The second condition is as follows, let$(\gamma_m)_{m\geq 1}$and$(s_m)_{m\geq 1}$be two positive sequences of real numbers such that:$\sum_1^{+\infty} \gamma_m =\gamma < +\infty$and$\sum_1^{+\infty} s_m =\sigma < +\infty$, then we define a condition$\mathcal{B}_{\gamma,\sigma}$by: \begin{equation} \label{eq:brunogamsig} \frac{\log q_{m+1}}{q_m}\leq s_m \log q_m +\gamma_m \quad \forall m\geq 1\, . \end{equation} \begin{proposition} \label{prop:differentbruno1s} Let$\omega \in (0,1)\setminus \Q$and let$s>0$then we have the following inclusions: \begin{enumerate} \item[1)] let$\omega \in \Tilde{\mathcal{B}}_s$, if$\omega$is not a Bruno number then$\omega \in \mathcal{B}_s^{\prime}$, otherwise$\omega \in \mathcal{B}^{\prime}_0$. \item[2)] Let$\sigma \leq s$and$\omega \in \mathcal{B}_{\gamma,\sigma}$then$\omega \in \Tilde{\mathcal{B}}_s$; \end{enumerate} \end{proposition} \proof To prove the first statement let us write the following identity: \begin{equation} \label{eq:step1} \sum_{l=0}^k \frac{\log q_{l+1}}{q_l}-s\log q_k = C + \sum_{l=2}^k \left[ \frac{\log q_{l+1}}{q_l}-s\left( \log q_l - \log q_{l-1} \right) \right] \, , \end{equation} where$C=(1-s)\log q_1+\frac{\log q_{2}}{q_1}$. By condition$\Tilde{\mathcal{B}}_s$, this series converges and then its generic term goes to zero, from which we get: \begin{equation} \label{eq:step3} \lim_{k\rightarrow +\infty}\frac{\log q_{k+1}}{q_k\log q_k}=s\left( 1-\lim_{k\rightarrow +\infty} \frac{\log q_{k-1}}{\log q_k} \right)\, . \end{equation} Let us denote by$s^{\prime}$be value of the right hand side of~\eqref{eq:step3}, then clearly$s^{\prime}\in \left[ 0, s\right]$. Let us suppose$s^{\prime}>0$, but then we have for all sufficiently large$k$: \begin{equation*} \frac{C_1}{q_{k}}\leq \frac{\log q_k}{\log q_{k+1}} \leq \frac{C_2}{q_{k}} \, , \end{equation*} for some positive constants$C_1,C_2$, from which we get$\frac{\log q_k}{\log q_{k+1}}\rightarrow 0$, and from~\eqref{eq:step3} we conclude that$s^{\prime}=s$. If$s^{\prime}=0$, namely$\frac{\log q_k}{\log q_{k+1}}\rightarrow 1$, then it is easy to check that$\omega$is a Bruno number. Let us prove the second statement. For any positive integer$k$, using the definition of$\mathcal{B}_{\gamma,\sigma}$we can write: \begin{equation} \sum_{l=0}^k \frac{\log q_{l+1}}{q_l}-s \log q_k \leq \sum_{l=0}^k s_l \log q_l +\sum_{l=0}^k \gamma_l - s \log q_k \, , \label{eq:primaeq} \end{equation} for all$0\leq l \leq k$we have$\log q_l \leq \log q_k$then the right hand side of~\eqref {eq:primaeq} is bounded by:$- \log q_k \left( s-\sum_{l=0}^k s_l \right)+\sum_{l=0}^k \gamma_l$. By hypothesis$\sum_{l=0}^k s_l \leq s$, for all$k$, then using$-\log q_k \leq -\log q_1$we obtain: \begin{equation*} \sum_{l=0}^k \frac{\log q_{l+1}}{q_l}-s \log q_k \leq - \log q_1 \left( s-\sum_{l=0}^k s_l \right)+\sum_{l=0}^k \gamma_l \, , \end{equation*} then passing to the limit on$k$we have: \begin{equation*} \lim_{k\rightarrow +\infty} \sum_{l=0}^k \frac{\log q_{l+1}}{q_l}-s \log q_k \leq - \log q_1 \left( s-\sigma \right) + \gamma <+\infty \, . \end{equation*} \endproof \begin{remark} These new arithmetical conditions are weaker than the Bruno one, for instance condition$\mathcal{B}^{\prime}_s$is verified by numbers$\omega$whose denominators$(q_k)_k$satisfy a growth condition like$q_{k+1}\sim q_k!^s$. Condition$\mathcal{B}_{\gamma,\sigma}$implies convergence of the series:$\sum_{k\geq 0}\frac{\log q_{k+1}}{q_k \log q_k}$. \end{remark} Let us conclude recalling a stability result of P\'erez--Marco~\cite{PerezMarco1,PerezMarco2} and compare it with our result. In~\cite{PerezMarco2} author proved (Theorem V.2.1 Annexe 2 \S f ) using a {\em geometric renormalization} scheme "\a la Yoccoz" valid in the one dimensional case, a stability result that can be stated as follows: % \begin{theorem}[P\'erez--Marco, Contr\^ole de la diffusion] \label{thm:pm} Let$\omega \in (0,1)\setminus \Q$and let$(q_k)_k$be the denominators of its convergents. Let$F$be an analytic and univalent function defined in the unit disk$\{ z\in \C: |z|<1 \}$such that$F(z)=\lambda z + \mathcal{O}(|z|^2)$, where$\lambda=e^{2\pi i \omega}$. There exist two positive constants$C_1,C_2$such that if: % \begin{equation} \label{eq:pm1} |z| \leq C_1 e^{-\sum_{j=0}^{k-1} \frac{\log q_{j+1}}{q_j}} \, , \end{equation} % then for all integer$0\leq m \leq q_k$we have: % \begin{equation} \label{eq:pm2} |F^m(z)| \leq C_2 e^{-\sum_{j=0}^{k-1} \frac{\log q_{j+1}}{q_j}} \, . \end{equation} % \end{theorem} The meaning of the Theorem is clear: if we start inside a disk of radius$r=C_1 e^{-\sum_{j=0}^{k-1} \frac{\log q_{j+1}}{q_j}}$then we can apply$F$, up to$q_k$times, without leaving a disk of radius$rC_2/C_1$. To compare this result with our effective stability result we have to make explicit the relation w.r.t.$r$and$q_k$, which give the time of "stability". Using our Bruno--$s$condition~\eqref{eq:newbruno1} we can say that$C\leq rq^s_{k-1} \leq C^{\prime}$for some positive constants$C,C^{\prime}$. But from~\eqref{eq:brunoprimes} we get$\log q_k \leq C_3 q_{k-1}\log q_{k-1}$for some positive constant$C_3$, namely there exist positive constants$C^{\prime}_3,C_4$such that: % \begin{equation*} q_k \leq exp \Big \{ \frac{C^{\prime}_3}{r^{1/s}} \log \frac{C_4}{r^{1/s}} \Big \} \, . \end{equation*} % We can then restate Theorem~\ref{thm:pm} as follows: if$|z|\leq r$, then$|F^m(z)|\leq rC_2/C_1$for all integer$0\leq m \leq exp \{ \frac{C^{\prime}_3}{r^{1/s}} \log \frac{C_4}{r^{1/s}} \}$, obtaining a better estimate on the time of effective stability. We end with a last remark related again to the work of P\'erez--Marco. % \begin{remark} P\'erez--Marco proved in~\cite{PerezMarco1,PerezMarco2} that any non--analytically linearizable analytic germ, univalent in the unit disk, whose multiplier at the fixed point, verifies the following arithmetical condition: % \begin{equation} \label{eq:perezemarco} \sum_{k\geq 0}\frac{\log \log q_{k+1}}{q_k}<+\infty \, , \end{equation} % has a sequence of periodic orbits accumulating the fixed point, whose periods,$(q_{n_k})_k$, make the Bruno series diverging. Our Bruno--$s$condition implies~\eqref{eq:perezemarco}, in fact from~\eqref{eq:newbruno1} we get: % \begin{equation*} \sum_{k= 0}^N\frac{\log \log q_{k+1}}{q_k} \leq \sum_{k= 0}^N \left(\frac{\log C_3}{q_k}+ \frac{\log q_k}{q_k}+\frac{\log\log q_k}{q_k}\right) \, , \end{equation*} % we can let$N$grow and using standard number theory results concerning the convergents, we obtain the P\'erez--Marco condition. Then we can suppose these periodic orbits accumulating the fixed point to produce the effective stability: preventing the orbits from a too fast escape'', a situation similar to the one holding in the Nekhoroshev Theorem for Hamiltonian systems~\cite{Nekhoroshev}, where the resonant web confines the flow for exponentially long times. It would be very interesting to know whether a similar phenomenon takes place in higher dimension. We conclude by pointing out that our method gives us a stability exponent depending on the Gevrey exponent and independent of the dimension: the bigger is the exponent, longer is the time interval of stability, we can always take$s$small enough to have a very long time of stability. \end{remark} % \begin{thebibliography}{XXXXX} \bibitem[Ba]{Balser} W. Balser: {\it From Divergent Power Series to Analytic Functions. Theory and Applications of Multisummable Power Series}, Lectures Notes in Mathematics,$\mathbf{1582}$, Springer$(1994)$. \bibitem[Br]{Bruno} A.D. 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