Content-Type: multipart/mixed; boundary="-------------0407131519836" This is a multi-part message in MIME format. ---------------0407131519836 Content-Type: text/plain; name="04-212.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-212.keywords" linearization vector field, Gevrey class, Bruno condition, effective stability, Nekhoroshev theorem ---------------0407131519836 Content-Type: application/x-tex; name="carletti.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="carletti.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} \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}[1]{\mathop{\mathbb{D}_{#1}}} \newcommand{\diag}[1]{\mathop{\it diag(#1_1,\dots,#1_n)}} \numberwithin{equation}{section} \begin{document} \title[Nekhoroshev--like estimate for non--linearizable analytic vector fields.]{Exponentially long time stability near an equilibrium point for non--linearizable analytic vector fields.} \author{Timoteo Carletti} \date{\today} \address[Timoteo Carletti]{Scuola Normale Superiore piazza dei Cavalieri 7, 56126 Pisa, Italia} \email[Timoteo Carletti]{t.carletti@sns.it} \subjclass{Primary 37C75, 34A25} \keywords{linearization vector field, Gevrey class, Bruno condition, effective stability, Nekhoroshev theorem} \begin{abstract} We study the orbit behavior of a germ of an analytic vector field of $(\C^n,0)$, $n \geq 2$. We prove that if its linear part is semisimple, non--resonant and verifies a Bruno--like condition, then the origin is effectively stable: stable for finite but exponentially long times. \end{abstract} \maketitle \section{Introduction} Let us consider the germ of analytic vector field, $X_F=\sum_{1\leq j \leq n}F_j(z)\frac{\partial}{\partial z_j}$, of $(\C^n,0)$ $n \geq 2$, whose components $(F_j)_{1\leq j \leq n}$ are analytic functions vanishing at $0\in \C^n$. Let us consider the associated Ordinary Differential Equation: % \begin{equation} \label{eq:ODE} \frac{dz}{dt}=F(z) \, ; \end{equation} % under the above assumptions $z(t;0)= 0$ for all $t$ is an equilibrium solution~\footnote{Here and throughout the paper by $z(t;z_0)$ we mean the solution at time $t$ of~\eqref{eq:ODE} s.t. $z(0;z_0)=z_0$. When the value of $z_0$ will not be relevant we'll just write $z(t)$.}. We are interested in studying the stability of orbits of $X_F$ in a neighborhood of this equilibrium point. We use the standard definition of {\em stability} (see~\cite{Moser}) for an equilibrium solution: $z=0$ it is stable is the past and in the future if for any neighborhood $U$ of $0$ there exists a neighborhood V, containing the origin, s.t. $z(0;z_0)\in V$ implies $z(t;z_0)\in U$ for all $t\in \R$. In a coordinates system centered at the equilibrium point the $j$--th component of the vector field will take the form: $F_j(z)=(Az)_j+f_j(z)$, with $A$ a $n\times n$ complex matrix and $f_j$ analytic function such that $f_j(0)=Df_j(0)=0$, for all $1\leq j \leq n$. Following the idea of Poincar\'e to study the orbit of~\eqref{eq:ODE} in a neighborhood of the origin, we will try to find an analytic change of coordinates, through an analytic diffeomorphisms $z\mapsto H(z)=w$ the {\em linearization}, s.t. in the new coordinates the vector field $X_F$ is conjugate to its linear part, $X_A=\sum (Az)_j \frac{\partial}{\partial z_j}$: $H^* X_F H^{-1}=X_A$. Hence equation~\eqref{eq:ODE} rewrites: % \begin{equation} \label{eq:ODE2} \frac{dw}{dt}=Aw \, . \end{equation} % This change of coordinates must solve: % \begin{equation} AH(z)=DH(z) \cdot \left( Az+f(z) \right) \, , \label{eq:conjugacy} \end{equation} % and it is unique by assuming $DH(0)=\mathbb{I}$. Clearly if the linear system~\eqref{eq:ODE2} is stable and~\eqref{eq:ODE} is analytically linearizable, then also the latter is stable. It is a remarkable result that this condition is also necessary, as the following Theorem states: \begin{theorem}[Carath\'eodory--Cartan 1932] \label{the:caratheodorycartan} Necessary and sufficient condition for the stability of the solution $z=0$ of~\eqref{eq:ODE} for all real $t$ is that: \begin{enumerate} \item $A$ is diagonalizable with purely imaginary eigenvalues; \item there exists an holomorphic function $z=K(w)=w+\mathcal{O}(|w|^2)$, $w\in\C^n$, which brings~\eqref{eq:ODE} into the linear system: % \begin{equation*} \frac{dw}{dt}=Aw \, . \end{equation*} % \end{enumerate} \end{theorem} So let us assume $A$ to verify hypothesis of Theorem~\ref{the:caratheodorycartan}: let $(\omega_j)_{1\leq j \leq n}\subset \R$ and $A=\diag{i\omega}$. Then $A$ belongs to the {\em Siegel domain}~\footnote{According to the classification of~\cite{Bruno} this case is {\em Poincar\'e domain 1.d}, but we prefer consider it as a Siegel case because the obstructions to the linearizability are very similar to those encountered in the Siegel domain.}: the origin is contained in the convex hull of the set of eigenvalues plotted as points in the complex plane (e segment in this case). This is the harder situation w.r.t. to the complementary case, {\em Poincar\'e domain}, because {\em small divisors} are involved: the existence of an analytic linearization is strictly related to the arithmetic property of approximation of the vector $\omega=(\omega_1,\dots,\omega_n)$, with vectors with integer entries. The first step is to assume $A$ to be {\em non--resonant}: $\alpha \cdot \omega \neq \omega_j$, for all $\alpha \in \N^n$ s.t. $|\alpha|=\alpha_1+\dots +\alpha_n \geq 2$ and for all $j\in \{ 1, \dots, n\}$. This ensures the existence of a {\em formal} change of variable which linearizes~\eqref{eq:ODE}. In~\cite{Bruno} author introduced the, today called, {\em Bruno condition}~\footnote{ The Bruno condition can be rewritten using a general increasing sequence of integer numbers, $(p_k)_k$. In~\cite{Bruno} pag. 222, author proved that~\eqref{eq:brunovf} is equivalent to: % \begin{equation*} \sum_{k \geq 0}\frac{\log \Hat{\Omega}^{-1}(p_{k+1})}{p_{k}}<+\infty \, , \end{equation*} % where $\Hat{\Omega}(p)= \min\{ |\alpha \cdot \omega - \omega_j|: j\in \{ 1,\dots, n\}, \alpha \in\Z^n, 0<|\alpha |0$, formal power series. Namely we are considering the {\em Gevrey linearization of analytic vector fields}. 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{Balser1994,Ramis1991}, $s>0$, 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 formal vector valued power series Gevrey--$s$ by $\mathcal{C}_s$. It is closed w.r.t. derivation and composition. In the Gevrey--$s$ case the arithmetical condition introduced in~\cite{Carletti2003}, called {\em Bruno}--$s$ condition, $s>0$, for short $\mathcal{B}_s$, reads: \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} for some increasing sequence of positive integer $(p_k)_k$ and $\kappa(\alpha)$ is defined by $p_{\kappa(\alpha)}\leq |\alpha| < p_{\kappa(\alpha)+1}$. \begin{remark} This definition recall the classical one of Bruno~\cite{Bruno}, where first one suppose the existence of a strictly increasing sequence of positive integer such that~\eqref{eq:brunosndim} holds, then one can prove (see~\cite{Bruno} \S IV page 222) that one can take an exponentially growing sequence, e.g. $p_k=2^k$. This holds also in our case, in fact we can prove that~\eqref{eq:brunosndim} is equivalent to: \begin{equation*} \limsup_{N\rightarrow +\infty}\left(\sum_{l=0}^N\frac{\log \Omega^{-1}(2^{l+1})}{2^l}-sN2\log2 \right) < +\infty \, . \end{equation*} A proof of this claim can be found in~\cite{Carletti2002}. \end{remark} When $n=2$, under the above condition (non--resonance and Siegel domain), rescaling time by $-\omega_2$ (assuming $\omega_2 \neq 0$), the ODE associated to the vector field can be rewritten as: % \begin{equation} \label{eq:ODEn2} \begin{cases} \dot z_1 = \omega z_1 + h.o.t. \\ \dot z_2 = - z_2 + h.o.t. \\ \end{cases}\, , \end{equation} % where $\omega = -\omega_1/\omega_2\in (\R\setminus\Q)^{+}$ and high order terms means $\mathcal{O}(|z|^{|\alpha|})$ with $|\alpha|\geq 2$, namely only the ratio of the eigenvalues enters. Then the Bruno--$s$ condition can be slightly weakened (see~\cite{CarlettiMarmi2000}): \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 $q_{k(n)}\leq n < q_{k(n)+1}$ and $(q_n)_n$ are the denominators of the convergents~\cite{HardyWright} to $\omega$. We remark that in both cases the new conditions are weaker than Bruno's condition, which is recovered when $s=0$. When $n=2$ we prove that the set $\bigcup_s \mathcal{B}_s$ is $PSL(2,\Z)$--invariant (see remark~\ref{rem:invariance}). The main result of~\cite{Carletti2003} in the case of Gevrey--$s$ classes reads: % \begin{theorem}[Gevrey--$s$ linearization] \label{thm:gevreylin} Let $\omega_1,\dots,\omega_n$ be real numbers and $A=\diag{i\omega}$; 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 condition~\eqref{eq:brunos1dim} if $n=2$), then there exists a formal Gevrey--$s$ linearization $\Hat{H}$. \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 vector field. Let $F(z)=Az+f(z)$ verify hypotheses of Theorem~\ref{thm:gevreylin}, assume moreover $X_F$ not to be analytically linearizable. We will show that even if there is not a {\em Stable domain}, where the dynamics of $X_F$ is conjugate to the dynamics of its linear part, we have an open neighborhood of the origin which ``behaves as a Stable domain'' for the flow of $X_F$ for finite but long time, which results exponentially long: the {\em effective stability}~\cite{GFGS,GiorgilliPosilicano} of the equilibrium solution. In the case of analytic linearization, $|H_j(z)|$, $j=1, \dots, n$, is {\em constant along the orbits}, namely it is a {\em first integral} and the flow of~\eqref{eq:ODE} is bounded for all $t$ 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 followed up to a time $T=\mathcal{O}(exp \{ r^{-1/s} \} )$, 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 $r$ during this interval of time. More precisely we prove the following \begin{theorem} \label{thm:maintheorem} Let $n\in\N$, $n\geq 2$. Given real $\omega_1,\dots,\omega_n$ consider $A=\diag{i\omega}$; 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} if $n=2$), then for all sufficiently small $0< r_{**} <1$, there exist positive constants $A_{**},B_{**},C_{**}$ such that for all $0<|z_0|0$ and $\tau > n-1$ such that for all $\alpha \in \N^n$ and all $j\in \{1,\dots,n\}$ one has: $|\alpha \cdot \omega - \omega_j| \geq \gamma |\alpha|^{-\tau}$. Let $A=\diag{\omega}$, then $A$ verifies a Diophantine condition if $\omega$ does.} $CD(\gamma,\tau)$, for some $\gamma>0$ and $\tau > n-1$, and the critical exponent of stability time is $1/\tau$. In our result, too, the critical exponent of stability time depends on some arithmetical property of the linear part of the vector field but in a more general way in fact we assume $A \in \mathcal{B}_s \supset CD(\gamma,\tau)$, for all $\gamma >0$ and $\tau \geq n-1$. The second remark is that in~\cite{GFGS,GiorgilliPosilicano} effective stability is obtained using some partial normal form, then working on it and using the Poincar\'e summation at the smallest term (see Lemma~\ref{lem:sumupsmallest}), the proof is done. Here the method used is completely different: we first linearize formally the system and then using properties of the formal linearization we conclude still using the Poincar\'e summation at the smallest term. This method introduce also our main drawback: we must assume $A$ to be non--resonant (to linearize) and this prevents us from considering real vector fields and hamiltonian ones, where an ``intrinsic'' resonance is present. In section~\ref{sec:conclusions} we discuss the relation between the Bruno--$s$ condition and other arithmetical conditions. %\indent %{\it Acknwoledgements.} %I am grateful to D. Sauzin for %a very stimulating discussion concerning Gevrey %classes and asymptotic analysis. \section{Proof of the main Theorem} \label{sec:proofmainthm} In this part we will prove our main result, Theorem~\ref{thm:maintheorem}. The proof will be divided into three steps: first we use the Gevrey--$s$ character of the formal linearization $\Hat H$, given by Theorem~\ref{thm:gevreylin}, to find an approximate solution of the conjugacy equation~\eqref{eq:conjugacy} up to a (exponentially) small correction (paragraph~\ref{ssec:firststep}); then we prove a Lemma allowing us to control how the small error introduced in the solution propagates (paragraph~\ref{ssec:thirdstep}). Finally we collect all the informations to conclude the proof (paragraph~\ref{ssec:endproof}). \subsection{Determination of an approximate solution} \label{ssec:firststep} Let $F$ verifies hypotheses of Theorem~\ref{thm:maintheorem} and let us consider the first order differential equation in $\mathbb{C}^n$, $n \geq 2$: % \begin{equation} \label{eq:ode} \frac{dz}{dt} = F(z) \, . \end{equation} % By Theorem~\ref{thm:gevreylin} this system can be put in linear form by a formal power series $\Hat{H}$ which belongs to $\mathcal{C}_s$ and it solves (formally): \begin{equation} \label{eq:forh} \frac{d}{dt}\Hat H(z) = A \Hat H(z) \, , \end{equation} we observe that one can choose $\Hat H(z)=z+\mathcal{O}(|z|^2)$. Since ${\Hat H}=\sum h_{\alpha} z^{\alpha}\in \mathcal{C}_s$, there exist positive constants $A_1$ and $B_1$ such that % \begin{equation} \label{eq:gevreyhm1} |h_{\alpha}| \leq A_1 B_1^{-s|\alpha|} (|\alpha|!)^s \quad \forall \,|\alpha| \geq 1 \, . \end{equation} % For any positive integer $N$ we consider the {\em vectorial polynomial}, sum of homogeneous vector monomials of degree $1\leq l \leq N$, defined by: $\mathcal{H}_N(z)=\sum_{l=1}^{N} \sum_{|\alpha|=l} h_{\alpha} z^{\alpha}$ and the {\em Remainder Function}: % \begin{equation} \label{eq:remainderfunction} \mathcal{R}_N(z)=D \mathcal{H}_N (z)\cdot F(z) - A\mathcal{H}_N(z) \, . \end{equation} % Clearly $\mathcal{H}_N(z)$ doesn't solve the linearization problem, but: \begin{equation} \label{eq:forhN} \frac{d}{dt}\mathcal{H}_N(z) = A \mathcal{H}_N(z)+ \mathcal{R}_N(z)\, , \end{equation} hence the remainder function gives the difference from the true solution and the approximate one. The following Proposition collects some properties of the remainder function. \begin{proposition} Let $\mathcal{R}_N(z)$ be the remainder function defined in~\eqref{eq:remainderfunction} and let $\alpha \in \N^n$, then: \begin{enumerate} \item[1)] $\partial_z^{\alpha} \mathcal{R}_N(0)=0$ if $|\alpha| \leq N$. \item[2)] For all $0Ra^{-1} e^{b/(2R)^{\alpha}}=T$, which gives a contradiction. Hence either $x(t_0)>2R$ for all $0< t 0$. Finally we can take $|z|$ sufficiently small, say $|z|0$, which in the case of $2$ dimensional vector fields can be put in the form: \begin{equation*} \limsup_{n\rightarrow +\infty} \left( \sum_{j=0}^{k(n)}\frac{\log q_{j+1}}{q_{j}} - s\log n \right) <+\infty \, . \end{equation*} \begin{remark}[Invariance of $\bigcup_{s>0}\mathcal{B}_s$, $n=1$ 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 : $1/2<\beta_kq_{k+1}<1$ and $q_n \beta_{n-1}+q_{n-1}\beta_{n}=1$, where $q_k$'s are the denominators of the continued fraction development of $\omega$. We claim 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} This can be proved by using the relations between $\beta_l$ and $q_l$, to obtain the bound, for all integer $k>0$: \begin{equation*} \Big\lvert \sum_{l=0}^k\left( \beta_{l-1}\log \omega_l +\frac{\log q_{l+1}}{q_l}\right)\Big\rvert \leq \sum_{l=0}^k\Big\lvert\beta_{l-1}\log{\beta_lq_{l+1}}\Big\rvert+\Big\lvert\beta_{l-1}\log \beta_{l-1}\Big\rvert+\Big\lvert\frac{q_{l-1}}{q_l}\beta_l \log q_{l+1} \Big\rvert\notag \leq 18\, , \end{equation*} where we used the convergence of series $\sum q_l^{-1}$ and $\sum q_l^{-1}\log q_l$ (see~\cite{MMY} page 272). To prove the invariance of $\bigcup_s \mathcal{B}_s$ under the action of $PSL(2,\Z)$, is enough to consider its generators: $T\omega = \omega +1$ 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 $S$ we have $\beta_k(\omega)=\omega \beta_{k-1}(S\omega)$ for all $k\geq 1$. Let $\omega$ be an irrational and let $\omega^{\prime}=\omega^{-1}$, let us also denote with a $\prime$ quantities given by the continued fraction algorithm applied to $\omega^{\prime}$, then using~\eqref{eq:brunosbeta} one can prove: \begin{equation*} \omega_0 \left( \sum_{j=0}^k \beta_{j-1}^{\prime} \log {\omega^{\prime}_j}^{-1} + s\omega_0^{-1} \log \beta_{k-1}^{\prime} \right)=C(\omega,s) + \sum_{j=0}^{k+1} \beta_{j-1} \log \omega_j^{-1} + s\log \beta_{k}\, , \end{equation*} where $C(\omega,s)=\omega_0\left(\log \omega_1^{-1}-s\log \omega_0\right)+\sum_{l=0}^1\beta_{l-1}\log \omega_l^{-1}$, from which the claim follows. \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 $\omega$, and let us introduce a second arithmetical condition denoted by $\mathcal{B}_s^{\prime}$ to be 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} We state without proof the following proposition, and we refer to~\cite{Carletti2002}, to all details: \begin{proposition} \label{prop:differentbruno1s} Let $s>0$ and let $\omega \in (0,1)\cap \Tilde{\mathcal{B}}_s$. Then if $\omega$ is not a Bruno number then $\omega \in \mathcal{B}_s^{\prime}$, otherwise $\omega \in \mathcal{B}^{\prime}_0$. \end{proposition} Therefore if $\omega \in \Tilde{\mathcal{B}}_s\setminus \mathcal{B}$ then the denominators of the convergent to $\omega$ can grow like a factorial, more precisely, $q_{k+1}=\mathcal{O}\Big( (q_k!)^s \Big)$, is allowed. \begin{thebibliography}{XXXXX} \bibitem[Ba]{Balser1994} 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. Bruno: {\it Analytical form of differential equations}, Transactions Moscow Math.Soc. $\mathbf{25}$, (1971), pp. 131--288. \bibitem[CM]{CarlettiMarmi2000} T. Carletti and S. Marmi:{\it Linearization of analytic and non--analytic germs of diffeomorphisms of $({\mathbb C},0)$}, Bull. Soc. Math. 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