%&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|