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Measure and Capacity of Wandering Domains in Gevrey Near-integrable Exact Symplectic Systems

About this Title

Laurent Lazzarini, Université Paris VI, UMR 7586, Analyse algébrique, 4 place Jussieu, 75252 Paris cedex 05, Jean-Pierre Marco, Université Paris VI, UMR 7586, Analyse algébrique, 4 place Jussieu, 75252 Paris cedex 05 and David Sauzin, CNRS UMR 8028 – IMCCE, Observatoire de Paris, 77 av. Denfert-Rochereau. 75014 Paris, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 257, Number 1235
ISBNs: 978-1-4704-3492-2 (print); 978-1-4704-4953-7 (online)
DOI: https://doi.org/10.1090/memo/1235
Published electronically: January 3, 2019
MSC: Primary 53D22, 70H08.; Secondary 26E10

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Presentation of the results
  • 3. Stability theory for Gevrey near-integrable maps
  • 4. A quantitative KAM result—proof of Part (i) of Theorem
  • 5. Coupling devices, multi-dimensional periodic domains, wandering domains
  • A. \texorpdfstring{Algebraic operations in ${\mathscr O}_k$}Algebraic operations in O
  • B. Estimates on Gevrey maps
  • C. Generating functions for exact symplectic $C^\infty$ maps
  • D. Proof of Lemma
  • Acknowledgements

Abstract

A wandering domain for a diffeomorphism $\Psi$ of $\mathbb {A}^n=T^*\mathbb {T}^n$ is an open connected set $W$ such that $\Psi ^k(W)\cap W=\emptyset$ for all $k\in \mathbb {Z}^*$. We endow $\mathbb {A}^n$ with its usual exact symplectic structure. An integrable diffeomorphism, i.e. the time-one map $\Phi ^h$ of a Hamiltonian $h: \mathbb {A}^n\to \mathbb {R}$ which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of $\Phi ^h$, in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to examples of Arnold diffusion. This is a contribution to the “quantitative Hamiltonian perturbation theory” initiated in previous works on the optimality of long term stability estimates and diffusion times; our emphasis here is on discrete systems because this is the natural setting to study wandering domains.

We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form $\exp \big ( - c \big (\frac {1}{\varepsilon }\big )^{\frac {1}{2n\alpha }} \big )$, where $\varepsilon$ is the size of the perturbation, $\alpha \geq 1$ is the Gevrey exponent ($\alpha =1$ for analytic systems) and $c$ is some positive constant depending mildly on $h$. This is obtained as a consequence of an exponential stability theorem for near-integrable exact symplectic maps, in the analytic or Gevrey category, for which we give a complete proof based on the most recent improvements of Nekhoroshev theory for Hamiltonian flows, and which requires the development of specific Gevrey suspension techniques.

The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for which the capacity (and thus the measure) can be estimated from below. We suppose $n\ge 2$, essentially because KAM theory precludes Arnold diffusion in too low a dimension. For any $\alpha >1$, we produce examples with lower bounds of the form $\exp \big ( - c \big (\frac {1}{\varepsilon }\big )^{\frac {1}{2(n-1)(\alpha -1)}} \big )$. This is done by means of a “coupling” technique, involving rescaled standard maps possessing wandering discs in $\mathbb {A}$ and near-integrable systems possessing periodic domains of arbitrarily large periods in $\mathbb {A}^{n-1}$. The most difficult part of the construction consists in obtaining a perturbed pendulum-like system on $\mathbb {A}$ with periodic islands of arbitrarily large periods, whose areas are explicitly estimated from below. Our proof is based on a version due to Herman of the translated curve theorem.

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