Measure and capacity of wandering domains in Gevrey near-integrable exact symplectic systems

TL;DR

This paper investigates the size of wandering domains in near-integrable exact symplectic systems, providing estimates that relate to stability and diffusion phenomena in Hamiltonian dynamics.

Contribution

It offers new upper and lower bounds on wandering domain sizes in Gevrey near-integrable systems, advancing quantitative Hamiltonian perturbation theory.

Findings

Upper bounds relate to Nekhoroshev stability

Lower bounds connect to Arnold diffusion examples

Focus on discrete symplectic systems for wandering domain analysis

Abstract

A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism {\Phi}^h, i.e. the time-one map of a Hamiltonian h 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.