An alternative proof of infinite dimensional Gromov's non-squeezing for compact perturbations of linear maps

TL;DR

This paper provides an alternative proof of Gromov's non-squeezing theorem extended to infinite dimensional Hilbert spaces, focusing on compact perturbations of linear symplectic maps.

Contribution

It offers a new proof approach inspired by finite dimensional methods and reformulates the problem via Hamiltonian action functionals.

Findings

Established infinite dimensional non-squeezing for compact perturbations

Connected non-squeezing to Palais-Smale sequences in Hamiltonian dynamics

Extended Gromov's theorem to a broader infinite dimensional setting

Abstract

This paper deals with the problem of generalising Gromov's non squeezing theorem to an infinite dimensional Hilbert phase space setting. By following the lines of the proof by Hofer and Zehnder of finite dimensional non-squeezing, we recover an infinite dimensional non-squeezing result by Kuksin for symplectic diffeomorphisms which are non-linear compact perturbations of a symplectic linear map. We also show that the infinite dimensional non-squeezing problem, in full generality, can be reformulated as the problem of finding a suitable Palais-Smale sequence for a distinguished Hamiltonian action functional.