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

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.