On the rigidity of integrable deformations of twist maps and Hamiltonian flows

TL;DR

This paper proves that certain deformations of integrable twist maps that preserve invariant circles for a range of rational rotation numbers are necessarily trivial, highlighting rigidity in such dynamical systems.

Contribution

It establishes a rigidity result for integrable area-preserving twist maps, showing that preserving invariant circles for a continuum of rational rotation numbers implies the deformation is trivial.

Findings

Deformations preserving invariant circles for rational rotation numbers are trivial.

Rigidity holds without restrictions on the size of the interval.

Analogous results are discussed for higher-dimensional Hamiltonian flows.

Abstract

In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e., homotopically non-trivial invariant curves of the cylinder on which the dynamics is conjugate to a rotation) for any rational rotation number in an open interval (without any further assumption on its size), then the deformation must be trivial. Analogue phenomena for integrable Tonelli Hamiltonian flows in higher dimensions are discussed.