Hyperbolicity for conservative twist maps of the 2-dimensional annulus

TL;DR

This paper discusses hyperbolicity in conservative twist maps of the 2D annulus, focusing on Aubry-Mather theory, Green bundles, and dynamics near instability zones, with open questions posed for future research.

Contribution

It provides a detailed analysis of hyperbolic behavior near Aubry-Mather sets and explores the relationship between hyperbolicity and the shape of these sets in twist maps.

Findings

Link between hyperbolicity and Aubry-Mather set shape

Behavior of dynamics near boundaries of instability zones

Definition and role of Green bundles in hyperbolic regions

Abstract

These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus and focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones. We will also give some open questions. This course is the second part of a minicourse that was begun by R. Potrie. Some topics of the part of R. Potrie will be useful for this part. Many thanks to E. Maderna and L. Rifford for the invitation to give the mini-course and to R. Potrie for accepting to share the course with me.