Lyapunov exponents for conservative twisting dynamics: a survey

TL;DR

This survey explores the relationship between Lyapunov exponents and the geometric support of minimizing measures in conservative twisting dynamical systems, building on classical and recent results in Aubry-Mather and weak KAM theories.

Contribution

It summarizes existing results on minimizing measures in conservative twisting dynamics and investigates the link between their Lyapunov exponents and the shape of their supports.

Findings

Existence of minimizing measures in conservative twisting dynamics.

Connections between Lyapunov exponents and the support shape of measures.

Open questions on deducing Lyapunov exponents from support geometry.

Abstract

Finding special orbits (as periodic orbits) of dynamical systems by variational methods and especially by minimization methods is an old method (just think to the geodesic flow). More recently, new results concerning the existence of minimizing sets and minimizing measures were proved in the setting of conservative twisting dynamics. These twisting dynamics include geodesic flows as well as the dynamics close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite . Two aspects of this theory are called the Aubry-Mather theory and the weak KAM theory. They were built by Aubry \& Mather in the '80s in the 2-dimensional case and by Mather, Ma{\~n}{\'e} and Fathi in the '90s in higher dimension. We will explain what are the conservative twisting dynamics and summarize the existence results of minimizing measures. Then we will explain more recent results concerning the link between different notions for minimizing measures for twisting dynamics: their Lyapunov exponents; their Oseledet's splitting; the shape of their support. The main question in which we are interested is: given some minimizing measure of a conservative twisting dynamics, is there a link between the geometric shape of its support and its Lyapunov exponents? Or : can we deduce the Lyapunov exponents of the measure from the shape of the support of this measure? Some proofs but not all of them will be provided. Some questions are raised in the last section.