Lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions

TL;DR

This paper explores the relationship between Green bundles, Lyapunov exponents, and the tangent cone of minimizing measures for globally positive diffeomorphisms in all dimensions, extending Aubry-Mather theory.

Contribution

It establishes a fundamental link between Green bundle angles, Lyapunov exponents, and the support structure of minimizing measures in high-dimensional symplectic dynamics.

Findings

Deep connection between Green bundle angles and Lyapunov exponents.

Characterization of the tangent cone to the support of minimizing measures.

Extension of Aubry-Mather theory to all dimensions.

Abstract

The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry-Mather theory was developed by Garibaldi \& Thieullen that provides the existence of some minimizing measures. Using the two Green bundles G- and G+ that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: -the angle between G- and G+ along the support of the considered measure m; -the size of the smallest positive Lyapunov exponent of m; -the tangent cone to the support of m.