Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures

TL;DR

This paper explores the connections between Green bundles, Lyapunov exponents, and regularity properties of minimizing measures in Tonelli Hamiltonian systems, revealing that measures with zero Lyapunov exponents have supports that are almost everywhere C1-regular.

Contribution

It establishes new relationships between Green bundles, Lyapunov exponents, and regularity of supports for minimizing measures in Hamiltonian dynamics.

Findings

Minimizing measures with zero Lyapunov exponents have supports that are C1-regular almost everywhere.

The study links Green bundles to the regularity properties of minimizing measures.

Results deepen understanding of the structure of minimizing measures in Tonelli Hamiltonian systems.

Abstract

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as: - the Lyapunov exponents of minimizing measures; -the weak KAM solutions. In particular, we deduce that the support of every minimizing measure all of whose Lyapunov exponents are zero is C1-regular almost everywhere.