Statistical regularity and linear response of Mather measures for Tonelli Lagrangian systems

TL;DR

This paper investigates how Mather measures in Tonelli Lagrangian systems change under small perturbations, establishing H"older continuity and exploring conditions for Lipschitz regularity based on Diophantine properties.

Contribution

It provides the first rigorous analysis of the statistical regularity of Mather measures under perturbations, linking regularity to Diophantine conditions and KAM theory.

Findings

H"older continuity of perturbed Mather measures with respect to perturbation parameter

Explicit dependence of H"older exponent on Diophantine index

Discussion on achieving Lipschitz regularity via KAM theory

Abstract

We study the statistical regularity of Mather measures associated with $C^1$ perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish H\"older continuity of the perturbed Mather measure with respect to the perturbation parameter. The H\"older exponent is shown to depend explicitly on the Diophantine index of the frequency. We also discuss the possibility of achieving Lipschitz regularity using KAM theory.