Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alfonso Sorrentino, Jianlu Zhang, Siyao Zhu

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 Hölder continuity of the perturbed Mather measure with respect to the perturbation parameter. The Hölder 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.

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

Abstract

We study the statistical regularity of Mather measures associated with perturbations of a Tonelli Lagrangian. When the unperturbed Mather measure is supported on a quasi-periodic torus with a Diophantine frequency, we establish Hölder continuity of the perturbed Mather measure with respect to the perturbation parameter. The Hölder 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.
Paper Structure (13 sections, 15 theorems, 179 equations)

This paper contains 13 sections, 15 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and setting
  4. Viscosity solutions
  5. Aubry-Mather theory for low-regularity Lagrangians
  6. Statistical regularity of the Mather measures for Mañé's Lagrangians
  7. Upper bound
  8. Lower bound in dimension 2
  9. Lower bound in higher dimensions ($n>2$)
  10. Statistical regularity of Mather measures for Tonelli Hamiltonians under perturbations
  11. Case I: Cohomological perturbations
  12. Case II: Mañé's perturbations
  13. Lipschitz regularity and Linear response under the KAM theory

Key Result

Lemma 2.4

Davini06 For any $y \in \mathcal{A}$, there exists an absolutely continuous curve $\gamma : \mathbb{R} \to \mathcal{A}$ with $\gamma(0) = y$ such that for all $a\le b$, Such a curve is called static.

Theorems & Definitions (36)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6: Mather measure
  • Lemma 2.7
  • ...and 26 more