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Rigidity of equilibrium states and unique quasi-ergodicity for horocyclic foliations

Pablo D. Carrasco, Federico Rodriguez-Hertz

TL;DR

The paper establishes a rigidity result for equilibrium states of topologically mixing metric Anosov flows with Hölder potentials, showing that such states are completely determined by their disintegration along unstable leaves and a transverse quasi-invariant structure. It develops a symbolic-model framework and Marcus-type averaging operators to transfer the problem to a shift system, proving convergence of averaging operators to constants and obtaining a unique equilibrium state from the unstable conditional data. This leads to unique quasi-ergodicity of horocyclic foliations under a Hölder Jacobian and provides an analytic, non-differential proof of Babillot-Ledrappier's result. The work connects transfer-operator methods with foliation holonomy and offers a broader perspective on rigidity phenomena beyond one-dimensional unstable distributions.

Abstract

In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.

Rigidity of equilibrium states and unique quasi-ergodicity for horocyclic foliations

TL;DR

The paper establishes a rigidity result for equilibrium states of topologically mixing metric Anosov flows with Hölder potentials, showing that such states are completely determined by their disintegration along unstable leaves and a transverse quasi-invariant structure. It develops a symbolic-model framework and Marcus-type averaging operators to transfer the problem to a shift system, proving convergence of averaging operators to constants and obtaining a unique equilibrium state from the unstable conditional data. This leads to unique quasi-ergodicity of horocyclic foliations under a Hölder Jacobian and provides an analytic, non-differential proof of Babillot-Ledrappier's result. The work connects transfer-operator methods with foliation holonomy and offers a broader perspective on rigidity phenomena beyond one-dimensional unstable distributions.

Abstract

In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.
Paper Structure (11 sections, 10 theorems, 57 equations)

This paper contains 11 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction and Main Results
  2. Equilibrium states for metric Anosov flows
  3. Uniqueness of the transversal measure and rigidity
  4. Comparison with other literature
  5. Prerequisites
  6. Metric Anosov flows
  7. Symbolic models for hyperbolic systems
  8. Suspension
  9. Ruelle-Perron-Frobenius operator and equilibrium states
  10. Marcus' operators
  11. Equicontinuity of the conditional expectations

Key Result

Lemma 2.1

If $\phi\in\mathcal{F}_A$ then there exists functions $\phi^{\ast},\gamma^{\ast}:\Sigma_A\to\mathbb R, \ast\in\{-, +\}$ such that $\phi^{\ast}-\phi=\gamma^{\ast}-\gamma^{\ast}\circ \sigma$, and

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2: Proposition $6.2$ in EqStatesCenter
  • Remark 2.2
  • ...and 13 more