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Measure of maximal entropy for minimal Anosov actions

Tristan Humbert

TL;DR

The paper addresses identifying and characterizing the measure of maximal entropy for higher-rank Anosov actions by linking Carrasco–Rodriguez-Hertz’s geometric construction to the Ruelle–Taylor resonances framework of Guedes Bonthonneau–Weich et al. It shows that the topological entropy appears as the first resonance on the bundle of $d_s$-forms, with leaf measures providing resonant and co-resonant states, and proves a Bowen-type formula that expresses the MME in terms of sums over periodic torii. A uniform construction of the MME across the Weyl chamber is established, and a corollary counting periodic torii is derived, via a joint-propagator analysis and trace formula. These results generalize rank-one resonance methods to higher rank, linking spectral data and periodic orbit counting to equilibrium states for Anosov actions. The work advances the understanding of entropy, resonances, and periodic-orbit statistics in the setting of higher-rank hyperbolic dynamics with potential applications to non-locally symmetric examples.

Abstract

For a minimal Anosov $\mathbb R^κ$-action on a closed manifold, we study the measure of maximal entropy constructed by Carrasco and Rodriguez-Hertz in \cite{CarHer} and show that it fits into the theory of Ruelle-Taylor resonances introduced by Guedes Bonthonneau, Guillarmou, Hilgert, and Weich in \cite{GBGHW}. More precisely, we show that the topological entropy corresponds to the first Ruelle-Taylor resonance for the action on a certain bundle of forms and that the measure of maximal entropy can be retrieved as the distributional product of the corresponding resonant and co-resonant states. As a consequence, we prove a Bowen-type formula for the measure of maximal entropy and a counting result on the number of periodic torii.

Measure of maximal entropy for minimal Anosov actions

TL;DR

The paper addresses identifying and characterizing the measure of maximal entropy for higher-rank Anosov actions by linking Carrasco–Rodriguez-Hertz’s geometric construction to the Ruelle–Taylor resonances framework of Guedes Bonthonneau–Weich et al. It shows that the topological entropy appears as the first resonance on the bundle of -forms, with leaf measures providing resonant and co-resonant states, and proves a Bowen-type formula that expresses the MME in terms of sums over periodic torii. A uniform construction of the MME across the Weyl chamber is established, and a corollary counting periodic torii is derived, via a joint-propagator analysis and trace formula. These results generalize rank-one resonance methods to higher rank, linking spectral data and periodic orbit counting to equilibrium states for Anosov actions. The work advances the understanding of entropy, resonances, and periodic-orbit statistics in the setting of higher-rank hyperbolic dynamics with potential applications to non-locally symmetric examples.

Abstract

For a minimal Anosov -action on a closed manifold, we study the measure of maximal entropy constructed by Carrasco and Rodriguez-Hertz in \cite{CarHer} and show that it fits into the theory of Ruelle-Taylor resonances introduced by Guedes Bonthonneau, Guillarmou, Hilgert, and Weich in \cite{GBGHW}. More precisely, we show that the topological entropy corresponds to the first Ruelle-Taylor resonance for the action on a certain bundle of forms and that the measure of maximal entropy can be retrieved as the distributional product of the corresponding resonant and co-resonant states. As a consequence, we prove a Bowen-type formula for the measure of maximal entropy and a counting result on the number of periodic torii.

Paper Structure

This paper contains 12 sections, 19 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Anosov actions
  3. Equilibrium states for partially hyperbolic flows
  4. Statement of results
  5. Measure of maximal entropy for Anosov actions
  6. Ruelle-Taylor resonances for the action on $d_s$-forms
  7. Ruelle-Taylor resonances
  8. Constructing the norm
  9. Critical axis
  10. Resonances on the critical axis
  11. Bowen-type formula
  12. Anosov actions for which the measure of maximal entropy and SRB measure do not coincide.

Key Result

Theorem 1

There exist families of leaf measures $\{m^u_{x}\mid x\in \mathcal{M}\}$ and $\{m^{cs}_{x}\mid x\in \mathcal{M}\}$ such that The product $m=cm^u\wedge m^{cs}$$($for some normalizing constant $c>0)$ defines a probability measure which is the measure of maximal entropy associated to any $A\in \mathcal{W}$. It is invariant for any $\varphi_1^A$ for $A\in \mathbf{a}$. Moreovoer, it is ergodic and has

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1: Common measure of maximal entropy
  • Theorem 2: First resonance
  • Theorem 3: Bowen-type formula
  • Corollary 3.1: Torii counting
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 26 more