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Measures of maximal entropy that are SRB

Fernando Micena, Ryo Moore, Jana Rodriguez Hertz, Raul Ures

TL;DR

The work analyzes when entropy-maximizing measures for $C^{1+\alpha}$ partially hyperbolic DA-diffeomorphisms of $\mathbb{T}^3$ are SRB and unique physical measures. By leveraging a semiconjugacy to the linear Anosov map $A$ and deploying Pesin theory, the authors translate entropy and Lyapunov constraints into a center-cohomological equation on the support of the measure. The central result, Theorem A, gives a sharp criterion: an entropy-maximizing measure $\mu$ is SRB (and unique) if and only if the sum of its positive Lyapunov exponents matches the corresponding sum for $A$ on all hyperbolic periodic points in $\mathrm{supp}(\mu)$, with the exact expression for $\lambda^+(p)$ determined by the sign of $\lambda_A^c$; conversely, when these equalities fail, rigidity is fragile, as shown by explicit DA examples. The paper also proves that the rigidity is not as strong as Katok-type conjectures by constructing diffeomorphisms that satisfy the entropy-Lyapunov conditions yet are not Anosov, highlighting the nuanced interplay among entropy, Lyapunov spectra, SRB properties, and partial hyperbolicity in dimension three.

Abstract

A smooth conservative DA-diffeomorphism is smoothly conjugated to its Anosov linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part. A more general result for entropy maximizing measures of $C^{1+α}$ partially hyperbolic diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on $T^3$ is that they are SRB measures if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure. This rigidity result is not as strong as in the A. Katok rigidity conjecture. Examples are provided.

Measures of maximal entropy that are SRB

TL;DR

The work analyzes when entropy-maximizing measures for partially hyperbolic DA-diffeomorphisms of are SRB and unique physical measures. By leveraging a semiconjugacy to the linear Anosov map and deploying Pesin theory, the authors translate entropy and Lyapunov constraints into a center-cohomological equation on the support of the measure. The central result, Theorem A, gives a sharp criterion: an entropy-maximizing measure is SRB (and unique) if and only if the sum of its positive Lyapunov exponents matches the corresponding sum for on all hyperbolic periodic points in , with the exact expression for determined by the sign of ; conversely, when these equalities fail, rigidity is fragile, as shown by explicit DA examples. The paper also proves that the rigidity is not as strong as Katok-type conjectures by constructing diffeomorphisms that satisfy the entropy-Lyapunov conditions yet are not Anosov, highlighting the nuanced interplay among entropy, Lyapunov spectra, SRB properties, and partial hyperbolicity in dimension three.

Abstract

A smooth conservative DA-diffeomorphism is smoothly conjugated to its Anosov linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part. A more general result for entropy maximizing measures of partially hyperbolic diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on is that they are SRB measures if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure. This rigidity result is not as strong as in the A. Katok rigidity conjecture. Examples are provided.
Paper Structure (11 sections, 14 theorems, 70 equations)

This paper contains 11 sections, 14 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminary rigidity results
  4. Strategy to prove the necessity in Theorem \ref{['thmA']}
  5. Proof of the necessity in Theorem \ref{['thmA']}
  6. The case $\lambda^{c}_{A}<0$
  7. The case $\lambda^{c}_{A}>0$.
  8. Proof of Theorem \ref{['thm.examples']}
  9. Proof of the Main Theorem
  10. Basic notations
  11. Strategy

Key Result

Theorem 2.5

If $f$ is a $C^{1+\alpha}$ diffeomorphism , $\mu$ is an SRB measure of $f$ if and only if ($h_{\mu}(f)$ is the sum of all positive Lyapunov exponents $\lambda_i$ counted with their multiplicity.)

Theorems & Definitions (41)

  • Conjecture 1.2: A. Katok, 1982 katok82
  • Remark 1.5
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: SRB measure
  • Theorem 2.5: Ledrappier-Young LY1, Brown B
  • Proposition 2.7
  • Definition 2.9
  • Theorem 1: SRB rigidity
  • Theorem 2: Rigidity, ma non tanto
  • ...and 31 more