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Entropy rigidity of $u$-Gibbs measures

Vítor Gomes, Bruno Santiago

TL;DR

The paper establishes entropy rigidity for $u$-Gibbs measures in partially hyperbolic diffeomorphisms by showing that the existence of a Margulis system of unstable measures forces constant unstable Jacobian data on periodic orbits. It develops a leafwise framework, constructing leafwise measures $\mu_x^u$ and, for $u$-Gibbs states, densities $g_x$ with respect to leaf Lebesgue measure, linking them to the dynamical density $\rho^u_x$. A key technical outcome is a formula relating unstable Jacobians to leafwise densities, plus a uniform bound on $\rho^u_x(y)$ derived from minimal unstable foliation, which yields rigidity. The results yield several applications, including constant unstable periodic data for center isometries and flow-type diffeomorphisms, and, in the jointly integrable 3-torus case, $C^{1+\alpha}$ conjugacy along the center-unstable foliation and uniqueness of $u$-Gibbs states. Overall, the work broadens entropy rigidity phenomena beyond uniformly hyperbolic systems and provides new tools for analyzing $u$-Gibbs dynamics in partially hyperbolic settings.

Abstract

We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be constant. We apply our result to center isometries and flow type diffeomorphisms showing that if a measure of maximal entropy is also $u$-Gibbs then Jacobian periodic data along the unstable bundle are constant. In the case of smooth jointly integrable partially hyperbolic diffeomorphisms of $\mathbb{T}^3$, assuming that there exists some $u$-Gibbs measure which is also a measure of maximal unstable entropy, we obtain smooth conjugacy along the center-unstable foliation and uniqueness of $u$-Gibbs measures in this case.

Entropy rigidity of $u$-Gibbs measures

TL;DR

The paper establishes entropy rigidity for -Gibbs measures in partially hyperbolic diffeomorphisms by showing that the existence of a Margulis system of unstable measures forces constant unstable Jacobian data on periodic orbits. It develops a leafwise framework, constructing leafwise measures and, for -Gibbs states, densities with respect to leaf Lebesgue measure, linking them to the dynamical density . A key technical outcome is a formula relating unstable Jacobians to leafwise densities, plus a uniform bound on derived from minimal unstable foliation, which yields rigidity. The results yield several applications, including constant unstable periodic data for center isometries and flow-type diffeomorphisms, and, in the jointly integrable 3-torus case, conjugacy along the center-unstable foliation and uniqueness of -Gibbs states. Overall, the work broadens entropy rigidity phenomena beyond uniformly hyperbolic systems and provides new tools for analyzing -Gibbs dynamics in partially hyperbolic settings.

Abstract

We obtain new entropy rigidity results for -Gibbs measures by showing that whenever a -Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be constant. We apply our result to center isometries and flow type diffeomorphisms showing that if a measure of maximal entropy is also -Gibbs then Jacobian periodic data along the unstable bundle are constant. In the case of smooth jointly integrable partially hyperbolic diffeomorphisms of , assuming that there exists some -Gibbs measure which is also a measure of maximal unstable entropy, we obtain smooth conjugacy along the center-unstable foliation and uniqueness of -Gibbs measures in this case.

Paper Structure

This paper contains 25 sections, 29 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some aplications of Theorem \ref{['main.technical']}
  3. Partially hyperbolic dynamics
  4. Subordinated partitions
  5. The dynamical density function
  6. Jacobian Periodic Data
  7. Joint integrability
  8. Unstable entropy
  9. Systems of unstable measures
  10. Proofs of the main results
  11. Proof of Theorem \ref{['teo.main']}
  12. Proof of Corollary \ref{['cor.casoji']}
  13. Proof of Theorem \ref{['main.teob']}
  14. Unstable leafwise measures
  15. Construction of the Leafwise Measures
  16. ...and 10 more sections

Key Result

Theorem A

Let $f:M\to M$ be a partially hyperbolic diffeomorphism with minimal unstable foliation. Assume that there exists some ergodic $u$-Gibbs measure $\mu\in\mathcal{P}^{\operatorname{erg}}_f(M)$ which admits a Margulis system of unstable measures. Then, $f$ has constant unstable Jacobian periodic data.

Figures (3)

  • Figure 1: Local geometrical picture of a partially hyperbolic diffeomorphism. In some cases, like those treated in Theorems \ref{['teo.main']} and \ref{['main.teob']} the bundle $E^c$ also uniquely integrates.
  • Figure 2: For a Borel set $A\subset B$ we have $\mu(A)=\int_M \mu_{B,x}^u(A)\, d\hat{\mu}(x)$, where $\hat{\mu}$ is the quotient measure on the quotient space under the partition into plaques.
  • Figure 3: Apllying minmality to find $y\in{\mathcal{W}}^u(p)$ which shadows $O(q)$ for a finite but long time.

Theorems & Definitions (56)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Subordinated Partition
  • Theorem 2.6
  • ...and 46 more