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Gibbs measures have local product structure

Vaughn Climenhaga

TL;DR

The work addresses the local product structure of equilibrium measures for uniformly hyperbolic systems by proving a direct implication from the Gibbs property to product structure. It develops a Bowen-property–based proof that for a topologically transitive $C^1$ Anosov diffeomorphism $f$ and a potential with the Bowen property, the unique equilibrium measure $\mu$ possesses local product structure with densities uniformly bounded relative to the leafwise measures $\mu^u$ and $\mu^s$. Central to the approach are leafwise and two-sided Bowen balls, their products, and Gibbs bounds that connect dynamical sums $S_n\varphi$ to measure estimates on product sets; the argument avoids transfer operators and leafwise constructions. The method aligns with Bowen’s thermodynamic formalism and offers a route potentially adaptable to non-uniform hyperbolicity, highlighting a direct link between the Gibbs property and local product structure with quantitative bounds. This contributes to a clearer, operator-free understanding of equilibrium measures’ stochastic properties and their geometric-measure structure in hyperbolic dynamics.

Abstract

It is well-known that equilibrium measures for uniformly hyperbolic dynamical systems have a local product structure, which plays an important role in their mixing properties. Existing proofs of this fact rely either on transfer operators or on leafwise constructions, and in particular are not well-suited to the approach to thermodynamic formalism based on Bowen's specification property. Here we provide an alternate proof based on the Gibbs property, which fits more comfortably in that approach.

Gibbs measures have local product structure

TL;DR

The work addresses the local product structure of equilibrium measures for uniformly hyperbolic systems by proving a direct implication from the Gibbs property to product structure. It develops a Bowen-property–based proof that for a topologically transitive Anosov diffeomorphism and a potential with the Bowen property, the unique equilibrium measure possesses local product structure with densities uniformly bounded relative to the leafwise measures and . Central to the approach are leafwise and two-sided Bowen balls, their products, and Gibbs bounds that connect dynamical sums to measure estimates on product sets; the argument avoids transfer operators and leafwise constructions. The method aligns with Bowen’s thermodynamic formalism and offers a route potentially adaptable to non-uniform hyperbolicity, highlighting a direct link between the Gibbs property and local product structure with quantitative bounds. This contributes to a clearer, operator-free understanding of equilibrium measures’ stochastic properties and their geometric-measure structure in hyperbolic dynamics.

Abstract

It is well-known that equilibrium measures for uniformly hyperbolic dynamical systems have a local product structure, which plays an important role in their mixing properties. Existing proofs of this fact rely either on transfer operators or on leafwise constructions, and in particular are not well-suited to the approach to thermodynamic formalism based on Bowen's specification property. Here we provide an alternate proof based on the Gibbs property, which fits more comfortably in that approach.
Paper Structure (8 sections, 12 theorems, 46 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Local product structure
  3. Stable and unstable manifolds
  4. Rectangles
  5. Measures with local product structure
  6. Leafwise Bowen balls and their products
  7. Further Gibbs bounds
  8. Completion of the proof

Key Result

Theorem 1.1

If $f\colon M\to M$ is a topologically transitive $C^1$ Anosov diffeomorphism and $\varphi \colon M\to \mathbb{R}$ is continuous and satisfies the Bowen property eqn:intro-Bow, then the unique equilibrium measure has local product structure with uniformly bounded densities in the sense of Definition

Figures (2)

  • Figure 1: Continuity of brackets and structure of rectangles
  • Figure 2: Distance estimates in Propositions \ref{['prop:Bow-balls-1']} and \ref{['prop:Bow-balls-2']}

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 17 more