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Constructing equilibrium states for some partially hyperbolic attractors via densities

David Parmenter, Mark Pollicott

Abstract

We shall describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalises our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin-Sinai and Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.

Constructing equilibrium states for some partially hyperbolic attractors via densities

Abstract

We shall describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalises our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin-Sinai and Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.
Paper Structure (13 sections, 13 theorems, 69 equations)

This paper contains 13 sections, 13 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Partial Hyperbolicity
  4. Local manifolds
  5. Examples
  6. Entropy
  7. Pressure and growth
  8. Proof of Theorem \ref{['thm:gibbsdiffeo_intro']}
  9. Proof of Theorem \ref{['thm:gibbsdiffeo_intro']}
  10. Examples
  11. Partially Hyperbolic Diffeomorphisms with Subexponential Contraction in Centre-Unstable Manifolds
  12. Growth of centre-unstable manifolds
  13. Constructing equilibrium states

Key Result

Theorem 1.1

Let $f: X \to X$ be a $C^{1+\alpha}$ topologically mixing partially hyperbolic attracting diffeomorphism satisfying Lyapunov stability (Definition LS) and let $G: X \to \mathbb R$ be a continuous function. Given $x\in X$ and $\delta > 0$ consider the sequence of probability measures $(\lambda_n)_{n= Then the weak* limit points of the averages (where $f_* \lambda_n(A) = \lambda_n(f^{-1}A)$ for Bor

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1: Partially hyperbolic set
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: Local unstable manifold
  • Theorem 2.5
  • Theorem 2.6
  • Example 2.7: Quasihyperbolic linear toral automorphism
  • Example 2.8: Compact group extensions
  • ...and 31 more