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Constructing equilibrium states for Smale spaces

David Parmenter, Mark Pollicott

Abstract

There are several known constructions of equilibrium states for Hölder continuous potentials in the context of both subshifts of finite type and uniformly hyperbolic systems. In this article we present another method of building such measures, formulated in the unified and more general setting of Smale spaces. This simultaneously extends the authors' previous work for hyperbolic attractors (modelled after Sinai's classical approach for SRB-measures) and gives a new and original construction of equilibrium states for subshifts of finite type.

Constructing equilibrium states for Smale spaces

Abstract

There are several known constructions of equilibrium states for Hölder continuous potentials in the context of both subshifts of finite type and uniformly hyperbolic systems. In this article we present another method of building such measures, formulated in the unified and more general setting of Smale spaces. This simultaneously extends the authors' previous work for hyperbolic attractors (modelled after Sinai's classical approach for SRB-measures) and gives a new and original construction of equilibrium states for subshifts of finite type.
Paper Structure (9 sections, 4 theorems, 60 equations)

This paper contains 9 sections, 4 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Examples
  4. Locally maximal hyperbolic diffeomorphisms
  5. Subshifts of finite type
  6. Constructing equilibrium states
  7. The main construction
  8. Growth of unstable manifolds for Smale spaces
  9. Proof of Theorem \ref{['transform main']}

Key Result

Theorem 3.5

Let $(X,d,f,[\cdot,\cdot])$ be a topologically mixing Smale space. Let $G_{1} : X \rightarrow \mathbb{R}$ be a Hölder continuous potential and let $G_{2} : X \rightarrow \mathbb{R}$ be a continuous potential. For $\mu_{G_1}$ a.e. $x \in X$ and $\delta>0$ small, we can define a family of measures sup where $A \subset X$ a measurable set. Then the measures supported on $f^n W_\delta^u(x)$ have weak

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.5
  • ...and 9 more