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Local large deviation principle for Smale spaces

David Parmenter

Abstract

Large deviation principles for hyperbolic systems are well studied and provide exponential rates for the deviations of Birkhoff averages from their limit. This short article presents a local large deviation principle for Smale spaces, in particular studying the rate functions of deviations with respect to conditional Gibbs measures supported on local unstable manifolds. The proof builds on a result due to Kifer and pressure growth estimates due to Parmenter and Pollicott.

Local large deviation principle for Smale spaces

Abstract

Large deviation principles for hyperbolic systems are well studied and provide exponential rates for the deviations of Birkhoff averages from their limit. This short article presents a local large deviation principle for Smale spaces, in particular studying the rate functions of deviations with respect to conditional Gibbs measures supported on local unstable manifolds. The proof builds on a result due to Kifer and pressure growth estimates due to Parmenter and Pollicott.

Paper Structure

This paper contains 4 sections, 2 theorems, 22 equations.

Table of Contents

  1. Introduction and main results
  2. Smale spaces
  3. Conditional Gibbs measures
  4. Proof of Theorem \ref{['LLDP']}

Key Result

Theorem 1.2

Let $(X, d, f, [\cdot , \cdot])$ be a mixing Smale space and $G : X \rightarrow \mathbb{R}$ a Hölder continuous potential. For $\mu_{G}$ a.e. $x \in X$, $\delta > 0$ sufficiently small and any closed $K \subset \mathcal{M}(X)$, where, Additionally, for any open $J \subset \mathcal{M}(X)$,

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • proof : Proof of Theorem \ref{['LLDP']}
  • Remark 4.1
  • Remark 4.2