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Closest Distance between Iterates of Typical Points

Boyuan Zhao

Abstract

The shortest distance between the first $n$ iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi entropy. For interval maps with the Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.

Closest Distance between Iterates of Typical Points

Abstract

The shortest distance between the first iterates of a typical point can be quantified with a log rule for some dynamical systems admitting Gibbs measures. We show this in two settings. For topologically mixing Markov shifts with at most countably infinite alphabet admitting a Gibbs measure with respect to a locally Hölder potential, we prove the asymptotic length of the longest common substring for a typical point converges and the limit depends on the Rényi entropy. For interval maps with the Gibbs-Markov structure, we prove a similar rule relating the correlation dimension of Gibbs measures with the shortest distance between two iterates in the orbit generated by a typical point.
Paper Structure (8 sections, 13 theorems, 194 equations)

This paper contains 8 sections, 13 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminaries for \ref{['Gibbs Cylinder']}
  3. General Definitions and Lemmas
  4. Thermodynamic Formalism for Gibbs Measures
  5. Proof of \ref{['Gibbs Cylinder']}
  6. Upper Bound (\ref{['ineqn:3.1.1']})
  7. Lower bound (\ref{['ineqn:3.1.2']})
  8. Gibbs-Markov interval maps.

Key Result

Theorem 1.1

For a one-sided subshift system $(\Sigma_A,\sigma,\mu,\mathcal{I})$ admitting a Gibbs measure $\mu$, for $\mu$-almost every $\underline{x}\in\Sigma_A$, where $M_n(\underline{x})$ is defined in ineqn:1.1.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 1.5: k-doubling maps
  • Example 1.6: Piecewise affine interval maps
  • Example 1.7: Gauss Map
  • Example 1.8: An induced map
  • Definition 2.1: Cylinders
  • Definition 2.2: Rényi entropy
  • ...and 40 more