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Topological entropy and Hausdorff dimension of shrinking target sets

Xiaobo Hou, Xueting Tian, Yiwei Zhang

Abstract

In this paper, we study the topological entropy and the Hausdorff dimension of a shrinking target set. We give lower and upper bounds of topological entropy and Hausdorff dimension for dynamical systems with exponential specification property and Lipschitz continuity for maps and homeomorphisms. It generally applies to uniformly hyperbolic systems, expanding systems, and some symbolic dynamics. We show that lower and upper bounds coincide for both topological entropy and Hausdorff dimension when the systems are hyperbolic automorphisms of torus induced from a matrix with only two different eigenvalues, expanding endomorphism of the torus induced from a matrix with only one eigenvalue or some symbolic systems including one or two-sided shifts of finite type and sofic shifts.

Topological entropy and Hausdorff dimension of shrinking target sets

Abstract

In this paper, we study the topological entropy and the Hausdorff dimension of a shrinking target set. We give lower and upper bounds of topological entropy and Hausdorff dimension for dynamical systems with exponential specification property and Lipschitz continuity for maps and homeomorphisms. It generally applies to uniformly hyperbolic systems, expanding systems, and some symbolic dynamics. We show that lower and upper bounds coincide for both topological entropy and Hausdorff dimension when the systems are hyperbolic automorphisms of torus induced from a matrix with only two different eigenvalues, expanding endomorphism of the torus induced from a matrix with only one eigenvalue or some symbolic systems including one or two-sided shifts of finite type and sofic shifts.

Paper Structure

This paper contains 21 sections, 25 theorems, 272 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Lower bounds for topological entropy and Hausdorff dimension
  4. Upper bounds for topological entropy and Hausdorff dimension
  5. Applications
  6. Hyperbolic set
  7. Expanding maps
  8. Symbolic dynamics
  9. Preliminaries
  10. Topological entropy and metric entropy
  11. Hausdorff dimension
  12. Exponential specification property
  13. Lower bounds
  14. Proof of Theorem \ref{['theorem-shrinking-1']}
  15. Proof of Corollary \ref{['theorem-shrinking-2']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $(X,f)$ is a dynamical system satisfying (A). If $\overline{\tau}<\lambda_1$, then the set $\cap_{j=1}^{\infty}\mathfrak{S}(f,\varphi_j,\mathcal{Z}_j,S_j)$ is non-empty and Moreover,

Figures (2)

  • Figure 1: The relationships between different specification properties
  • Figure 2: Illustration of canonical points in $G(\mathbf{y}_1,\dots,\mathbf{y}_k)$

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 1.9
  • proof
  • ...and 37 more