Topological entropy and Hausdorff dimension of shrinking target sets

TL;DR

This paper investigates the topological entropy and Hausdorff dimension of shrinking target sets in various dynamical systems, providing bounds and exact values for specific hyperbolic and symbolic systems.

Contribution

It establishes bounds and exact values for topological entropy and Hausdorff dimension in systems with exponential specification, including hyperbolic automorphisms and symbolic dynamics.

Findings

Bounds for topological entropy and Hausdorff dimension are provided.

Exact values are obtained for certain hyperbolic and symbolic systems.

Results apply to systems with exponential specification and Lipschitz maps.

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.