Inequalities for Entropy, Hausdorff Dimension, and Lipschitz Constants

TL;DR

This paper develops new metrics for certain dynamical systems to establish lower bounds on topological entropy using Hausdorff dimensions and Lipschitz constants, offering new insights and invariants.

Contribution

It introduces a novel approach to relate entropy, dimensions, and Lipschitz constants, reversing previous inequalities and proposing a new conjugacy invariant.

Findings

Established lower bounds for topological entropy

Reversed an existing inequality by Dai, Zhou, and Gheng

Proposed a new invariant for topological conjugacy

Abstract

We construct suitable metrics for two classes of topological dynamical systems (linear maps on the torus and non-invertible expansive maps on compact spaces) in order to get a lower bound for topological entropy in terms of the resulting Hausdorff dimensions and Lipschitz constants. This reverses an old inequality of Dai, Zhou, and Gheng and leads to a short proof of a well-known theorem on expansive mappings. It also suggests a new invariant of topological conjugacy for dynamical systems on compact metric spaces.