Inequalities for entropy, Hausdorff dimension, and Lipschitz constants
Volume 250 / 2020
Studia Mathematica 250 (2020), 253-264
MSC: Primary 37B40, 54F45.
DOI: 10.4064/sm180705-2-11
Published online: 7 August 2019
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 Geng 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.
