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Inequalities for entropy, Hausdorff dimension, and Lipschitz constants

Volume 250 / 2020

Samuel Roth, Zuzana Roth 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.

Authors

  • Samuel RothMathematical Institute
    Silesian University in Opava
    Na Rybničku 1
    74601 Opava, Czech Republic
    e-mail
  • Zuzana RothMathematical Institute
    Silesian University in Opava
    Na Rybničku 1
    74601 Opava, Czech Republic
    e-mail

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