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July  2004, 10(3): 827-833. doi: 10.3934/dcds.2004.10.827

On Bowen's definition of topological entropy

Michał Misiurewicz 1,

1. 

Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, United States

Received  March 2003 Revised  June 2003 Published  January 2004

About 5 years ago, Dai, Zhou and Geng proved the following result. If $X$ is a metric compact space and $f:X\to X$ a Lipschitz continuous map, then the Hausdorff dimension of $X$ is bounded from below by the topological entropy of $f$ divided by the logarithm of its Lipschitz constant. We show that this is a simple consequence of a 30 years old Bowen's definition of topological entropy for noncompact sets. Moreover, a modification of this definition provides a new insight into the entropy of subshifts of finite type.
Keywords: Hausdorff dimension., Topological entropy.
Mathematics Subject Classification: 37B40, 37F35, 54F4.
Citation: Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
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