# American Institute of Mathematical Sciences

November  2018, 38(11): 5711-5733. doi: 10.3934/dcds.2018249

## Local correlation entropy

 1 Slovanet a.s., Záhradnícka 151, Bratislava, Slovakia 2 Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica, Slovakia

Received  December 2017 Revised  June 2018 Published  August 2018

Local correlation entropy, introduced by Takens in 1983, represents the exponential decay rate of the relative frequency of recurrences in the trajectory of a point, as the embedding dimension grows to infinity. In this paper we study relationship between the supremum of local correlation entropies and the topological entropy. For dynamical systems on topological graphs we prove that the two quantities coincide. Moreover, there is an uncountable set of points with local correlation entropy arbitrarily close to the topological entropy. On the other hand, we construct a strictly ergodic subshift with positive topological entropy having all local correlation entropies equal to zero. As a necessary tool, we derive an expected relationship between the local correlation entropies of a system and those of its iterates.

Citation: Vladimír Špitalský. Local correlation entropy. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5711-5733. doi: 10.3934/dcds.2018249
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