Local correlation entropy

Vladimír Špitalský

doi:10.3934/dcds.2018249

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2018, Volume 38, Issue 11: 5711-5733. Doi: 10.3934/dcds.2018249

This issue Previous Article Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations Next Article L¹ semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs

Local correlation entropy

Vladimír Špitalský^1,2,

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: November 30, 2017

Revised: May 31, 2018

Published: August 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B40, 28D20; Secondary: 54H20.

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