Local correlation entropy
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 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.
Cite
@article{arxiv.1612.02592,
title = {Local correlation entropy},
author = {Vladimír Špitalský},
journal= {arXiv preprint arXiv:1612.02592},
year = {2021}
}
Comments
24 pages. In this replacement, a flaw in Lemma 16 is corrected, and minor changes in Lemma 18 and the proof of Theorem A are given. Further, a simple proof of Proposition 3 is added