Local correlation entropy

TL;DR

This paper explores the relationship between local correlation entropy and topological entropy in dynamical systems, establishing conditions under which they coincide or differ, with implications for systems on graphs and subshifts.

Contribution

It proves that for systems on graphs, local correlation entropy equals topological entropy, and constructs examples with contrasting entropy properties.

Findings

Local correlation entropy and topological entropy coincide on graphs.

Uncountable set of points can have local correlation entropy close to topological entropy.

Existence of a strictly ergodic subshift with positive topological entropy but zero 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.