Chain recurrence rates and topological entropy

TL;DR

This paper explores the properties of chain recurrence, transitivity, and mixing in dynamical systems, establishing bounds on recurrence times and linking chain mixing time to topological entropy.

Contribution

It provides a detailed analysis of chain recurrence structures and introduces bounds on recurrence times, connecting chain mixing time with topological entropy.

Findings

Bounds on chain recurrence and mixing times

Relation between chain mixing time and topological entropy

Structural description of chain transitive maps

Abstract

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.