Topological entropy and irregular recurrence

TL;DR

This paper investigates the relationship between various types of recurrent points and topological entropy in continuous maps, providing new results especially for one-dimensional spaces and clarifying when these relations are positive or negative.

Contribution

It offers new insights into the connection between recurrence properties and topological entropy, with specific results for interval maps and one-dimensional spaces.

Findings

Negative relation between recurrence types and entropy in general cases

Positive relations for continuous maps of the interval and certain one-dimensional spaces

Clarification of conditions under which recurrence properties influence entropy

Abstract

This paper is devoted to problems stated by Z. Zhou and F. Li in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map f and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases, the answer is positive.