Polynomial Entropy and Expansivity

TL;DR

This paper investigates polynomial entropy in compact metric spaces, constructing examples with low entropy and demonstrating that expansive maps have entropy at least 1, highlighting the relationship between expansivity and entropy.

Contribution

It introduces new examples of homeomorphisms with low polynomial entropy and establishes a lower bound for entropy in expansive maps.

Findings

Constructed a homeomorphism with vanishing polynomial entropy that is not equicontinuous

Provided examples with arbitrarily small polynomial entropy

Proved expansive homeomorphisms have polynomial entropy ≥ 1

Abstract

In this paper we study the polynomial entropy of homeomorphism on compact metric space. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater or equal than 1.