Polynomial entropy on regular curves

TL;DR

This paper investigates the polynomial entropy of homeomorphisms on regular curves, establishing bounds and conditions for when the entropy equals one, and extends results to local dendrites, graphs, and dendrites.

Contribution

It proves that polynomial entropy on regular curves is at most one and characterizes when it equals one, providing a rigidity result for local dendrites.

Findings

Polynomial entropy on regular curves is bounded above by one.

Polynomial entropy equals one if the homeomorphism has a wandering point.

On local dendrites, polynomial entropy is either zero or one.

Abstract

We show that the polynomial entropy of homeomorphisms on regular curves is bounded above by one. Moreover, the polynomial entropy equals one under the fairly mild condition that the homeomorphism possesses a wandering point. We obtain a rigidity result for homeomorphisms on local dendrites (and therefore on graphs and dendrites): their polynomial entropy is either zero or one.