Rigidity and Flexibility of Polynomial Entropy

TL;DR

This paper introduces the concept of one-way horseshoes to analyze polynomial entropy of interval maps, establishing a rigidity result that finite polynomial entropy must be an integer, and explores the flexibility of polynomial entropy on continua.

Contribution

It develops a new framework using one-way horseshoes for polynomial entropy, proves a rigidity theorem for interval maps, and characterizes the entropy values on various continua.

Findings

Polynomial entropy of interval maps is characterized by one-way horseshoes.

Finite polynomial entropy of an interval map is always an integer.

Polynomial entropy on continua can take any non-negative value, showing high flexibility.

Abstract

We introduce the notion of a one-way horseshoe and show that the polynomial entropy of an interval map is given by one-way horseshoes of iterates of the map, obtaining in such a way an analogue of Misiurewicz's theorem on topological entropy and standard `two-way' horseshoes. Moreover, if the map is of Sharkovskii type 1 then its polynomial entropy can also be computed by what we call chains of essential intervals. As a consequence we get a rigidity result that if the polynomial entropy of an interval map is finite, then it is an integer. We also describe the possible values of polynomial entropy of maps of all Sharkovskii types. As an application we compute the polynomial entropy of all maps in the logistic family. On the other hand, we show that in the class of all continua the polynomial entropy of continuous maps is very flexible. For every value $\alpha \in [0,\infty]$ there is a homeomorphism on a continuum with polynomial entropy $\alpha$. We discuss also possible values of the polynomial entropy of continuous maps on dendrites.