On the structure of isentropes of polynomial maps

TL;DR

This paper investigates the structure of isentropes in polynomial maps, revealing that they can lack local connectivity and that entropy may not vary monotonically with critical values, challenging previous assumptions.

Contribution

It demonstrates that isentropes of multimodal polynomial families are not necessarily locally connected and that entropy does not always depend monotonically on a single critical value.

Findings

Isentropes can be non-locally connected

Entropy may not depend monotonically on a single critical value

Challenges previous assumptions about polynomial map entropy structure

Abstract

The structure of isentropes (i.e. level sets of constant topological entropy) including the monotonicity of entropy, has been studied for polynomial interval maps since the 1980s. We show that isentropes of multimodal polynomial families need not be locally connected and that entropy does in general not depend monotonically on a single critical value.