Entropy of homeomorphisms on unimodal inverse limit spaces

TL;DR

This paper establishes a precise formula for the topological entropy of homeomorphisms on inverse limit spaces of tent maps with slopes in (√2, 2], linking entropy to isotopy classes and extending results to quadratic maps.

Contribution

It proves that all self-homeomorphisms on these inverse limit spaces have entropy determined by their isotopy class, providing a clear formula and extending to quadratic maps.

Findings

Entropy of homeomorphisms is |R| log s, with R integral and h isotopic to σ^R.

Classifies possible entropy values for homeomorphisms of quadratic map inverse limits.

Connects isotopy classes with entropy values in the context of unimodal inverse limits.

Abstract

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of the tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ has topological entropy $\htop(h) = |R| \log s$, where $R \in \Z$ is such that $h$ and $\sigma^R$ are isotopic. Conclusions on the possible values of the entropy of homeomorphisms of the inverse limit space of a (renormalizable) quadratic map are drawn as well.