Constant slope, entropy and horseshoes for a map on a tame graph

TL;DR

This paper characterizes when certain monotone maps on tame graphs are conjugate to constant slope maps, linking topological entropy to horseshoes and providing conditions based on Markov properties.

Contribution

It establishes a necessary and sufficient condition for conjugacy to constant slope maps on tame graphs, connecting entropy, horseshoes, and Markov properties.

Findings

Conjugacy to constant slope maps depends on Markov and recurrence conditions.

Topological entropy is realized through horseshoes in the studied class.

Recurrent transition matrices satisfy the conjugacy condition for constant slope maps.

Abstract

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a constant slope map $g$ of a countably affine tame graph. In particular, we show that in the case of a Markov map $f$ that corresponds to recurrent transition matrix, the condition is satisfied for constant slope $e^{h_{\operatorname{top}}(f)}$, where $h_{\operatorname{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\operatorname{top}}(f)$ is achievable through horseshoes of the map $f$.