Thurston's Theorem: Entropy in Dimension One

TL;DR

This paper revises Thurston's theorem linking topological entropy of traintrack maps to weak Perron numbers, providing a clearer, modernized proof including ergodicity, which was previously omitted.

Contribution

It offers a modern, cohesive proof of Thurston's traintrack theorem, filling gaps and clarifying the original complex machinery.

Findings

Proves the equivalence between entropy and weak Perron numbers for traintrack maps.

Provides a complete, accessible proof including ergodicity of traintrack representatives.

Clarifies the connection between entropy and algebraic properties of expansion constants.

Abstract

In his paper, Thurston shows that a positive real number $h$ is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant $\lambda = e^h$ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.