Symbolic dynamics for the Teichmueller flow

TL;DR

This paper constructs a symbolic coding for the Teichmueller flow on certain moduli spaces, proving the uniqueness of the measure of maximal entropy among invariant measures.

Contribution

It introduces a finite-type subshift and a semi-conjugacy to analyze the Teichmueller flow, establishing measure uniqueness.

Findings

Constructed a symbolic coding for the flow.

Proved the Lebesgue measure is the unique measure of maximal entropy.

Established a semi-conjugacy between the flow and a subshift.

Abstract

Let Q be a component of a stratum of abelian or quadratic differentials on an oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq 2$. We construct a subshift of finite type $(\Omega,\sigma)$ and a Borel suspension of $(\Omega,\sigma)$ which admits a finite-to-one semi-conjugacy into the Teichmueller flow $\Phi^t$ on Q. This is used to show that the $\Phi^t$-invariant Lebesgue measure on Q is the unique measure of maximal entropy.