Symbolic dynamics for the Teichmueller flow

TL;DR

This work develops a symbolic coding for the Teichmüller flow on any component $\mathcal{Q}$ of a stratum of quadratic or abelian differentials by combining train-track technology with a subshift of finite type. A finite-to-one semi-conjugacy from a suspension of a transitive shift to the Teichmüller flow is first established on a dense set, and then extended to a countable-alphabet system to control non-compactness and apply thermodynamic formalism. The paper proves that the $\Phi^t$-invariant Lebesgue measure on $\mathcal{Q}$ is the unique measure of maximal entropy, using Sarig’s theory for countable Markov shifts and prior entropy-maximization results, with implications for the dynamics near the principal boundary of strata. The methodology provides a robust framework for entropy analysis in moduli spaces and sets the stage for potential extensions to affine invariant manifolds and boundary behavior in subsequent work.

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 $(Ω,σ)$ and a Borel suspension of $(Ω,σ)$ which admits a finite-to-one semi-conjugacy into the Teichmueller flow $Φ^t$ on Q. This is used to show that the $Φ^t$-invariant Lebesgue measure on Q is the unique measure of maximal entropy.

Figures (1)

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Theorems & Definitions (58)

• Theorem 1
• Theorem 2
• Definition 2.1: Definition 2.1 of H22
• Remark 2.2
• Definition 2.3: Definition 2.8 of H22
• Proposition 2.4: Proposition 3.2 and Proposition 3.3 of H22
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.4