Coding Teichm\"uller flow using veering triangulations

TL;DR

This paper introduces a coding method for the Teichmüller flow on moduli spaces of quadratic differentials using veering triangulations, enabling detailed dynamical analysis.

Contribution

It develops veering triangulations for surfaces and applies them to code the Teichmüller flow, revealing its structure and properties.

Findings

Coding has an approximate product structure

Roof function exhibits exponential tails

Facilitates dynamical study of Teichmüller flow

Abstract

We develop the theory of veering triangulations on oriented surfaces adapted to moduli spaces of half-translation surfaces. We use veering triangulations to give a coding of the Teichm\"uller flow on connected components of strata of quadratic differentials. We prove that this coding, given by a countable shift, has an approximate product structure and a roof function with exponential tails. This makes it conducive to the study of the dynamics of Teichm\"uller flow.