Discretizations of Teichm\"uller Geodesic Flow and Enumeration of Pseudo-Anosov Diffeomorphisms

TL;DR

This thesis introduces two discrete dynamical systems for studying Teichmüller geodesic flow, enabling enumeration of pseudo-Anosov diffeomorphisms and advancing understanding of translation surface dynamics.

Contribution

It presents a novel discretization called diagonal changes for counting pseudo-Anosov mapping classes, extending previous methods like Rauzy-Veech induction.

Findings

Diagonal changes effectively enumerate pseudo-Anosov classes.

The method applies to hyperelliptic components and general strata.

Complete lists of pseudo-Anosov diffeomorphisms ordered by dilatation are produced.

Abstract

In this thesis, we study the Teichm\"uller geodesic flow on the space of translation surfaces by introducing two related discrete-time dynamical systems. First, we discuss the Rauzy-Veech induction, highlighting its connections to interval exchange transformations and continued fraction expansions. While effective for addressing ergodic properties, this method faces challenges in counting closed orbits. Second, we introduce diagonal changes, a discretization better suited for counting and enumeration problems, initially applied to hyperelliptic components - subsets of translation surfaces with additional symmetries. Understanding closed orbits is significant due to the one-to-one correspondence with pseudo-Anosov mapping classes. After detailing this connection, we demonstrate how diagonal changes can produce a complete list of pseudo-Anosov mapping classes ordered by dilatation, and explain how to extend the algorithm to general components of strata.