A classification of pseudo-Anosov homeomorphisms I: the geometric type is a complete conjugacy invariant

TL;DR

This paper establishes that the geometric type of a geometric Markov partition uniquely determines the topological conjugacy class of pseudo-Anosov homeomorphisms, providing a foundation for an algorithmic classification.

Contribution

It proves that the geometric type is a complete conjugacy invariant for pseudo-Anosov homeomorphisms, linking symbolic dynamics with topological conjugacy.

Findings

Two pseudo-Anosov homeomorphisms are conjugate iff they have the same geometric type.

The geometric type serves as a complete invariant for classification.

The approach uses symbolic dynamics to establish the conjugacy criterion.

Abstract

Every pseudo-Anosov homeomorphism $f$ admits infinitely many Markov partitions. A \textit{geometric Markov partition} is a Markov partition $\mathcal{R}$ in which each rectangle is equipped with a vertical orientation. To each pair $(f, \mathcal{R})$, consisting of a pseudo-Anosov homeomorphism $f$ and a geometric Markov partition $\mathcal{R}$, there is a naturally associated combinatorial object called its \textit{geometric type} $\mathbf{T}(f, \mathcal{R})$. We prove, using symbolic dynamics, that two pseudo-Anosov homeomorphisms are topologically conjugate via an orientation-preserving homeomorphism if and only if they admit geometric Markov partitions with the same geometric type. This result lays the groundwork for the algorithmic classification we will develop in subsequent work.