Primitive Geometric Markov Partitions for pseudo-Anosov Homeomorphisms

TL;DR

This paper presents an effective algorithm for constructing primitive Markov partitions for pseudo-Anosov homeomorphisms on surfaces, providing a finite classification of their geometric types and a canonical form for conjugacy classes.

Contribution

It introduces a combinatorial criterion and an explicit algorithm to produce primitive Markov partitions, establishing their finiteness and canonical nature for conjugacy classification.

Findings

An explicit algorithm for constructing Markov partitions.

Existence of a finite set of primitive geometric types for each map.

Canonical Markov partitions characterize conjugacy classes.

Abstract

Let $f$ be a pseudo-Anosov homeomorphism on a closed, oriented surface. We give an effective construction of Markov partitions for $f$ based on a simple combinatorial criterion deciding when an immersed graph bounds a Markov partition. This yields an explicit algorithm: from a point $z$ at the intersection of stable and unstable separatrices of a singularity of $f$, and a sufficiently large integer $n$, it produces a partition $\mathcal{R}(f,z,n)$. Applying the algorithm to the first intersection points of $f$ we produces the set of primitive Markov partitions. We prove the existence of an integer $n(f)$, the compatibility order of $f$, depending only on the conjugacy class of $f$, such that $\mathcal{R}(f,z,n)$ exists for all $n\ge n(f)$ and all first intersection points $z$. Each geometric Markov partition $\mathcal{R}$ has an associated geometric type $T(f,\mathcal{R})$, extending the incidence matrix; it result the geometric type is constant along orbits of primitive partitions, and for $n\ge n(f)$ the set $\mathcal{T}(f,n)$ of primitive geometric types is finite. By \cite{IntiThesis}, this family is canonical: two maps are topologically conjugate by an orientation-preserving homeomorphism iff they share the compatibility order and the primitive geometric types for some $n\ge n(f)$. The types in $\mathcal{T}(f,n(f))$ are minimal and are the canonical Markov partitions of $f$.