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

Inti Cruz Diaz

TL;DR

This work establishes that the geometric type $\\mathcal{T}(f,\\mathcal{R})$ of a pseudo-Anosov map, realized via a geometric Markov partition with a vertical orientation, is a complete invariant for topological conjugacy under orientation-preserving homeomorphisms. By embedding the dynamics in a symbolic framework, it introduces a binary refinement to ensure a binary incidence matrix, defines sector codes and a quotient surface model $(\\Sigma_T, \\sigma_T)$, and constructs an explicit equivalence relation $\\sim_T$ on the shift space that captures when two codes project to the same point. The central result shows that two pseudo-Anosov maps are conjugate by an orientation-preserving homeomorphism if and only if they admit geometric Markov partitions with the same geometric type, providing a foundation for an algorithmic classification in future work. The approach combines geometric partitions, symbolic dynamics, and carefully designed boundary/interior decompositions to translate surface dynamics into a robust, computable invariant with potential applications to mapping class groups and algorithmic recognition of pseudo-Anosov maps.

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.

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

TL;DR

This work establishes that the geometric type of a pseudo-Anosov map, realized via a geometric Markov partition with a vertical orientation, is a complete invariant for topological conjugacy under orientation-preserving homeomorphisms. By embedding the dynamics in a symbolic framework, it introduces a binary refinement to ensure a binary incidence matrix, defines sector codes and a quotient surface model , and constructs an explicit equivalence relation on the shift space that captures when two codes project to the same point. The central result shows that two pseudo-Anosov maps are conjugate by an orientation-preserving homeomorphism if and only if they admit geometric Markov partitions with the same geometric type, providing a foundation for an algorithmic classification in future work. The approach combines geometric partitions, symbolic dynamics, and carefully designed boundary/interior decompositions to translate surface dynamics into a robust, computable invariant with potential applications to mapping class groups and algorithmic recognition of pseudo-Anosov maps.

Abstract

Every pseudo-Anosov homeomorphism admits infinitely many Markov partitions. A \textit{geometric Markov partition} is a Markov partition in which each rectangle is equipped with a vertical orientation. To each pair , consisting of a pseudo-Anosov homeomorphism and a geometric Markov partition , there is a naturally associated combinatorial object called its \textit{geometric type} . 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.

Paper Structure

This paper contains 28 sections, 54 theorems, 142 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric Markov partitions
  4. The Arnold' Cat map
  5. The Plykin attractor
  6. Abstract geometric types
  7. The geometric type of a geometric Markov partition
  8. The symbolic dynamics of a geometric type
  9. The Binary refinement.
  10. The projection and the sector codes
  11. The sectors of a point
  12. Sector codes
  13. The quotient space is a surface
  14. The gemetric type induce a decomposition of the shift space.
  15. Periodic points and codes
  16. ...and 13 more sections

Key Result

Theorem 1

A pair of pseudo-Anosov homeomorphisms admits geometric Markov partitions with the same geometric type if and only if they are topologically conjugate via an orientation-preserving homeomorphism.

Figures (8)

  • Figure 1: Binary refinement
  • Figure 2: Regular neighborhood of a $3$-prong
  • Figure 3: Projections of the stable leaf codes
  • Figure 4: The $s$-generation function
  • Figure 5: The stable identification mechanism
  • ...and 3 more figures

Theorems & Definitions (136)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: Geometric Markov partition
  • Definition 7
  • Definition 8
  • ...and 126 more