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Primitive Geometric Markov Partitions for pseudo-Anosov Homeomorphisms

Inti Cruz Diaz

TL;DR

This work develops an explicit, algorithmic framework for primitive geometric Markov partitions of generalized pseudo-Anosov maps on closed orientable surfaces. By leveraging a simple combinatorial criterion for when immersed graphs bound Markov partitions and introducing first intersection points, the authors define a canonical, finite family of primitive partitions $\mathcal{M}(f,n)$ and primitive geometric types $\mathcal{T}(f,n)$ that are invariant under orientation-preserving conjugacy. A key construct is the compatibility order $n(f)$, ensuring that for $n\ge n(f)$ each first-intersection-based partition exists and yields finite, analyzable geometric types. Consequently, two p-A maps are conjugate if and only if they share the same $n(f)$ and primitive geometric types for some $n\ge n(f)$, with canonical partitions arising at $n=n(f)$. The framework lays groundwork for an effective, algorithmic approach to the conjugacy problem in the p-A setting and suggests connections to broader geometric structures via primitive partitions and potential triangulation-based links.

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 $f$.

Primitive Geometric Markov Partitions for pseudo-Anosov Homeomorphisms

TL;DR

This work develops an explicit, algorithmic framework for primitive geometric Markov partitions of generalized pseudo-Anosov maps on closed orientable surfaces. By leveraging a simple combinatorial criterion for when immersed graphs bound Markov partitions and introducing first intersection points, the authors define a canonical, finite family of primitive partitions and primitive geometric types that are invariant under orientation-preserving conjugacy. A key construct is the compatibility order , ensuring that for each first-intersection-based partition exists and yields finite, analyzable geometric types. Consequently, two p-A maps are conjugate if and only if they share the same and primitive geometric types for some , with canonical partitions arising at . The framework lays groundwork for an effective, algorithmic approach to the conjugacy problem in the p-A setting and suggests connections to broader geometric structures via primitive partitions and potential triangulation-based links.

Abstract

Let be a pseudo-Anosov homeomorphism on a closed, oriented surface. We give an effective construction of Markov partitions for based on a simple combinatorial criterion deciding when an immersed graph bounds a Markov partition. This yields an explicit algorithm: from a point at the intersection of stable and unstable separatrices of a singularity of , and a sufficiently large integer , it produces a partition . Applying the algorithm to the first intersection points of we produces the set of primitive Markov partitions. We prove the existence of an integer , the compatibility order of , depending only on the conjugacy class of , such that exists for all and all first intersection points . Each geometric Markov partition has an associated geometric type , extending the incidence matrix; it result the geometric type is constant along orbits of primitive partitions, and for the set 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 . The types in are minimal and are the canonical Markov partitions of .

Paper Structure

This paper contains 14 sections, 27 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Presentation of results
  3. Remarks and Perspectives for Further Research
  4. Geometric Markov partition and their geometric types
  5. Geometric Markov partitions
  6. Geometric types
  7. An Algorithmic Construction of Markov Partitions
  8. Graphs adapted to a p-A Homeomorphism
  9. Compatible Graphs
  10. The existence of Markov partitions
  11. Primitive Markov partitions and their geometric types
  12. First intersection points
  13. Primitive geometric Markov partitions
  14. Primitive geometric types

Key Result

Theorem 1

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

Figures (6)

  • Figure 1: Boundary identification in $R=\overline{r}$
  • Figure 2: Rectangles $A$ and $C$ with boundary identifications; rectangle $A$ is embedded.
  • Figure 3: The arcs $[x,y_1]^s$ and $[x,y_1]^u$.
  • Figure 4: The vertical directions in $[0,1]\times [0,1]$
  • Figure 5: Item $(3)$ and $(4)$ in Definition \ref{['Defi: Geo Markov partition']}
  • ...and 1 more figures

Theorems & Definitions (69)

  • Theorem 1
  • Proposition : \ref{['Prop:Compatibles implies Markov partition']}
  • Proposition : \ref{['Prop: Conjugates then primitive Markov partition']}
  • Corollary : \ref{['Coro: Finite orbits of primitive Markov partitions']}
  • Theorem : \ref{['Theo: finite geometric types']}
  • Corollary 1
  • Definition 1: p-A homeomorphism
  • Definition 2
  • Definition 3: Rectangle
  • Definition 4: Subrectangle
  • ...and 59 more