Geometric Markov partitions for pseudo-Anosov homeomorphisms with prescribed combinatorics

TL;DR

This paper develops an algorithmic framework for constructing and refining geometric Markov partitions for pseudo-Anosov homeomorphisms, enabling classification and analysis of their combinatorial and geometric properties.

Contribution

It introduces systematic methods for creating adapted and primitive Markov partitions, including new refinement techniques and a classification approach based on geometric types.

Findings

Finite set of primitive geometric types for each order.

Construction of Markov partitions with binary incidence matrices.

Algorithmic classification of pseudo-Anosov homeomorphisms.

Abstract

In this paper, we focus on constructing and refining geometric Markov partitions for pseudo-Anosov homeomorphisms that may contain spines. We introduce a systematic approach to constructing \emph{adapted Markov partitions} for these homeomorphisms. Our primary result is an algorithmic construction of \emph{adapted Markov partitions} for every generalized pseudo-Anosov map, starting from a single point. This algorithm is applied to the so-called \emph{first intersection points} of the homeomorphism, producing \emph{primitive Markov partitions} that behave well under iterations. We also prove that the set of \emph{primitive geometric types} of a given order is finite, providing a canonical tool for classifying pseudo-Anosov homeomorphisms. We then construct new geometric Markov partitions from existing ones, maintaining control over their combinatorial properties and preserving their geometric types. The first geometric Markov partition we construct has a binary incidence matrix, which allows for the introduction of the sub-shift of finite type associated with any Markov partition's incidence matrix -- this is known as the \emph{binary refinement}. We also describe a process that cuts any Markov partition along stable and unstable segments prescribed by a finite set of periodic codes, referred to as the $s$ and $U$-boundary refinements. Finally, we present an algorithmic construction of a Markov partition where all periodic boundary points are located at the corners of the rectangles in the partition, called the \emph{corner refinement}. Each of these Markov partitions and their intrinsic combinatorial properties plays a crucial role in our algorithmic classification of pseudo-Anosov homeomorphisms up to topological conjugacy.