An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions

TL;DR

This thesis develops an algorithmic method to classify generalized pseudo-Anosov homeomorphisms using geometric Markov partitions and their types, providing a complete invariant for conjugacy classes.

Contribution

It introduces a new classification approach based on geometric types, including an algorithm to determine conjugacy of generalized pseudo-Anosov homeomorphisms.

Findings

Geometric type is a complete invariant of conjugation.

An algorithm to decide conjugacy of generalized pseudo-Anosov homeomorphisms.

Criteria for realizing geometric types by pseudo-Anosov homeomorphisms.

Abstract

This thesis provides a classification of generalized pseudo-Anosov homeomorphisms up to topological conjugacy using an algorithmic approach. A Markov partition of a generalized pseudo-Anosov homeomorphism is a decomposition of the surface into a finite number of rectangles with disjoint interiors, such that their images intersect with any other rectangle in the Markov partition along a finite number of horizontal sub-rectangles. Every generalized pseudo-Anosov homeomorphism has a Markov partition, and, by using the surface's orientation, we can endow any Markov partition with a geometrization. The geometric type of a geometric Markov partition was defined by Bonatti and Langevin in their book, "Diffeomorphismes de Smale des surfaces", to classify saddle-type basic pieces for structurally stable diffeomorphisms on surfaces. A geometric type is an abstract combinatorial object that generalizes the incidence matrix of a Markov partition. It takes into account not only the number of times the image of a rectangle intersects with any other rectangle in the family but also the order and change of orientation induced by the homeomorphisms. This thesis employs the geometric type of a geometric Markov partition to classify conjugacy classes of pseudo-Anosov homeomorphisms. The classification is provided by the three main results in this manuscript: I) The geometric type is a complete invariant of conjugation. II) A criterion is provided for determining whether an abstract geometric type is realized by a geometric Markov partition of a pseudo-Anosov homeomorphism. III) An algorithm is described for determining whether two geometric types in the pseudo-Anosov class are realized by generalized pseudo-Anosov homeomorphisms that are topologically conjugated or not.