An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions
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arXiv
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| Format: | Preprint |
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2024
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| author | Diaz, Inti Cruz |
| author_facet | Diaz, Inti Cruz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions Diaz, Inti Cruz Dynamical Systems Geometric Topology 37E30, 37C86, 37C83, 37C15, 37B10 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. |
| title | An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions |
| topic | Dynamical Systems Geometric Topology 37E30, 37C86, 37C83, 37C15, 37B10 |
| url | https://arxiv.org/abs/2405.07954 |