A classification of 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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| _version_ | 1866917662642143232 |
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| author | Diaz, Inti Cruz |
| author_facet | Diaz, Inti Cruz |
| contents | We continue with the ideas of Bonatti-Langevin-Jeandenans towards a constructive and algorithmic classification of pseudo-Anosov homeomorphisms (possibly with spines), up to topological conjugacy. We begin by indicating how to assign to every pseudo-Anosov homeomorphism an abstract geometric type through a Markov partition whose rectangles have been endowed with a vertical direction; these are known as geometric Markov partitions. Such assignment is not unique, as it depends on the specific geometric Markov partition, and not every abstract geometric type is realized by a pseudo-Anosov homeomorphism. This poses interesting problems related to the algorithmic computability of the classification. The article contains three main results: $i)$ the geometric type is a total invariant of conjugacy, $(ii)$ a pictorial and combinatorial criterion for determining when a given geometric type is realizable by a pseudo-Anosov homeomorphism is provided, and $(iii)$ we describe an algorithmic procedure for determining when two geometric types correspond to topologically conjugate pseudo-Anosov homeomorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A classification of pseudo-Anosov homeomorphisms via geometric Markov partitions Diaz, Inti Cruz Dynamical Systems 37B10, 37C15, 37C83, 37C86, 37E30 We continue with the ideas of Bonatti-Langevin-Jeandenans towards a constructive and algorithmic classification of pseudo-Anosov homeomorphisms (possibly with spines), up to topological conjugacy. We begin by indicating how to assign to every pseudo-Anosov homeomorphism an abstract geometric type through a Markov partition whose rectangles have been endowed with a vertical direction; these are known as geometric Markov partitions. Such assignment is not unique, as it depends on the specific geometric Markov partition, and not every abstract geometric type is realized by a pseudo-Anosov homeomorphism. This poses interesting problems related to the algorithmic computability of the classification. The article contains three main results: $i)$ the geometric type is a total invariant of conjugacy, $(ii)$ a pictorial and combinatorial criterion for determining when a given geometric type is realizable by a pseudo-Anosov homeomorphism is provided, and $(iii)$ we describe an algorithmic procedure for determining when two geometric types correspond to topologically conjugate pseudo-Anosov homeomorphisms. |
| title | A classification of pseudo-Anosov homeomorphisms via geometric Markov partitions |
| topic | Dynamical Systems 37B10, 37C15, 37C83, 37C86, 37E30 |
| url | https://arxiv.org/abs/2405.06160 |