Unimodal Measurable Pseudo-Anosov Maps
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arXiv
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| Format: | Preprint |
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2023
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| author | Boyland, Philip de Carvalho, André Hall, Toby |
| author_facet | Boyland, Philip de Carvalho, André Hall, Toby |
| contents | We exhibit a continuously varying family $F_λ$ of homeomorphisms of the sphere $S^2$, for which each $F_λ$ is a measurable pseudo-Anosov map. Measurable pseudo-Anosov maps are generalizations of Thurston's pseudo-Anosov maps, and also of the generalized pseudo-Anosov maps of [19]. They have a transverse pair of invariant full measure turbulations, consisting of streamlines which are dense injectively immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure.
Each map $F_λ$ is semi-conjugate to the inverse limit of the core tent map with slope $λ$: it is topologically transitive, ergodic with respect to a background Oxtoby-Ulam measure, has dense periodic points, and has topological entropy $h(F_λ) = \log λ$ (so that no two $F_λ$ are topologically conjugate). For a full measure, dense $G_δ$ set of parameters, $F_λ$ is a measurable pseudo-Anosov map but not a generalized pseudo-Anosov map, and its turbulations are nowhere locally regular. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_16059 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unimodal Measurable Pseudo-Anosov Maps Boyland, Philip de Carvalho, André Hall, Toby Dynamical Systems Geometric Topology 37E30 (Primary) 57K20 (Secondary) We exhibit a continuously varying family $F_λ$ of homeomorphisms of the sphere $S^2$, for which each $F_λ$ is a measurable pseudo-Anosov map. Measurable pseudo-Anosov maps are generalizations of Thurston's pseudo-Anosov maps, and also of the generalized pseudo-Anosov maps of [19]. They have a transverse pair of invariant full measure turbulations, consisting of streamlines which are dense injectively immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure. Each map $F_λ$ is semi-conjugate to the inverse limit of the core tent map with slope $λ$: it is topologically transitive, ergodic with respect to a background Oxtoby-Ulam measure, has dense periodic points, and has topological entropy $h(F_λ) = \log λ$ (so that no two $F_λ$ are topologically conjugate). For a full measure, dense $G_δ$ set of parameters, $F_λ$ is a measurable pseudo-Anosov map but not a generalized pseudo-Anosov map, and its turbulations are nowhere locally regular. |
| title | Unimodal Measurable Pseudo-Anosov Maps |
| topic | Dynamical Systems Geometric Topology 37E30 (Primary) 57K20 (Secondary) |
| url | https://arxiv.org/abs/2306.16059 |