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Natural extensions of
unimodal maps: virtual sphere homeomorphisms and prime ends
of basin boundaries
Philip Boyland, André de Carvalho and Toby Hall |
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Geometry & Topology 25 (2021) 111–228
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Abstract |
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Let be a family of unimodal maps with topological entropies , and be their natural extensions, where . Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge–Martin embeddings of into the sphere. We also construct a family of sphere homeomorphisms with the property that each is a factor of , by a semiconjugacy for which all fibers except one contain at most three points, and for which the exceptional fiber carries no topological entropy; that is, unimodal natural extensions are virtually sphere homeomorphisms. In the case where is the tent family, we show that is a generalized pseudo-Anosov map for the dense set of parameters for which is postcritically finite, so that is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188). |
Keywords
natural extensions, inverse limits, unimodal maps, prime
ends, sphere homeomorphisms
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Mathematical Subject Classification 2010
Primary: 37B45, 37E05, 37E30
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References |
Publication
Received: 21 September 2018
Revised: 4 November 2019
Accepted: 12 January 2020
Published: 2 March 2021
Proposed: Benson Farb
Seconded: Paul Seidel, David M Fisher
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Authors |
| Philip Boyland | |
| Department of Mathematics University of Florida Gainesville, FL United States |
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| André de Carvalho | |
| Departamento de Matemática
Aplicada IME-USP São Paulo SP Brazil |
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| Toby Hall | |
| Department of Mathematical
Sciences University of Liverpool Liverpool United Kingdom |
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