Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries

Boyland, Philip; de Carvalho, André; Hall, Toby

doi:10.2140/gt.2021.25.111

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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

Geometry & Topology 25 (2021) 111–228

DOI: 10.2140/gt.2021.25.111

Abstract

Let ${f_{t} : I \to I}$ be a family of unimodal maps with topological entropies $h (f_{t}) > \frac{1}{2} log 2$ , and ${\hat{f}}_{t} : Î_{t} \to Î_{t}$ be their natural extensions, where $Î_{t} = lim_{\leftarrow} (I, f_{t})$ . 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 $Î_{t}$ into the sphere. We also construct a family ${χ_{t} : S^{2} \to S^{2}}$ of sphere homeomorphisms with the property that each $χ_{t}$ is a factor of ${\hat{f}}_{t}$ , 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 ${f_{t}}$ is the tent family, we show that $χ_{t}$ is a generalized pseudo-Anosov map for the dense set of parameters for which $f_{t}$ is postcritically finite, so that ${χ_{t}}$ is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188).

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Keywords

natural extensions, inverse limits, unimodal maps, prime ends, sphere homeomorphisms

Mathematical Subject Classification 2010

Primary: 37B45, 37E05, 37E30

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

Authors

Philip Boyland
Department of Mathematics University of Florida Gainesville, FL United States

André de Carvalho
Departamento de Matemática Aplicada IME-USP São Paulo SP Brazil

Toby Hall
Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom

Volume 25, issue 1 (2021)