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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
Abstract

Let {ft: I I} be a family of unimodal maps with topological entropies h(ft) > 1 2 log2, and f̂t: Ît Ît be their natural extensions, where Ît = lim(I,ft). 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 Ît into the sphere. We also construct a family {χt: S2 S2} of sphere homeomorphisms with the property that each χt is a factor of f̂t, 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 {ft} is the tent family, we show that χt is a generalized pseudo-Anosov map for the dense set of parameters for which ft is postcritically finite, so that {χt} 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
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