Volume 25, issue 1 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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

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).

natural extensions, inverse limits, unimodal maps, prime ends, sphere homeomorphisms
Mathematical Subject Classification 2010
Primary: 37B45, 37E05, 37E30
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
Philip Boyland
Department of Mathematics
University of Florida
Gainesville, FL
United States
André de Carvalho
Departamento de Matemática Aplicada
São Paulo SP
Toby Hall
Department of Mathematical Sciences
University of Liverpool
United Kingdom