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

Boyland, Philip¹ ; de Carvalho, André² ; Hall, Toby³

¹ Department of Mathematics, University of Florida, Gainesville, FL, United States
² Departamento de Matemática Aplicada, IME-USP, São Paulo SP, Brazil
³ Department of Mathematical Sciences, University of Liverpool, Liverpool, United Kingdom

Geometry & topology, Tome 25 (2021) no. 1, pp. 111-228.

Résumé

Let $\left\{f_t:I\to I\right\}$ be a family of unimodal maps with topological entropies $h\left(f_t\right)>\frac{1}{2}log2$, and ${\widehat{f}}_t:{Î}_t\to {Î}_t$ be their natural extensions, where ${Î}_t=\underleftarrow{lim}\left(I,f_t\right)$. 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 $\left\{{\chi }_t:S^2\to S^2\right\}$ of sphere homeomorphisms with the property that each ${\chi }_t$ is a factor of ${\widehat{f}}_t$, 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 $\left\{f_t\right\}$ is the tent family, we show that ${\chi }_t$ is a generalized pseudo-Anosov map for the dense set of parameters for which $f_t$ is postcritically finite, so that $\left\{{\chi }_t\right\}$ is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188).

Détail

DOI : 10.2140/gt.2021.25.111

Classification : 37B45, 37E05, 37E30
Keywords: natural extensions, inverse limits, unimodal maps, prime ends, sphere homeomorphisms

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