#### Volume 25, issue 1 (2021)

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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 $\left\{{f}_{t}:I\to I\right\}$ be a family of unimodal maps with topological entropies $h\left({f}_{t}\right)>\frac{1}{2}log2$, and ${\stackrel{̂}{f}}_{t}:{Î}_{t}\to {Î}_{t}$ be their natural extensions, where ${Î}_{t}=\underset{←}{lim}\left(I,{f}_{t}\right)$. 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 ${\stackrel{̂}{f}}_{t}$, 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).

##### Keywords
natural extensions, inverse limits, unimodal maps, prime ends, sphere homeomorphisms
##### Mathematical Subject Classification 2010
Primary: 37B45, 37E05, 37E30