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

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 ${\widehat{f}}_t:{Î}_t\to {Î}_t$ 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).