Abstract
Let be a family of unimodal maps with topological entropies , and be their natural extensions, where . 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 into the sphere. We also construct a family of sphere homeomorphisms with the property that each is a factor of , 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 is the tent family, we show that is a generalized pseudo-Anosov map for the dense set of parameters for which is postcritically finite, so that is the completion of the unimodal generalized pseudo-Anosov family introduced by de Carvalho and Hall (Geom. Topol. 8 (2004) 1127–1188).
Citation
Philip Boyland. André Carvalho. Toby Hall. "Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries." Geom. Topol. 25 (1) 111 - 228, 2021. https://doi.org/10.2140/gt.2021.25.111
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