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• W3134550525 abstract "Let ${f_tcolon Ito I}$ be a family of unimodal maps with topological entropies $h(f_t)>frac12log 2$, and ${widehat{f}}_tcolon{widehat{I}}_tto{widehat{I}}_t$ be their natural extensions, where ${widehat{I}}_t=varprojlim(I,f_t)$. 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 ${widehat{I}}_t$ into the sphere. We also construct a family ${chi_tcolon S^2to S^2}$ of sphere homeomorphisms with the property that each $chi_t$ is a factor of ${widehat{f}}_t$, by a semi-conjugacy 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 ${f_t}$ 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 post-critically finite, so that ${chi_{t}}$ is the completion of the unimodal generalized pseudo-Anosov family introduced in [21]." @default.
• W3134550525 created "2021-03-15" @default.
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• W3134550525 date "2021-03-02" @default.
• W3134550525 modified "2023-09-25" @default.
• W3134550525 title "Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries" @default.
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• W3134550525 doi "https://doi.org/10.2140/gt.2021.25.111" @default.
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