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Natural extensions of unimodal maps: prime ends of planar embeddings and semi-conjugacy to sphere homeomorphisms. (arXiv:1704.06624v1 [math.DS])
来源于:arXiv
Let $f\colon I\to I$ be a unimodal map with topological entropy
$h(f)>\frac12\log2$, and let $\widehat{f}\colon\widehat{I}\to\widehat{I}$ be
its natural extension, where $\widehat{I}=\varprojlim(I,f)$. Subject to some
regularity conditions, which are satisfied for tent maps and quadratic maps, we
give a complete description of the prime ends of the Barge-Martin embedding of
$\widehat{I}$ into the disk, and identify the prime ends rotation number with
the height of $f$. We also show that $\widehat{f}$ is semi-conjugate to a
sphere homeomorphism by a semi-conjugacy for which all fibers except one
contain at most three points. In the case where $f$ is a post-critically finite
tent map, we show that the corresponding sphere homeomorphism is a generalized
pseudo-Anosov map. 查看全文>>