| When An Infinitely-Renormalizable Endomorphism Of The Interval Can Be Smoothed (2007) | |||||||||||||
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| Let K be a closed subset of a smooth manifold M , and let f : K ! K be a continuous selfmap of K. We say that f is smoothable if it is conjugate to the restriction of a smooth map by a homeomorphism of the ambient space M . We give a necessary condition for the smoothability of the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets in the unit interval. This class contains, in particular, all minimal homeomorphisms of Cantor sets in the line which extend to continuous maps of an interval with zero topological entropy. I. Introduction We consider the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets embedded in the unit interval. We begin with some definitions (a detailed topological description of these maps is given in [BORT]). If K is a nonempty topological space and f : K ! K is continuous, we say that (K; f) is a dynamical system. A morphism from (K; f) to (K 0 ; f 0 ) is defined by a continuous map OE : K ! K 0 such that f... | |||||||||||||
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