An auto-homeomorphism of a Cantor set with derivative zero everywhere

We construct a closed bounded subset X of R with no isolated points which admits a differentiable bijection f:X→X such that f′(x)=0 for all x∈X. We also show that any such function admits a restriction f↾P to an uncountable closed P⊆X forming a minimal dynamical system. The existence of such a map f...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2016-02, Vol.434 (2), p.1267-1280
Main Authors: Ciesielski, Krzysztof Chris, Jasinski, Jakub
Format: Article
Language:English
Subjects:
Differentiable minimal dynamical systems
Fixed point theorem
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Summary:We construct a closed bounded subset X of R with no isolated points which admits a differentiable bijection f:X→X such that f′(x)=0 for all x∈X. We also show that any such function admits a restriction f↾P to an uncountable closed P⊆X forming a minimal dynamical system. The existence of such a map fseems to contradict several well know results. The map f marks a limit beyond which Banach Fixed-Point Theorem cannot be generalized.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.09.076