An auto-homeomorphism of a Cantor set with derivative zero everywhere. (English) Zbl 1331.37012

Ciesielski, Krzysztof Chris; Jasinski, Jakub

J. Math. Anal. Appl. 434, No. 2, 1267-1280 (2016).

By a delicate construction the authors prove the following:
Theorem. There exists a nonempty compact subset \(X\subset \mathbb R\) with no isolated points and a differentiable bijection \(f:X\to X\), which is extendable to a differentable function \(F:\mathbb R \to \mathbb R\), and such the derivative \(f'(x)=0\) for all \(x\in X\) and \(f(P)\neq P\) for all proper subsets of \(X\) (thus \((X,f)\) is a minimal dynamical system).
Moreover \(f\) satisfies certain local contractivity properties and has no fixed points. Thus it shows some borders for generalizations of the Banach fixed point Theorem to local versions.

Reviewer: Zdzisław Dzedzej (Gdansk)

MSC:

37B05	Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
26A30	Singular functions, Cantor functions, functions with other special properties
54C30	Real-valued functions in general topology
54H20	Topological dynamics (MSC2010)
54H25	Fixed-point and coincidence theorems (topological aspects)
47H10	Fixed-point theorems

Keywords:

differentiable minimal dynamical systems; fixed point theorem; Cantor set

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