An auto-homeomorphism of a Cantor set with derivative zero everywhere (Q891393)

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.