Some cohomological aspects of the Banach fixed point principle. (English) Zbl 1249.54081

Janoš, Ludvík

Math. Bohem. 136, No. 3, 333-336 (2011).

Summary: Let \(T X\to X\) be a continuous selfmap of a compact metrizable space \(X\). We prove the equivalence of the following two statements: (1) The mapping \(T\) is a Banach contraction relative to some compatible metric on \(X\). (2) There is a countable point separating family \(\mathcal {F}\subset \mathcal {C}(X)\) of non-negative functions \(f\in \mathcal {C}(X)\) such that for every \(f\in \mathcal {F}\) there is \(g\in \mathcal {C}(X)\) with \(f=g-g\circ T\).

MSC:

54H25	Fixed-point and coincidence theorems (topological aspects)
54H20	Topological dynamics (MSC2010)

Keywords:

Banach contraction; cohomology; cocycle; coboundary; separating family; core