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

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)
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