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The Banach contraction mapping principle and cohomology. (English) Zbl 1038.54014
For a metrizable topological space \(X\) and a continuous map \(T:X\to X\), it is shown that \(T\) has a unique fixed point and is a contraction with respect to a compatible metric if there is a compatible metric that, regarded as a cocycle in the system \((X,T)\times (X,T)\), is a coboundary. Some examples and more relations to modified versions of the result are given.

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