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Keywords:
Banach contraction; cohomology; cocycle; coboundary; separating family; core
Banach contraction; cohomology; cocycle; coboundary; separating family; core
Summary:
Let $T\colon X\to X$ 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 $f=g-g\circ T$.
References:
[1] Bakakhanian, A.: Cohomological Methods in Group Theory. Marcel Dekker, New York (1972).
[2] Janoš, L.: The Banach contraction mapping principle and cohomology. Comment. Math. Univ. Carolin. 41 (2000), 605-610. MR 1795089
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