Geodesic
The Banach contraction mapping principle and cohomology
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 605-610

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By a dynamical system $(X,T)$ we mean the action of the semigroup $(\Bbb Z^+,+)$ on a metrizable topological space $X$ induced by a continuous selfmap $T:X\rightarrow X$. Let $M(X)$ denote the set of all compatible metrics on the space $X$. Our main objective is to show that a selfmap $T$ of a compact space $X$ is a Banach contraction relative to some $d_1\in M(X)$ if and only if there exists some $d_2\in M(X)$ which, regarded as a $1$-cocycle of the system $(X,T)\times (X,T)$, is a coboundary.
Classification : 37B25, 37B99, 47H10, 54H15, 54H20, 54H25
Keywords: $B$-system; $E$-system 
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Janoš, Ludvík. The Banach contraction mapping principle and cohomology. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 605-610. http://geodesic.mathdoc.fr/item/CMUC_2000__41_3_a15/