Ludv\'\i k Jano\v s
The Banach contraction mapping principle and cohomology

Comment.Math.Univ.Carolinae 41,3 (2000) 605-610.

Abstract: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.

Keywords: $B$-system, $E$-system
AMS Subject Classification: 54H15, 54H25, 37B99