The Banach contraction mapping principle and cohomology (Q1567440)

If \(X\) is a metrizable topological space, \(M(X)\) denotes the set of all compatible metrics on \(X\). In the paper it is proved that if \(X\) is compact and \(T:X\to X\) continuous then \(T\) 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 is a coboundary of the system \((X\times X, T\times T)\), i.e. is of the form \(d_2(x,y)= f(x,y)- f(Tx,Ty)\), \(x,y\in X\), where \(f\) is a continuous map \(X\times X\to \mathbb{R}\).