Cohomology of dynamical systems

In this paper criteria for the cohomological nullity of functions on phase spaces of various dynamical systems (U-systems, topological Markov chains, Smale systems) with coefficients in certain groups are formulated and proved, and applications of these criteria are studied. (In the simplest case of a transformation $T\colon M\to M$, the cohomological nullity of a real function $f$ means that $f(x)=g(Tx)-g(x)$.)