Geodesic
Cohomology of dynamical systems
Izvestiya. Mathematics , Tome 6 (1972) no. 6, pp. 1278-1301

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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 $f(x)=g(Tx)-g(x)$.) 
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     author = {A. N. Livshits},
     title = {Cohomology of dynamical systems},
     journal = {Izvestiya. Mathematics },
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A. N. Livshits. Cohomology of dynamical systems. Izvestiya. Mathematics , Tome 6 (1972) no. 6, pp. 1278-1301. http://geodesic.mathdoc.fr/item/IM2_1972_6_6_a5/