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Let T be a measure-preserving and mixing action of a count- able abelian group G on a probability space (X,S,µ) and A a locally compact second countable abelian group. A cocycle c: G ◊ X 7! A for T disperses if limg!1 c(g,·) (g) = 1 in measure for every map : G ! A. We prove that such a cocycle c does not disperse if and only if there exists a compact subgroup A0 A such that the composition c: G ◊ X ! A/A0 of c with the quotient map : A ! A/A0 is trivial (i.e. cohomologous to a homomorphism : G ! A/A0). This result extends a number of earlier characterizations of cobound- aries and trivial cocycles by tightness conditions on the distributions of the maps {c(g,·) : g 2 G} and has implications for flows under functions: let T be a measure-preserving and ergodic automorphism of a probab- ility space (X,S,µ), f : X ! R a positive Borel map with R f dµ = 1, and let T f be the flow under the function f with base T. Our main result implies that, if T is mixing and T f is weakly mixing, or if T is ergodic and T f is mixing, then the cocycle f : Z ◊ X ! R defined by f disperses. The latter statement answers a question raised by Mariusz Lemanczyk in (7).

Dispersing cocycles and mixing flows under functions