Dispersing cocycles and mixing flows under functions

Volume 173 / 2002

Klaus Schmidt Fundamenta Mathematicae 173 (2002), 191-199 MSC: 37A05, 37A10, 37A20, 37A25, 37H05. DOI: 10.4064/fm173-2-6


Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ disperses if $\lim_{g\to\infty }c(g,\cdot )-\alpha (g)=\infty $ in measure for every map $\alpha \colon G\to A$. We prove that such a cocycle $c$ does not disperse if and only if there exists a compact subgroup $A_0\subset A$ such that the composition $\theta \circ c\colon G\times X\rightarrow A/A_0$ of $c$ with the quotient map $\theta \colon A\rightarrow A/A_0$ is trivial (i.e. cohomologous to a homomorphism $\eta \colon G\rightarrow A/A_0$). This result extends a number of earlier characterizations of coboundaries and trivial cocycles by tightness conditions on the distributions of the maps $\{c(g,\cdot ):g\in G\}$ and has implications for flows under functions: let $T$ be a measure-preserving ergodic automorphism of a probability space $(X,\mathscr S,\mu )$, $f\colon X\rightarrow \mathbb{R}$ be a nonnegative Borel map with $\int f\,d\mu =1$, and $T^f$ 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 ${\bf f}\colon \mathbb{Z}\times X\rightarrow \mathbb{R}$ defined by $f$ disperses. The latter statement answers a question raised by Mariusz Lemańczyk in [7].


  • Klaus SchmidtMathematics Institute
    University of Vienna
    Strudlhofgasse 4
    A-1090 Wien, Austria
    Erwin Schrödinger Institute
    for Mathematical Physics
    Boltzmanngasse 9
    A-1090 Vienna, Austria

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