Dispersing cocycles and mixing flows under functions

Klaus Schmidt¹

¹ Mathematics Institute University of Vienna Strudlhofgasse 4 A-1090 Wien, Austria Erwin Schrödinger Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Vienna, Austria

Fundamenta Mathematicae, Tome 173 (2002) no. 2, pp. 191-199.

Résumé

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].

Détail

DOI : 10.4064/fm173-2-6

Keywords: measure preserving mixing action countable abelian group probability space mathscr locally compact second countable abelian group cocycle colon times disperses lim infty cdot alpha infty measure every map alpha colon prove cocycle does disperse only there exists compact subgroup subset composition theta circ colon times rightarrow quotient map theta colon rightarrow trivial cohomologous homomorphism eta colon rightarrow result extends number earlier characterizations coboundaries trivial cocycles tightness conditions distributions maps cdot has implications flows under functions measure preserving ergodic automorphism probability space mathscr colon rightarrow mathbb nonnegative borel map int flow under function base main result implies mixing weakly mixing ergodic mixing cocycle colon mathbb times rightarrow mathbb defined disperses latter statement answers question raised mariusz lema czyk