Hillel Furstenberg (Hebrew University)

$(G,\mu)$-dynamical systems: structure theory and multiple recurrence.

Abstract. A $(G,\mu)$-dynamical system is a pair $(X,\nu)$ where $X$ is a $G$-space and $\nu$ is a "stationary" probability measure on $X$ for the measure $\mu$ on $G$: $\mu*\nu = \nu$.   We discuss some of the contexts in which these appear and pose some general questions regarding these systems. As an example we discuss a structure theorem valid for all such systems and present an application to multiple recurrence for any minimal action of $G$ where $G$ is the group $Sl_2(\mathbb{R})$.