Stationary dynamical systems

TL;DR

This paper develops a general theory of stationary dynamical systems for groups with probability measures, offering a simple structure theory and applications like a Szemeredi-type theorem for SL(2,R).

Contribution

It introduces an abstract framework for stationary systems and highlights new applications, including results in ergodic theory and group actions.

Findings

Proposes a simple structure theory for stationary systems

Extends the theory to general acting groups with probability measures

Provides a Szemeredi-type theorem for SL(2,R)

Abstract

Following works of Furstenberg and Nevo and Zimmer we present an outline of a theory of stationary (or m-stationary) dynamical systems for a general acting group G equipped with a probability measure m. Our purpose is two-fold: First to suggest a more abstract line of development, including a simple structure theory. Second, to point out some interesting applications; one of these is a Szemeredi type theorem for SL(2,R).