Partial actions and orbit equivalence relations

Abstract: In this talk, we will discuss the framework of partial actions for constructing orbit equivalent actions of Polish groups. While related ideas have been employed in ergodic theory and Borel dynamics for many years, the particular viewpoint of partial actions simplifies construction of orbit equivalent actions of distinct groups.

As an application, we will present a Borel version of Katok's representation theorem for multidimensional Borel flows. One-dimensional flows are closely connected to actions of $\mathbb{Z}$ via the so-called "flow under a function" construction. This appealing geometric picture does not generalize to higher dimensions. Within the ergodic theoretical framework, Katok introduced the concept of a special flow as a way to connect multidimensional $\mathbb{R}^d$ and $\mathbb{Z}^d$ actions. We will show that similar connections continue to hold in Borel dynamics.

Another illustration of the partial actions techniques that we intend to touch is the following result: a Borel equivalence relation generated by a free R-flow can also be generated by a free action of any non-discrete and non-compact Polish group. This is in contrast with the situation for discrete groups, where amenability distinguishes groups that can and cannot generate free finite measure-preserving hyperfinite actions.

dynamical systemslogic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic