Turbulence, representations, and trace-preserving actions

TL;DR

This paper investigates turbulence in spaces of C*-algebra representations and applies it to classify group actions on probability spaces and the hyperfinite II_1 factor, revealing complex dynamical behaviors.

Contribution

It establishes criteria for turbulence in representation spaces and demonstrates turbulence in conjugacy actions on various group actions and flows.

Findings

Conjugacy action on free actions of amenable groups on R is turbulent.

Conjugacy action on ergodic measure-preserving flows is generically turbulent.

Certain classification problems are nonclassifiable by countable structures.

Abstract

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent.