Orbit equivalence superrigidity for type III$_0$ actions

TL;DR

This paper establishes the first orbit equivalence superrigidity results for certain type III$_0$ actions, expanding understanding of rigidity phenomena in ergodic theory and operator algebras.

Contribution

It introduces new superrigidity results for type III$_ eq 1$ actions, specifically for skew product actions of dense subgroups of SL(n,R) on spheres.

Findings

Proves superrigidity for type III$_0$ actions with prescribed flows

Extends rigidity results beyond previously known type III$_1$ cases

Demonstrates actions can have any prescribed associated flow

Abstract

We prove the first orbit equivalence superrigidity results for actions of type III$_\lambda$ when $\lambda \neq 1$. These actions arise as skew products of actions of dense subgroups of $SL(n,\mathbb{R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.