Invariant Distributions for homogeneous flows

TL;DR

This paper establishes that homogeneous flows on finite-volume manifolds typically have countably many invariant distributions unless they are conjugate to linear toral flows, confirming related conjectures.

Contribution

It proves the countability of invariant distributions for a broad class of homogeneous flows and affine transformations, extending previous conjectures.

Findings

Homogeneous flows have countably many invariant distributions unless conjugate to toral flows.

Smooth partially hyperbolic flows have countably many minimal sets and ergodic measures.

The results confirm the Katok and Greenfield-Wallach conjectures in these cases.

Abstract

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.