Parabolic perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$

TL;DR

This paper studies smooth perturbations of unipotent flows on compact quotients of SL(3,R), showing that certain measure-preserving perturbations are parabolic and mixing, with implications for cocycle rigidity.

Contribution

It introduces a new class of non-time-change perturbations of unipotent flows and characterizes when these perturbations preserve measure and exhibit mixing.

Findings

Perturbations preserving Haar measure are parabolic and mixing.

Characterization of trivial versus non-trivial perturbations.

Connection between perturbations and cocycle rigidity failure.

Abstract

We consider a family of smooth perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in $\text{SL}(3,\mathbb{R})$.