Polynomial mixing for time-changes of unipotent flows

TL;DR

This paper proves that smooth time-changes of unipotent flows on certain homogeneous manifolds exhibit polynomial decay of correlations, extending previous results to more general settings with spectral gap assumptions.

Contribution

It establishes polynomial decay of correlations for smooth time-changes of unipotent flows under spectral gap conditions, generalizing prior work on horocycle flows.

Findings

Polynomial decay of correlations for smooth time-changes.

Applicability to compact and certain non-compact quotients.

Extension of previous results to broader classes of flows.

Abstract

Let $G$ be a connected semisimple Lie group with finite centre, and let $M= \Gamma \backslash G$ be a compact homogeneous manifold. Under a spectral gap assumption, we show that smooth time-changes of any unipotent flow on $M$ have polynomial decay of correlations. Our result applies also in the case where $M$ is a finite volume, non-compact quotient under some additional assumptions on the generator of the time-change. This generalizes a result by Forni and Ulcigrai (JMD, 2012) for smooth time-changes of horocycle flows on compact surfaces.