On rigidity properties of time-changes of unipotent flows

TL;DR

This paper investigates the rigidity of time-changes of unipotent flows on certain homogeneous spaces, showing that under spectral gap conditions, such flows exhibit strong structural constraints and limited possible isomorphisms.

Contribution

It extends Ratner's work by establishing rigidity results for time-changed unipotent flows with spectral gap assumptions, characterizing their isomorphisms as affine transformations.

Findings

Any measurable isomorphism induces a non-trivial joining on the graph.

Under spectral gap, limit points of joinings are trivial or affine.

Isomorphisms are either trivial or algebraically affine.

Abstract

We study time-changes of unipotent flows on finite volume quotients of semisimple linear groups, generalising previous work by Ratner on time-changes of horocycle flows. Any measurable isomorphism between time-changes of unipotent flows gives rise to a non-trivial joining supported on its graph. Under a spectral gap assumption on the groups, we show the following rigidity result: either the only limit point of this graph joining under the action of a one-parameter renormalising subgroup is the trivial joining, or the isomorphism is "affine", namely it is obtained composing an algebraic isomorphism with a (non-constant) translation along the centraliser.