On rigidity properties of time-changes of unipotent flows

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.