Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL (3 , R)

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We consider a family of smooth perturbations of unipotent flows on compact quotients of SL(3 , 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 SL(3 , R).

Original languageEnglish
Pages (from-to)331–351
Number of pages21
JournalCommunications in Mathematical Physics
Issue number1
Publication statusPublished - 2019
Externally publishedYes

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