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

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).