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arXiv · 1711.04821

Parabolic perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$

Davide Ravotti
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Abstract

We consider a family of smooth perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$ 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 $\text{SL}(3,\mathbb{R})$.

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Davide Ravotti. 2018-12-02. Parabolic perturbations of unipotent flows on compact quotients of $\text{SL}(3,\mathbb{R})$. https://arxiv.org/abs/1711.04821

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