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Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL\({(3,\mathbb{R})}\)

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Abstract

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

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Acknowledgments

I am grateful to Giovanni Forni for many enlightening suggestions. I would also like to thank Danijela Damjanovic, Livio Flaminio, Vinay Kumaraswamy, and my supervisor Corinna Ulcigrai for several helpful conversations. I thank the referees for their valuable comments and suggestions on a previous version of the paper. The research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 335989.

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  1. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Davide Ravotti

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  1. Davide Ravotti
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Correspondence to Davide Ravotti.

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Communicated by C. Liverani

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Ravotti, D. Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL\({(3,\mathbb{R})}\). Commun. Math. Phys. 371, 331–351 (2019). https://doi.org/10.1007/s00220-019-03348-0

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