Mixing for time-changes of heisenberg nilflows

Artur Avila Institut de Math´ematiques de Jussieu Giovanni Forni University of Maryland Corinna Ulcigrai University Walk

Differential Geometry mathscidoc:1609.10242

Journal of Differential Geometry, 89, (3), 369-410, 2011
We consider reparametrizations of Heisenberg nilflows.We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial timechanges within a dense subspace of smooth time-changes are mixing. Equivalently, in the language of special flows, we consider special flows over linear skew-shift extensions of irrational rotations of the circle. Without assuming any Diophantine condition on the frequency, we define a dense class of smooth roof functions for which the corresponding special flows are mixing whenever the roof function is not a coboundary. Mixing is produced by a mechanism known as stretching of ergodic sums. The complement of the set of mixing time-changes (or, equivalently, of mixing roof functions) has countable codimension and can be explicitly described in terms of the invariant distributions for the nilflow (or, equivalently,for the skew-shift), producing concrete examples of mixing time-changes.
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@inproceedings{artur2011mixing,
  title={MIXING FOR TIME-CHANGES OF HEISENBERG NILFLOWS},
  author={Artur Avila, Giovanni Forni, and Corinna Ulcigrai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913210846672009903},
  booktitle={Journal of Differential Geometry},
  volume={89},
  number={3},
  pages={369-410},
  year={2011},
}
Artur Avila, Giovanni Forni, and Corinna Ulcigrai. MIXING FOR TIME-CHANGES OF HEISENBERG NILFLOWS. 2011. Vol. 89. In Journal of Differential Geometry. pp.369-410. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913210846672009903.
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