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arXiv · 1003.4636

Mixing for Time-Changes of Heisenberg Nilflows

Artur Avila·Giovanni Forni·Corinna Ulcigrai
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Abstract

We consider reparametrizations of Heisenberg nilflows. We show that if a Heisenberg nilflow is uniquely ergodic, all non-trivial time-changes 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. Mixing is produced by a mechanism known as stretching of Birkhoff sums. The complement of the set of mixing time-changes (or, equivalently, of mixing roof functions) has countable codimension and can be explicitely described in terms of the invariant distributions for the nilflow (or, equivalently, for the skew-shift), allowing to produce concrete examples of mixing time-changes.

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Artur Avila, Giovanni Forni, Corinna Ulcigrai. 2010-04-28. Mixing for Time-Changes of Heisenberg Nilflows. https://arxiv.org/abs/1003.4636

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