Mixing for the Time-Changes of Heisenberg Nilflows

Artur Avila, Giovanni Forni, C Ulcigrai

Research output: Contribution to journalArticle (Academic Journal)peer-review

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.
Translated title of the contributionMixing for Time-Changes of Heisenberg Nilflows
Original languageEnglish
Pages (from-to)369-410
Number of pages42
JournalJournal of Differential Geometry
Volume89
Issue number3
Publication statusPublished - 2011

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