Mixing for Time-Changes of Heisenberg Nilflows

TL;DR

This paper demonstrates that certain smooth reparametrizations of Heisenberg nilflows are mixing, expanding understanding of ergodic properties without Diophantine restrictions, through the stretching of Birkhoff sums.

Contribution

It introduces a dense class of smooth time-changes for Heisenberg nilflows that are mixing, using a new mechanism applicable without Diophantine conditions.

Findings

All non-trivial smooth time-changes in a dense subspace are mixing.

Mixing is achieved via stretching of Birkhoff sums.

The set of non-mixing time-changes has countable codimension and is explicitly characterized.

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.