 
Summary: MIXING FOR TIMECHANGES
OF HEISENBERG NILFLOWS
ARTUR AVILA, GIOVANNI FORNI, AND CORINNA ULCIGRAI
Abstract. We consider reparametrizations of Heisenberg nilflows. We show
that if a Heisenberg nilflow is uniquely ergodic, all nontrivial timechanges
within a dense subspace of smooth timechanges are mixing. Equivalently, in
the language of special flows, we consider special flows over linear skewshifts
over an irrational rotation 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 set of mixing time
changes (or equivalently roof functions) has countable codimension and can
be explicitely described (in terms of invariant distributions for the nilflow),
allowing to produce concrete examples of mixing timechanges.
1. Introduction
In this paper we give a contribution to the smooth ergodic theory of parabolic
flows. We prove that for any uniquely ergodic Heisenberg nilflow all nontrivial
timechanges, within a dense subspace of timechanges, are mixing. The set of
trivial timechanges has countable codimension and can be explicitly described in
terms of invariant distributions for the nilflow.
