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Summary: MIXING FOR TIME-CHANGES
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 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-shifts
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 time-changes.
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 non-trivial
time-changes, within a dense subspace of time-changes, are mixing. The set of
trivial time-changes has countable codimension and can be explicitly described in
terms of invariant distributions for the nilflow.
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