abstract: Nilflows on nilmanifolds are classical examples of parabolic homogeneous flows and have been intensively studied also because of their connections with number theory. Their ergodic properties are well-understood: although almost every nilflow is uniquely ergodic, they are never weak mixing.
The absence of mixing is the result of an algebraic obstruction, namely a factor isomorphic to a linear flow on a torus. When the nilflows is perturbed by performing a smooth time-change, we should expect this toral factor to disappear. Indeed, we show that for generic time-changes of any uniquely ergodic nilflow, mixing holds unless the time-change function is measurably trivial. This result confirms some heuristic principles on the ergodic properties of smooth parabolic flows.
This is a joint work with Artur Avila, Giovanni Forni and Corinna Ulcigrai.