Weak mixing for reparameterized linear flows on the torus

Abstract

In this paper, we study the display of weak mixing by reparameterized linear flows on the torus \mathbb{T}^d, d\geq 2. We show that if the vector of the translation flow is Liouvillian (i.e. well approximated by rationals), then for a residual set of time change functions in the C^\infty topology, the reparameterized flow is weak mixing. If this is not the case, i.e. if the vector of the linear flow is Diophantine, it follows from a result of Kolmogorov on the two torus, and its generalization to any dimension by Herman, that any C^\infty reparameterization of the flow is C^\infty conjugate to a linear flow. More generally, in any given class of differentiability C^r for the time change function \phi, we give the optimal arithmetical condition on the vector of the translation flow that guarantees the existence of a residual set in the C^r topology of weak mixing reparameterizations. In the real analytic case, the optimal arithmetical condition for the generic display of weak mixing under time change is also given.

As a consequence of our results on reparameterizations of Liouvillian linear flows, we obtain that an aperiodic smooth flow on the two-dimensional torus is in general weak mixing. We also deduce the existence on the torus of analytic diffeomorphisms that are rank one and weak mixing.