Commutator criteria for strong mixing

Abstract

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^{N}\}_{N\in \mathbb{Z}}$ and continuous flows $\{e^{-itH}\}_{t\in \mathbb{R}}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space ${\mathcal{H}}$. Our approach put into light a general definition for the topological degree of the maps $N\mapsto U^{N}$ and $t\mapsto e^{-itH}$ with values in the unitary group of ${\mathcal{H}}$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on graphs.