  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
$\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and
self-adjoint operators $H$ in a Hilbert space $\mathcal H$. Our approach put
into evidence a general definition for the topological degree of the curves
$N\mapsto U^N$ and $t\mapsto{\rm e}^{-itH}$ 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.
