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Commutator criteria for strong mixing

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.