Commutator criteria for strong mixing

TL;DR

This paper introduces new commutator-based criteria to determine strong mixing properties of discrete and continuous flows generated by unitary and self-adjoint operators, with applications to various dynamical systems.

Contribution

It develops a general framework for assessing strong mixing using commutator methods and defines a topological degree for operator curves, extending analysis to diverse systems.

Findings

Criteria applicable to skew products of compact Lie groups

Applicable to time-changed horocycle flows

Useful for adjacency operators on graphs

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 $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ 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.