Multiple mixing and local rank of dynamical systems (Q1910995)

An automorphism \(T\) of a Lebesgue space \((X, {\mathcal B}, \mu)\) defines a measure-preserving \(\mathbb{Z}^n\)-action on \(X\). The action has the \(k\)-fold mixing property if for any \(A_0,\dots, A_k\in {\mathcal B}\) we have: \[ \mu(T^{n_0} A_0\cap T^{n_1} A_1\cap\cdots \cap T^{n_k} A_k)\to \mu(A_0) \mu(A_1)\cdots \mu(A_k) \] as \(|n_p- n_q|\to \infty\), \(0\leq p< q\leq k\). The paper under review gives new answers to the question: Which invariants of mixing dynamical systems imply the multiple mixing property? Namely, it is proved that each mixing \(\mathbb{Z}^n\)-action with \(D\)-approximation (a notion defined in the paper) of positive local rank has the mixing property of all orders. The author also defines the \(1+ \varepsilon\)-mixing property and proves that this one implies the multiple mixing property for systems of positive local rank.