Joinings, intertwining operators, factors, and mixing properties of dynamical systems

V. V. Ryzhikov

Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 91-114

Résumé

This paper is mostly devoted to the following problem. If the Markov (stochastic) centralizer of a measure-preserving action $\Psi$ is known, what can be said about the Markov centralizer of the action $\Psi\otimes\Psi$? For a mixing flow with minimal Markov centralizer the author proves the triviality of the Markov centralizer of a Cartesian power of it, from which it follows that this flow possesses mixing of arbitrary multiplicity. For actions of the groups $\mathbf Z^n$ the analogous assertion holds if their tensor product with themselves does not possess three pairwise independent factors. In particular, this is true for actions of $\mathbf Z^n$ admitting a partial approximation and possessing mixing of multiplicity 2.