Disjointness properties for Cartesian products of weakly mixing systems

Abstract

For n 1 we consider the class JP(n) of dynamical systems each of whose ergodic joinings with a Cartesian product of k weakly mixing automorphisms (k n) can be represented as the independent extension of a joining of the system with only n coordinate factors. For n 2 we show that, whenever the maximal spectral type of a weakly mixing automorphism T is singular with respect to the convolution of any n continuous measures, i.e. T has the so-called convolution singularity property of order n, then T belongs to JP(n 1). To provide examples of such automorphisms, we exploit spectral simplicity on symmetric Fock spaces. This also allows us to show that for any n 2 the class JP(n) is essentially larger than JP(n 1). Moreover, we show that all members of JP(n) are disjoint from ergodic automorphisms generated by infinitely divisible stationary processes. 1. Introduction. In this paper we deal with several properties of dy- namical systems which are related to the notion of disjointness. This notion was introduced by Furstenberg (8) and, among other motivations, bore fruit in the development of tools to classify dynamical systems and construct ex- amples of dierent behaviour. We will devote our attention to classes of