## Disjointness properties for Cartesian products of weakly mixing systems

### Volume 128 / 2012

#### Abstract

For $n\geq 1$ we consider the class ${\rm JP}(n)$ of dynamical systems each of whose ergodic joinings with a Cartesian product of $k$ weakly mixing automorphisms ($k\geq n$) can be represented as the independent extension of a joining of the system with only $n$ coordinate factors. For $n\geq 2$ 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 ${\rm JP} (n-1)$. To provide examples of such automorphisms, we exploit spectral simplicity on symmetric Fock spaces. This also allows us to show that for any $n\geq 2$ the class ${\rm JP}(n)$ is essentially larger than ${\rm JP}(n-1)$. Moreover, we show that all members of ${\rm JP}(n)$ are disjoint from ergodic automorphisms generated by infinitely divisible stationary processes.