Disjointness properties for Cartesian products of weakly mixing systems

TL;DR

This paper introduces a hierarchy of dynamical systems based on their disjointness properties with Cartesian products of weakly mixing systems, linking spectral singularity conditions to membership in these classes.

Contribution

It establishes a new classification of systems (JP(n)) based on disjointness properties and spectral singularity, providing examples and demonstrating the hierarchy's strictness.

Findings

JP(n) classes are strictly larger than JP(n-1) for n≥2

Systems with convolution singularity property of order n belong to JP(n-1)

All JP(n) systems are disjoint from automorphisms from infinitely divisible processes

Abstract

For $n\geq 1$ we consider the class JP($n$) of dynamical systems whose every ergodic joining 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 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 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.