Disjointness properties for Cartesian products of weakly mixing systems

Joanna Kułaga-Przymus¹ ; François Parreau²

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland
² Laboratoire d'Analyse, Géométrie et Applications UMR 7539 Université Paris 13 Sorbonne Paris Cité et CNRS 99 av. J.-B. Clément 94430 Villetaneuse, France

Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 153-177.

Résumé

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

Détail

DOI : 10.4064/cm128-2-2

Mots-clés : geq consider class dynamical systems each whose ergodic joinings cartesian product weakly mixing automorphisms geq represented independent extension joining system only coordinate factors geq whenever maximal spectral type weakly mixing automorphism singular respect convolution continuous measures has so called convolution singularity property order belongs n provide examples automorphisms exploit spectral simplicity symmetric fock spaces allows geq class essentially larger n moreover members disjoint ergodic automorphisms generated infinitely divisible stationary processes