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Disjointness properties for Cartesian products of weakly mixing systems

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