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On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

100%

Kułaga-Przymus J.

Fundamenta Mathematicae

2015

tom 230

nr 1

15-76

We compare self-joining and embeddability properties. In particular, we prove that a measure preserving flow $(T_t)_{t∈ℝ}$ with T₁ ergodic is 2-fold quasi-simple (resp. 2-fold distally simple) if and only if T₁ is 2-fold quasi-simple (resp. 2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow $(T_t)_{t∈ℝ}$ with T₁ ergodic with respect to any flow factor is the same for $(T_t)_{t∈ℝ}$ and for T₁. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is simple. We describe two classes of flows (flows with minimal self-joining property and flows with the so-called Ratner property) whose time-one maps have unique embeddings into measurable flows. We also give an example of a 2-fold simple flow whose time-one map has more than one embedding.

Disjointness properties for Cartesian products of weakly mixing systems

63%

Kułaga-Przymus J., Parreau F.

Colloquium Mathematicum

2012

tom 128

nr 2

153-177

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Fundamenta Mathematicae

2 Kułaga-Przymus J.

1 Parreau F.

1 2015

1 2012

Wyniki wyszukiwania

On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

Disjointness properties for Cartesian products of weakly mixing systems