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Isometric extensions, 2-cocycles and ergodicity of skew products

Alexandre DanilenkoMariusz Lemańczyk — 1999

Studia Mathematica

We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension T α and admits a prescribed subgroup in the centralizer of T α .

On disjointness properties of some smooth flows

Krzysztof FrączekMariusz Lemańczyk — 2005

Fundamenta Mathematicae

Special flows over some locally rigid automorphisms and under L² ceiling functions satisfying a local L² Denjoy-Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that ∙ special flows built over ergodic interval exchange...

A cut salad of cocycles

Jon AaronsonMariusz LemańczykDalibor Volný — 1998

Fundamenta Mathematicae

We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.

Gaussian automorphisms whose ergodic self-joinings are Gaussian

Mariusz LemańczykF. ParreauJ. Thouvenot — 2000

Fundamenta Mathematicae

 We study ergodic properties of the class of Gaussian automorphisms whose ergodic self-joinings remain Gaussian. For such automorphisms we describe the structure of their factors and of their centralizer. We show that Gaussian automorphisms with simple spectrum belong to this class.  We prove a new sufficient condition for non-disjointness of automorphisms giving rise to a better understanding of Furstenberg's problem relating disjointness to the lack of common factors. This...

Simple systems are disjoint from Gaussian systems

Andrés del JuncoMariusz Lemańczyk — 1999

Studia Mathematica

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

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