Ergodic measures for non-abelian compact group extensions

H. B. Keynes; D. Newton

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We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general ergodic aperiodic systems, we also show that the set of continuous ergodic cocycles is residual in the class of all continuous cocycles provided the range group G is a compact connected Lie group. The first construction is based on...

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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 $T_{α}$ .

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On measure theoretical analogues of the Takesaki structure theorem for type III factors

Alexandre Danilenko, Toshihiro Hamachi (2000)

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The orbit equivalence of type $I I I_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $I I I_{0}$ cocycles with values in Abelian groups.

Skew products in the centralizer of compact abelian group extensions.

Goodson, G.R. (1999)

Acta Mathematica Universitatis Comenianae. New Series

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On subrelations of ergodic measured type III equivalence relations

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We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.