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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On extension of cocycles
to normalizer elements, outer conjugacy,
and related problems


Authors: Alexandre I. Danilenko and Valentin Ya. Golodets
Journal: Trans. Amer. Math. Soc. 348 (1996), 4857-4882
MSC (1991): Primary 46L55; Secondary 28D15, 28D99
MathSciNet review: 1376544
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Abstract: Let $T$ be an ergodic automorphism of a Lebesgue space and $\alpha $ a cocycle of $T$ with values in an Abelian locally compact group $G$. An automorphism $\theta $ from the normalizer $N[T]$ of the full group $[T]$ is said to be compatible with $\alpha $ if there is a measurable function $\varphi : X \to G$ such that $\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)$ at a.e. $x$. The topology on the set $D(T, \alpha )$ of all automorphisms compatible with $\alpha $ is introduced in such a way that $D(T , \alpha )$ becomes a Polish group. A complete system of invariants for the $\alpha $-outer conjugacy (i.e. the conjugacy in the quotient group $D(T, \alpha )/[T])$ is found. Structure of the cocycles compatible with every element of $N[T]$ is described.


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Additional Information

Alexandre I. Danilenko
Affiliation: Department of Mechanics and Mathematics, Kharkov State University, Freedom Square 4, Kharkov, 310077, Ukraine
Email: danilenko@ilt.kharkov.ua

Valentin Ya. Golodets
Affiliation: Mathematics Department, Institute for Low Temperature Physics, Lenin Avenue 47, Kharkov, 310164, Ukraine
Email: golodets@ilt.kharkov.ua

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01753-9
PII: S 0002-9947(96)01753-9
Keywords: Ergodic dynamical system, cocycle, outer conjugacy
Received by editor(s): January 4, 1995
Additional Notes: The work was supported in part by the International Science Foundation Grant No U2B000.
Article copyright: © Copyright 1996 American Mathematical Society