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Isometric extensions, 2-cocycles and ergodicity of skew products

Alexandre Danilenko; Mariusz Lemańczyk

Studia Mathematica (1999)

  • Volume: 137, Issue: 2, page 123-142
  • ISSN: 0039-3223

Abstract

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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 .

How to cite

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Danilenko, Alexandre, and Lemańczyk, Mariusz. "Isometric extensions, 2-cocycles and ergodicity of skew products." Studia Mathematica 137.2 (1999): 123-142. <http://eudml.org/doc/216679>.

@article{Danilenko1999,
abstract = {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 $T_α$.},
author = {Danilenko, Alexandre, Lemańczyk, Mariusz},
journal = {Studia Mathematica},
keywords = {ergodic transformation; cocycle; isometric extension; nonsingular transformation; centralizer; ergodicity},
language = {eng},
number = {2},
pages = {123-142},
title = {Isometric extensions, 2-cocycles and ergodicity of skew products},
url = {http://eudml.org/doc/216679},
volume = {137},
year = {1999},
}

TY - JOUR
AU - Danilenko, Alexandre
AU - Lemańczyk, Mariusz
TI - Isometric extensions, 2-cocycles and ergodicity of skew products
JO - Studia Mathematica
PY - 1999
VL - 137
IS - 2
SP - 123
EP - 142
AB - 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 $T_α$.
LA - eng
KW - ergodic transformation; cocycle; isometric extension; nonsingular transformation; centralizer; ergodicity
UR - http://eudml.org/doc/216679
ER -

References

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  3. [D2] A. I. Danilenko, On cocycles with values in group extensions. Generic results, Mat. Analiz Geom., to appear. Zbl0963.22004
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  10. [Kw] J. Kwiatkowski, Factors of ergodic group extensions of rotations, Studia Math. 103 (1992), 123-131. Zbl0809.28014
  11. [Le] M. Lemańczyk, Cohomology groups, multipliers and factors in ergodic theory, ibid. 122 (1997), 275-288. Zbl0884.28012
  12. [Me] M. K. Mentzen, Ergodic properties of group extensions of dynamical systems with discrete spectra, ibid. 101 (1991), 20-31. 
  13. [Ne] D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. 19 (1979), 129-136. Zbl0425.28012
  14. [Pa] K. R. Parthasarathy, Multipliers on Locally Compact Groups, Lecture Notes in Math. 93, Springer, 1969. 
  15. [Sc] K. Schmidt, Lectures on Cocycles of Ergodic Transformation Groups, Lecture Notes in Math. 1, Macmillan, 1977. 
  16. [Z1] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. Zbl0334.28015
  17. [Z2] R. Zimmer, Ergodic Theory and Semisimple Lie Groups, Birkhäuser, Boston, 1984. 

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