# Factors of ergodic group extensions of rotations

Studia Mathematica (1992)

• Volume: 103, Issue: 2, page 123-131
• ISSN: 0039-3223

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## Abstract

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Diagonal metric subgroups of the metric centralizer $C\left({T}_{\phi }\right)$ of group extensions are investigated. Any diagonal compact subgroup Z of $C\left({T}_{\phi }\right)$ is determined by a compact subgroup Y of a given metric compact abelian group X, by a family ${v}_{y}:y\in Y$, of group automorphisms and by a measurable function f:X → G (G a metric compact abelian group). The group Z consists of the triples $\left(y,{F}_{y},{v}_{y}\right)$, y ∈ Y, where ${F}_{y}\left(x\right)={v}_{y}\left(f\left(x\right)\right)-f\left(x+y\right)$, x ∈ X.

## How to cite

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Kwiatkowski, Jan. "Factors of ergodic group extensions of rotations." Studia Mathematica 103.2 (1992): 123-131. <http://eudml.org/doc/215940>.

@article{Kwiatkowski1992,
abstract = {Diagonal metric subgroups of the metric centralizer $C(T_φ)$ of group extensions are investigated. Any diagonal compact subgroup Z of $C(T_φ)$ is determined by a compact subgroup Y of a given metric compact abelian group X, by a family $\{v_y : y ∈ Y\}$, of group automorphisms and by a measurable function f:X → G (G a metric compact abelian group). The group Z consists of the triples $(y,F_y,v_y)$, y ∈ Y, where $F_y(x) = v_y(f(x)) - f(x+y)$, x ∈ X.},
author = {Kwiatkowski, Jan},
journal = {Studia Mathematica},
keywords = {group extension; cocycle; ergodic group extensions; rotations; metric subgroups; metric centralizer},
language = {eng},
number = {2},
pages = {123-131},
title = {Factors of ergodic group extensions of rotations},
url = {http://eudml.org/doc/215940},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Kwiatkowski, Jan
TI - Factors of ergodic group extensions of rotations
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 2
SP - 123
EP - 131
AB - Diagonal metric subgroups of the metric centralizer $C(T_φ)$ of group extensions are investigated. Any diagonal compact subgroup Z of $C(T_φ)$ is determined by a compact subgroup Y of a given metric compact abelian group X, by a family ${v_y : y ∈ Y}$, of group automorphisms and by a measurable function f:X → G (G a metric compact abelian group). The group Z consists of the triples $(y,F_y,v_y)$, y ∈ Y, where $F_y(x) = v_y(f(x)) - f(x+y)$, x ∈ X.
LA - eng
KW - group extension; cocycle; ergodic group extensions; rotations; metric subgroups; metric centralizer
UR - http://eudml.org/doc/215940
ER -

## References

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4. [4] M. Lemańczyk and M. K. Mentzen, Compact subgroups in the centralizer of natural factors of an ergodic group extension of a rotation determine all factors, Ergodic Theory Dynamical Systems 10 (1990), 763-776. Zbl0725.54030
5. [5] D. Newton, On canonical factors of ergodic dynamical systems, J. London Math. Soc. (2) 19 (1979), 129-136. Zbl0425.28012
6. [6] W. Parry, Compact abelian group extensions of discrete dynamical systems, Z. Wahrsch. Verw. Gebiete 13 (1969), 95-113. Zbl0184.26901
7. [7] M. Rychlik, The Wiener lemma and cocycles, Proc. Amer. Math. Soc. 104 (1988), 932-933. Zbl0687.58018
8. [8] W. A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Trans. Amer. Math. Soc. 140 (1969), 1-33. Zbl0201.05601

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