Construction of non-constant and ergodic cocycles

Mahesh Nerurkar

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 395-419
  • ISSN: 0010-1354

Abstract

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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 the "closure of coboundaries technique", whereas the second result is proved by developing in addition a new approximation technique.

How to cite

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Nerurkar, Mahesh. "Construction of non-constant and ergodic cocycles." Colloquium Mathematicae 84/85.2 (2000): 395-419. <http://eudml.org/doc/210822>.

@article{Nerurkar2000,
abstract = {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 the "closure of coboundaries technique", whereas the second result is proved by developing in addition a new approximation technique.},
author = {Nerurkar, Mahesh},
journal = {Colloquium Mathematicae},
keywords = {cocycles; rigid dynamical systems; ergodicity},
language = {eng},
number = {2},
pages = {395-419},
title = {Construction of non-constant and ergodic cocycles},
url = {http://eudml.org/doc/210822},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Nerurkar, Mahesh
TI - Construction of non-constant and ergodic cocycles
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 395
EP - 419
AB - 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 the "closure of coboundaries technique", whereas the second result is proved by developing in addition a new approximation technique.
LA - eng
KW - cocycles; rigid dynamical systems; ergodicity
UR - http://eudml.org/doc/210822
ER -

References

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  9. [M] I. Melbourne, Symmetric ω-limit sets for smooth Γ-equivariant dynamical systems with Γ 0 abelian, Nonlinearity 10 (1997), 1551-1567. Zbl0908.58044
  10. [N1] M. Nerurkar, Ergodic continuous skew product actions of amenable groups, Pacific J. Math. 119 (1985), 343-363. Zbl0563.28013
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  12. [Sch] K. Schmidt, Cocycles and Ergodic Transformation Groups, MacMillan of India, 1977. Zbl0421.28017
  13. [Z] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. Zbl0334.28015

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