Construction of non-constant and ergodic cocycles
Nerurkar, Mahesh
Colloquium Mathematicae, Tome 84/85 (2000), p. 395-419 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:210822 
@article{bwmeta1.element.bwnjournal-article-cmv84i2p395bwm,
     author = {Mahesh Nerurkar},
     title = {Construction of non-constant and ergodic cocycles},
     journal = {Colloquium Mathematicae},
     volume = {84/85},
     year = {2000},
     pages = {395-419},
     zbl = {0979.37003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p395bwm}
}

Nerurkar, Mahesh. Construction of non-constant and ergodic cocycles. Colloquium Mathematicae, Tome 84/85 (2000) pp. 395-419. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i2p395bwm/

[000] [AK] D. Anosov and A. Katok, New examples in smooth ergodic theory, Trans. Moscow Math. Soc. 23 (1970), 1-35. | Zbl 0255.58007

[001] [GM] S. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems 9 (1989), 309-320. | Zbl 0661.58027

[002] [GW] S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), 321-336. | Zbl 0434.54032

[003] [H] P. Halmos, Lectures on Ergodic Theory, Math. Soc. Japan, Tokyo, 1956.

[004] [HSY] B. Hunt, T. Sauer and J. Yorke, Prevalence: A translation-invariant 'almost every' on infinite-dimensional spaces, Bull. Amer. Math. Soc. 27 (1992), 217-238. | Zbl 0763.28009

[005] [IS] A. Iwanik and J. Serafin, Most monothetic extensions are rank-1, Colloq. Math. 66 (1993), 63-76. | Zbl 0833.28009

[006] [JP] R. Jones and W. Parry, Compact abelian group extensions of dynamical systems II, Compositio Math. 25 (1972), 135-147. | Zbl 0243.54039

[007] [K] A. Katok, Constructions in Ergodic Theory, Part II, unpublished notes. | Zbl 1030.37001

[008] [M] I. Melbourne, Symmetric ω-limit sets for smooth Γ-equivariant dynamical systems with Γ0 abelian, Nonlinearity 10 (1997), 1551-1567. | Zbl 0908.58044

[009] [N1] M. Nerurkar, Ergodic continuous skew product actions of amenable groups, Pacific J. Math. 119 (1985), 343-363. | Zbl 0563.28013

[010] [N2] M. Nerurkar, On the construction of smooth ergodic skew-products, Ergodic Theory Dynam. Systems 8 (1988), 311-326. | Zbl 0662.58028

[011] [Sch] K. Schmidt, Cocycles and Ergodic Transformation Groups, MacMillan of India, 1977. | Zbl 0421.28017

[012] [Z] R. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. | Zbl 0334.28015