# Generic smooth cocycles of degree zero over irrational rotations

Studia Mathematica (1995)

- Volume: 115, Issue: 3, page 241-250
- ISSN: 0039-3223

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topIwanik, A.. "Generic smooth cocycles of degree zero over irrational rotations." Studia Mathematica 115.3 (1995): 241-250. <http://eudml.org/doc/216210>.

@article{Iwanik1995,

abstract = {If a rotation α of has unbounded partial quotients then “most” of its skew-product diffeomorphic extensions to the 2-torus × defined by $C^1$ cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for $C^r$ cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic $C^1$ cocycles the speed can be improved to o(1/(nlogn)). For generic α and generic $C^r$ cocycles (r = 1,...,∞) the spectral measure of the skew product has a continuous component and Hausdorff dimension zero.},

author = {Iwanik, A.},

journal = {Studia Mathematica},

keywords = {Anzai skew product; weakly mixing cocycle; cyclic approximation; simple spectrum; cocycles; skew product diffeomorphisms},

language = {eng},

number = {3},

pages = {241-250},

title = {Generic smooth cocycles of degree zero over irrational rotations},

url = {http://eudml.org/doc/216210},

volume = {115},

year = {1995},

}

TY - JOUR

AU - Iwanik, A.

TI - Generic smooth cocycles of degree zero over irrational rotations

JO - Studia Mathematica

PY - 1995

VL - 115

IS - 3

SP - 241

EP - 250

AB - If a rotation α of has unbounded partial quotients then “most” of its skew-product diffeomorphic extensions to the 2-torus × defined by $C^1$ cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for $C^r$ cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic $C^1$ cocycles the speed can be improved to o(1/(nlogn)). For generic α and generic $C^r$ cocycles (r = 1,...,∞) the spectral measure of the skew product has a continuous component and Hausdorff dimension zero.

LA - eng

KW - Anzai skew product; weakly mixing cocycle; cyclic approximation; simple spectrum; cocycles; skew product diffeomorphisms

UR - http://eudml.org/doc/216210

ER -

## References

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- [B] L. Baggett, On functions that are trivial cocycles for a set of irrationals, Proc. Amer. Math. Soc. 104 (1988), 1212-1217. Zbl0692.28008
- [BM1] L. Baggett and K. Merill, Equivalence of cocycles under irrational rotation, ibid., 1050-1053. Zbl0695.10046
- [BM2] L. Baggett and K. Merill, Smooth cocycles for an irrational rotation, preprint.
- [CFS] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1982.
- [GLL] P. Gabriel, M. Lemańczyk et P. Liardet, Ensemble d'invariants pour les produits croisés de Anzai, Mémoire no. 47, Suppl. Bull. Soc. Math. France 119 (3) (1991), 1-102. Zbl0754.28011
- [I1] A. Iwanik, Cyclic approximation of irrational rotations, Proc. Amer. Math. Soc. 121 (1994), 691-695. Zbl0807.28008
- [I2] A. Iwanik, Cyclic approximation of ergodic step cocycles over irrational rotations, Acta Univ. Carolin. Math. Phys. 34 (2) (1993), 59-65. Zbl0817.28008
- [I3] A. Iwanik, Approximation by periodic transformations and diophantine approximation of the spectrum, in: Proc. Warwick Sympos. 1994, to appear.
- [ILR] A. Iwanik, M. Lemańczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel J. Math. 83 (1993), 73-95. Zbl0786.28011
- [IS] A. Iwanik and J. Serafin, Most monothetic extensions are rank-1, Colloq. Math. 66 (1993), 63-76. Zbl0833.28009
- [K] A. Katok, Constructions in Ergodic Theory, unpublished lecture notes.
- [KS] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (5) (1967), 81-106 (in Russian). Zbl0172.07202
- [Kh] A. Ya. Khintchin, Continued Fractions, Univ. of Chicago Press, 1964.
- [R] A. Robinson, Non-abelian extensions have nonsimple spectrum, Compositio Math. 65 (1988), 155-170. Zbl0641.28011

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