Generic smooth cocycles of degree zero over irrational rotations

A. Iwanik

Studia Mathematica (1995)

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

Abstract

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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.

How to cite

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Iwanik, 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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  1. [A] H. Anzai, Ergodic skew product transformations on the torus, Osaka Math. J. 3 (1951), 83-99. Zbl0043.11203
  2. [B] L. Baggett, On functions that are trivial cocycles for a set of irrationals, Proc. Amer. Math. Soc. 104 (1988), 1212-1217. Zbl0692.28008
  3. [BM1] L. Baggett and K. Merill, Equivalence of cocycles under irrational rotation, ibid., 1050-1053. Zbl0695.10046
  4. [BM2] L. Baggett and K. Merill, Smooth cocycles for an irrational rotation, preprint. 
  5. [CFS] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, 1982. 
  6. [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
  7. [I1] A. Iwanik, Cyclic approximation of irrational rotations, Proc. Amer. Math. Soc. 121 (1994), 691-695. Zbl0807.28008
  8. [I2] A. Iwanik, Cyclic approximation of ergodic step cocycles over irrational rotations, Acta Univ. Carolin. Math. Phys. 34 (2) (1993), 59-65. Zbl0817.28008
  9. [I3] A. Iwanik, Approximation by periodic transformations and diophantine approximation of the spectrum, in: Proc. Warwick Sympos. 1994, to appear. 
  10. [ILR] A. Iwanik, M. Lemańczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel J. Math. 83 (1993), 73-95. Zbl0786.28011
  11. [IS] A. Iwanik and J. Serafin, Most monothetic extensions are rank-1, Colloq. Math. 66 (1993), 63-76. Zbl0833.28009
  12. [K] A. Katok, Constructions in Ergodic Theory, unpublished lecture notes. 
  13. [KS] A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspekhi Mat. Nauk 22 (5) (1967), 81-106 (in Russian). Zbl0172.07202
  14. [Kh] A. Ya. Khintchin, Continued Fractions, Univ. of Chicago Press, 1964. 
  15. [R] A. Robinson, Non-abelian extensions have nonsimple spectrum, Compositio Math. 65 (1988), 155-170. Zbl0641.28011

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