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1995 | 115 | 3 | 241-250
Tytuł artykułu

Generic smooth cocycles of degree zero over irrational rotations

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Języki publikacji
EN
Abstrakty
EN
If a rotation α of 𝕋 has unbounded partial quotients then "most" of its skew-product diffeomorphic extensions to the 2-torus 𝕋 × 𝕋 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.
Czasopismo
Rocznik
Tom
115
Numer
3
Strony
241-250
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-07-04
poprawiono
1995-04-11
Twórcy
  • Institute of Mathematics, Technical University of Wrocław, 50-370 Wrocław, Poland, iwanik@im.pwr.wroc.pl
Bibliografia
  • [A] H. Anzai, Ergodic skew product transformations on the torus, Osaka Math. J. 3 (1951), 83-99.
  • [B] L. Baggett, On functions that are trivial cocycles for a set of irrationals, Proc. Amer. Math. Soc. 104 (1988), 1212-1217.
  • [BM1] L. Baggett and K. Merill, Equivalence of cocycles under irrational rotation, ibid., 1050-1053.
  • [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.
  • [I1] A. Iwanik, Cyclic approximation of irrational rotations, Proc. Amer. Math. Soc. 121 (1994), 691-695.
  • [I2] A. Iwanik, Cyclic approximation of ergodic step cocycles over irrational rotations, Acta Univ. Carolin. Math. Phys. 34 (2) (1993), 59-65.
  • [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.
  • [IS] A. Iwanik and J. Serafin, Most monothetic extensions are rank-1, Colloq. Math. 66 (1993), 63-76.
  • [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).
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv115i3p241bwm
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