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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.