Cyclic approximation of irrational rotations
HTML articles powered by AMS MathViewer
- by A. Iwanik PDF
- Proc. Amer. Math. Soc. 121 (1994), 691-695 Request permission
Abstract:
We prove that an irrational number $\alpha$ admits a rational approximation $|\alpha - p/q| = o(f(q))$ iff the irrational rotation $Tx = \{ x + \alpha \}$ admits cyclic approximation with speed $o(f(n))$. As an application to earlier results we obtain that a generic Anzai skew product over every irrational rotation is rank-1 and for a.e. $\alpha$ most skew products admit cyclic approximation with speed $o(1/{n^2}\log n)$.References
- Andrés del Junco, Transformations with discrete spectrum are stacking transformations, Canadian J. Math. 28 (1976), no. 4, 836–839. MR 414822, DOI 10.4153/CJM-1976-080-3 A. del Junco, A. Fieldsteel, and K. Park, Residual properties of $\alpha$-flows, preprint.
- A. Iwanik and J. Serafin, Most monothetic extensions are rank-$1$, Colloq. Math. 66 (1993), no. 1, 63–76. MR 1242646, DOI 10.4064/cm-66-1-63-76 A. Katok, Constructions in ergodic theory, preprint.
- A. B. Katok and A. M. Stepin, Approximations in ergodic theory, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 81–106 (Russian). MR 0219697 A. Ya. Khintchin, Continued fractions, Univ. of Chicago Press, Chicago, 1964.
- E. A. Robinson Jr., Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), no. 2, 299–314. MR 700773, DOI 10.1007/BF01389325
- E. Arthur Robinson Jr., Nonabelian extensions have nonsimple spectrum, Compositio Math. 65 (1988), no. 2, 155–170. MR 932641
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 691-695
- MSC: Primary 28D05; Secondary 58F11
- DOI: https://doi.org/10.1090/S0002-9939-1994-1221724-X
- MathSciNet review: 1221724