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Spectral types of skewed Bernoulli shift


Authors: Youngho Ahn and Geon Ho Choe
Journal: Proc. Amer. Math. Soc. 128 (2000), 503-510
MSC (1991): Primary 28D05, 47A35
DOI: https://doi.org/10.1090/S0002-9939-99-04990-4
Published electronically: June 21, 1999
MathSciNet review: 1622769
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Abstract: For the transformation $T: x \mapsto kx \pmod 1$ for $k \geq 2 $, it is proved that a real-valued function $f(x)$ of modulus $1$ is not a multiplicative coboundary if the discontinuities $0 < x_1< \cdots < x_n \leq 1$ of $f(x)$ are $k$-adic points and $x_1 \ge \frac 1k$. It is also proved that the weakly mixing skew product transformations arising from Bernoulli shifts have Lebesgue spectrum.


References [Enhancements On Off] (What's this?)

  • 1. Y. Ahn and G. H. Choe, On normal numbers mod 2, Colloq. Math. 76 (1998), 161-170. CMP 98:11
  • 2. G. H. Choe, Spectral types of uniform distribution, Proc. Amer. Math. Soc. 120 (1994), 715-722. MR 94e:47014 MR 94e:47014
  • 3. -, Ergodicity and irrational rotations, Proc. R. Ir. Acad. 93A (1993), 193-202. MR 94m:28017
  • 4. -, Products of operators with singular continuous spectra, Proc. Sympos. Pure Math., 51, Part 2 (1990), 65-68. MR 91h:47006
  • 5. J. P. Conze, Remarques sur les transformations cylindriques et les equations fonctionnelles, Séminaire de Probabilité I, Rennes, France (1976). MR 58:28427
  • 6. J. Feldman, D. J. Rudolph and C. C. Moore, Affine extensions of a Bernoulli shift, Trans. Amer. Math. Soc. 257 (1980), 171-191. MR 81a:28014
  • 7. H. Helson and W. Parry, Cocycles and spectra, Ark. Mat. 16 (1978), 195-206. MR 80d:28036
  • 8. A. Iwanik, M. Lema\'{n}czyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel J. Math. 83 (1993), 73-95. MR 94i:58108
  • 9. R. B. Kirk, Sets which split families of measurable sets, Amer. Math. Monthly 79 (1972), 884-886. MR 47:5201
  • 10. H. A. Medina, Spectral types of unitary operators arising from irrational rotations on the circle, Michigan. Math. J. 41(1) (1994), 39-49. MR 95a:28014
  • 11. W. Parry, A Cocycle equation for shift, Contemp. Math. 135 (1992), 327-333. MR 93j:28028
  • 12. K. Petersen, Ergodic Theory, Cambridge Univ. Press London, 1983. MR 87i:28002
  • 13. W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, 1987. MR 88k:00002
  • 14. D. J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math. 34 (1978), 36-60. MR 80g:28020
  • 15. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag New York, 1982. MR 84e:28017

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Additional Information

Youngho Ahn
Affiliation: Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-gu, 305-701 Taejon, Korea
Email: ahntau@math.kaist.ac.kr

Geon Ho Choe
Affiliation: Korea Advanced Institute of Science and Technology, Kusong-dong, Yusong-gu, 305-701 Taejon, Korea
Email: choe@euclid.kaist.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-99-04990-4
Keywords: Coboundary, metric density, weakly mixing, Lebesgue spectrum, Bernoulli shift
Received by editor(s): July 25, 1997
Received by editor(s) in revised form: March 31, 1998
Published electronically: June 21, 1999
Additional Notes: The second author’s research was supported by GARC-SRC and KOSEF 95-07-01-02-01-3
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

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