Spectral types of skewed Bernoulli shift
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- by Youngho Ahn and Geon Ho Choe PDF
- Proc. Amer. Math. Soc. 128 (2000), 503-510 Request permission
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
- Y. Ahn and G. H. Choe, On normal numbers mod 2, Colloq. Math. 76 (1998), 161–170.
- Geon H. Choe, Spectral types of uniform distribution, Proc. Amer. Math. Soc. 120 (1994), no. 3, 715–722. MR 1169880, DOI 10.1090/S0002-9939-1994-1169880-6
- Nikolaos S. Papageorgiou, On Fatou’s lemma and parametric integrals for set-valued functions, Proc. Indian Acad. Sci. Math. Sci. 103 (1993), no. 2, 181–195. MR 1249905, DOI 10.1007/BF02837240
- Geon H. Choe, Products of operators with singular continuous spectra, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 65–68. MR 1077421, DOI 10.3934/cpaa.2015.14.2453
- J.-P. Conze, Remarques sur les transformations cylindriques et les équations fonctionnelles, Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976) Dépt. Math. Informat., Univ. Rennes, Rennes, 1976, pp. 13 (French). MR 0584021
- J. Feldman, D. J. Rudolph, and C. C. Moore, Affine extensions of a Bernoulli shift, Trans. Amer. Math. Soc. 257 (1980), no. 1, 171–191. MR 549160, DOI 10.1090/S0002-9947-1980-0549160-4
- Henry Helson and William Parry, Cocycles and spectra, Ark. Mat. 16 (1978), no. 2, 195–206. MR 524748, DOI 10.1007/BF02385994
- A. Iwanik, M. Lemańczyk, and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel J. Math. 83 (1993), no. 1-2, 73–95. MR 1239717, DOI 10.1007/BF02764637
- R. B. Kirk, Sets which split families of measurable sets, Amer. Math. Monthly 79 (1972), 884–886. MR 316654, DOI 10.2307/2317668
- Herbert A. Medina, Spectral types of unitary operators arising from irrational rotations on the circle group, Michigan Math. J. 41 (1994), no. 1, 39–49. MR 1260607, DOI 10.1307/mmj/1029004913
- William Parry, A cocycle equation for shifts, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 327–333. MR 1185098, DOI 10.1090/conm/135/1185098
- Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Daniel J. Rudolph, Classifying the isometric extensions of a Bernoulli shift, J. Analyse Math. 34 (1978), 36–60 (1979). MR 531270, DOI 10.1007/BF02790007
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
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
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1622769