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A new class of rank-one transformations with singular spectrum

Published online by Cambridge University Press:  01 October 2007

EL HOUCEIN EL ABDALAOUI
EL HOUCEIN EL ABDALAOUI*
Affiliation:
Université de Rouen-Mathématiques, Laboratoire de Maths Raphael Salem, UMR 60 85 CNRS, Avenue de l’Université, BP 12, 76801 Saint Etienne du Rouvray, France (email: [email protected])
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Abstract

We introduce a new tool to study the spectral type of rank-one transformations using the method of the central limit theorem for trigonometric sums. We get some new applications.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1541 - 1555
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Adams, T. R.. On Smorodinsky conjecture. Proc. Amer. Math. Soc. 126 (1998), 739–744.CrossRefGoogle Scholar
[2]Adams, T. R. and Friedman, N. A.. Mixing staircase. Preprint, 1992.Google Scholar
[3]Berkes, I.. On the central limit theorem for Lacunary trigonometric series. Anal. Math. 4(3) (1978), 159–180.CrossRefGoogle Scholar
[4]Bourgain, J.. On the spectral type of Ornstein class one transformations. Israel J. Math. 84 (1993), 53–63.CrossRefGoogle Scholar
[5]Choksi, J. R. and Nadkarni, M. G.. The maximal spectral type of rank one transformation. Canad. Math. Bull. 37(1) (1994), 29–36.CrossRefGoogle Scholar
[6]Creutz, D. and Silva, C. E.. Mixing on a class of rank-one transformations. Ergod. Th. & Dynam. Sys. 24(2) (2004), 407–440.CrossRefGoogle Scholar
[7]Dooley, A. H. and Eigen, S. J.. A family of generalized Riesz products. Canad. J. Math. 48(2) (1996), 302–315.CrossRefGoogle Scholar
[8]El Abdalaoui, E. H.. La singularité mutuelle presque sûre du spectre des transformations d’Ornstein. Israel J. Math. 112 (1999), 135–155.Google Scholar
[9]El Abdalaoui, E. H., Parreau, F. and Prikhod’ko, A. A.. A new class of Ornstein transformations with singular spectrum. Ann. Inst. H. Poincaré Probab. Statist. 42(6) (2006), 671–681.CrossRefGoogle Scholar
[10]Erdös, P.. On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kutató Int. Közl 7 (1962), 37–42.Google Scholar
[11]Friedman, N. A.. Introduction to Ergodic Theory. Van Nostrand Reinhold, New York, 1970.Google Scholar
[12]Friedman, N. A.. Replication and stacking in ergodic theory. Amer. Math. Monthly 99 (1992), 31–34.CrossRefGoogle Scholar
[13]Fukuyama, K. and Takahashi, S.. The central limit theorem for lacunary series. Proc. Amer. Math. Soc. 127(2) (1999), 599–608.CrossRefGoogle Scholar
[14]Kac, M.. Statistical Independence in Probability, Analysis and Number Theory (The Carus Mathematical Monographs, 12). Mathematical Association of America. John Wiley and Sons, New York, 1959.CrossRefGoogle Scholar
[15]Kahane, J.-P.. Lacunary Taylor and Fourier series. Bull. Amer. Math. Soc. 70 (1964), 199–213.CrossRefGoogle Scholar
[16]Kilmer, S. J. and Saeki, S.. On Riesz products measures, mutual absolute continuity and singularity. Ann. Inst. Fourier (Grenoble) 38 (1988), 63–93.CrossRefGoogle Scholar
[17]Klemes, I.. The spectral type of staircase transformations. Tohoku Math. J. 48 (1994), 247–258.Google Scholar
[18]Klemes, I. and Reinhold, K.. Rank one transformations with singular spectrum type. Israel J. Math. 98 (1997), 1–14.CrossRefGoogle Scholar
[19]McLeish, D. L.. Dependent central limit theorems and invariance principles. Ann. Probab. 2 (1974), 620–628.CrossRefGoogle Scholar
[20]Murai, T.. The central limit theorem for trigonometric series. Nagoya Math. J. 87 (1982), 79–94.CrossRefGoogle Scholar
[21]Nadkarni, M. G.. Spectral Theory of Dynamical Systems. Birkhäuser, Cambridge, MA, 1998.Google Scholar
[22]Ornstein, D. S.. On the root problem in ergodic theory. Proc. Sixth Berkeley Symp. in Mathematical Statistics and Probability. University of California Press, Berkeley, 1971, pp. 347–356.Google Scholar
[23]Petit, B.. Le théorème limite central pour des sommes de Riesz–Raĭkov. Probab. Theory Related Fields 93(4) (1992), 407–438.CrossRefGoogle Scholar
[24]Prikhod’ko, A. A.. Stochastic constructions of flows of rank 1. Sb. Math. 192(11–12) (2001), 1799–1828.CrossRefGoogle Scholar
[25]Takahashi, S.. Probability limit theorems for trigonometric series. Limit Theorems of Probability Theory (Colloq., Keszthely, 1974) (Colloq. Math. Soc. Janos Bolyai, 11). North-Holland, Amsterdam, 1975, pp. 381–397.Google Scholar
[26]Zygmund, A.. Trigonometric Series. Vol. II, 2nd edn. Cambridge University Press, Cambridge, 1959, pp. 263–264.Google Scholar