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A non-singular transformation whose spectrum has Lebesgue component of multiplicity one

Published online by Cambridge University Press:  06 November 2014

E. H. EL ABDALAOUI Open the ORCID record for E. H. EL ABDALAOUI [Opens in a new window]  and
M. G. NADKARNI
E. H. EL ABDALAOUI
Affiliation:
Normandie University, University of Rouen, Department of Mathematics, LMRS UMR 60 85 CNRS, Avenue de l’Université, BP 12, 76801 Saint Etienne du Rouvray, France email [email protected]
M. G. NADKARNI
Affiliation:
Department of Mathematics, University of Mumbai, Vidyanagari, Kalina, Mumbai 400098, India email [email protected]
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Abstract

In this note we give an example of an ergodic non-singular map whose unitary operator admits a Lebesgue component of multiplicity one in its spectrum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 3 , May 2016 , pp. 671 - 681
Copyright
© Cambridge University Press, 2014 

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References

el Abdalaoui, E. H. and Nadkarni, M.. Calculus of generalized Riesz products. Contemporary Mathematics (Proc. Conf. in Honour of Professor S. G. Dani). 2014, accepted for publication.Google Scholar
el Abdalaoui, E. H. and Nadkarni, M.. Some notes on flat polynomials. Preprint, 2014, http://fr.arxiv.org/ abs/1402.5457, submitted for publication.Google Scholar
Ageev, O. Dynamical system with an even-multiplicity Lebesgue component in the spectrum. Math. USSR 64 (1987), 305–317.Google Scholar
Brown, B.. Singular infinitely divisible distributions whose characteristic functions vanish at infinity. Math. Proc. Cambridge Philos. Soc. 82(2) (1977), 277–287.CrossRefGoogle Scholar
Choksi, J. R. and Nadkarni, M. G.. On the question of transformations with simple Lebesgue spectrum. Lie Groups and Ergodic Theory (Mumbai, 1996) (Tata Institute of Fundamental Research Studies in Mathematics, 14). Tata Institute of Fundamental Research, Bombay, 1998, pp. 33–57.Google Scholar
Downarowicz, T. and Lacroix, Y.. Merit factors and Morse sequences. Theor. Comput. Sci. 209 (1998), 377–387.Google Scholar
Friedman, N.. Introduction to Ergodic Theory. Van Nostrand-Reinhold, New York, 1970.Google Scholar
Guenais, M.. Morse cocycles and simple Lebesgue spectrum. Ergod. Th. & Dynam. Sys. 19(2) (1999), 437–446.Google Scholar
Ismagilov, R. S.. Riesz products and the spectrum of the Mackey action. Funktsional. Anal. i Prilozhen. 20(3) (1986), 86–87 (in Russian).Google Scholar
Ismagilov, R. S.. The spectrum of dynamical systems and the Riesz products. Mat. Sb. 180(7) (1989), 888–912, 991 (in Russian); Translated in Math. USSR-Sb. 67(2) (1990), 341–366.Google Scholar
Ismagilov, R. S.. Riesz products, random walk, and spectrum. Funktsional. Anal. i Prilozhen. 36(1) (2002), 16–29, 96 (in Russian); Engl. Transl. Funct. Anal. Appl. 36(1) (2002), 13–24.Google Scholar
Kahane, J.-P.. Sur les polynômesà coefficients unimodulaires. Bull. Lond. Math. Soc. 12(5) (1980), 321–342.CrossRefGoogle Scholar
Kirillov, A. A.. Dynamical systems, factors and group representations. Uspekhi Mat. Nauk 22 (1967), 67–80 (in Russian).Google Scholar
Littlewood, J. E.. On polynomials ∑nz m, ∑m=0ne 𝛼iz m, z = e 𝜃i. J. Lond. Math. Soc. 41 (1966), 367–376.Google Scholar
Mathew, J. and Nadkarni, M. G.. A measure-preserving transformation whose spectrum has Lebesgue component of multiplicity two. Bull. Lond. Math. Soc. 16 (1984), 402–406.Google Scholar
Nadkarni, M. G.. Spectral theory of dynamical systems. Hindustan Book Agency, New Delhi, (1998) (Birkhäuser Advanced Texts: Basler LehrbÆcher [Birkhäuser Advanced Texts: Basel Textbooks]). Birkhäuser Verlag, Basel, 1998.Google Scholar
Peyrière, J.. Étude de quelques propriétés des produits de Riesz. Ann. Inst. Fourier (Grenoble), 25 25(2) (1975), 127–169.Google Scholar
Queffélec, M.. Une nouvelle propriété des suites de Rudin-Shapiro. Ann. Inst. Fourier 37 (1987), 115–138.CrossRefGoogle Scholar
Rokhlin, V. A.. Selected topics in the metric theory of dynamical systems. Uspekhi Mat. Nauk 4 (1949), 57–128 (in Russian); Engl. transl. Amer. Math. Soc. Transl. Ser. 2 40 (1966), 171–240.Google Scholar
Ulam, S. M.. Problems in Modern Mathematics (Science Editions). John Wiley & Sons, Inc, New York, 1964.Google Scholar