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

  • E. H. EL ABDALAOUI (a1) and M. G. NADKARNI (a2)

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.

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[1]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.
[2]el Abdalaoui, E. H. and Nadkarni, M.. Some notes on flat polynomials. Preprint, 2014, http://fr.arxiv.org/ abs/1402.5457, submitted for publication.
[3]Ageev, O. Dynamical system with an even-multiplicity Lebesgue component in the spectrum. Math. USSR 64 (1987), 305317.
[4]Brown, B.. Singular infinitely divisible distributions whose characteristic functions vanish at infinity. Math. Proc. Cambridge Philos. Soc. 82(2) (1977), 277287.
[5]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. 3357.
[6]Downarowicz, T. and Lacroix, Y.. Merit factors and Morse sequences. Theor. Comput. Sci. 209 (1998), 377387.
[7]Friedman, N.. Introduction to Ergodic Theory. Van Nostrand-Reinhold, New York, 1970.
[8]Guenais, M.. Morse cocycles and simple Lebesgue spectrum. Ergod. Th. & Dynam. Sys. 19(2) (1999), 437446.
[9]Ismagilov, R. S.. Riesz products and the spectrum of the Mackey action. Funktsional. Anal. i Prilozhen. 20(3) (1986), 8687 (in Russian).
[10]Ismagilov, R. S.. The spectrum of dynamical systems and the Riesz products. Mat. Sb. 180(7) (1989), 888912, 991 (in Russian); Translated in Math. USSR-Sb. 67(2) (1990), 341–366.
[11]Ismagilov, R. S.. Riesz products, random walk, and spectrum. Funktsional. Anal. i Prilozhen. 36(1) (2002), 1629, 96 (in Russian); Engl. Transl. Funct. Anal. Appl. 36(1) (2002), 13–24.
[12]Kahane, J.-P.. Sur les polynômesà coefficients unimodulaires. Bull. Lond. Math. Soc. 12(5) (1980), 321342.
[13]Kirillov, A. A.. Dynamical systems, factors and group representations. Uspekhi Mat. Nauk 22 (1967), 6780 (in Russian).
[14]Littlewood, J. E.. On polynomials ∑nz m, ∑m=0ne 𝛼iz m, z = e 𝜃i. J. Lond. Math. Soc. 41 (1966), 367376.
[15]Mathew, J. and Nadkarni, M. G.. A measure-preserving transformation whose spectrum has Lebesgue component of multiplicity two. Bull. Lond. Math. Soc. 16 (1984), 402406.
[16]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.
[17]Peyrière, J.. Étude de quelques propriétés des produits de Riesz. Ann. Inst. Fourier (Grenoble), 25 25(2) (1975), 127169.
[18]Queffélec, M.. Une nouvelle propriété des suites de Rudin-Shapiro. Ann. Inst. Fourier 37 (1987), 115138.
[19]Rokhlin, V. A.. Selected topics in the metric theory of dynamical systems. Uspekhi Mat. Nauk 4 (1949), 57128 (in Russian); Engl. transl. Amer. Math. Soc. Transl. Ser. 2 40 (1966), 171–240.
[20]Ulam, S. M.. Problems in Modern Mathematics (Science Editions). John Wiley & Sons, Inc, New York, 1964.

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