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1997 | 122 | 3 | 289-298
Tytuł artykułu

Product $ℤ^d$-actions on a Lebesgue space and their applications

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We define a class of $ℤ^d$-actions, d ≥ 2, called product $ℤ^d$-actions. For every such action we find a connection between its spectrum and the spectra of automorphisms generating this action. We prove that for any subset A of the positive integers such that 1 ∈ A there exists a weakly mixing $ℤ^d$-action, d≥2, having A as the set of essential values of its multiplicity function. We also apply this class to construct an ergodic $ℤ^d$-action with Lebesgue component of multiplicity $2^d k$, where k is an arbitrary positive integer.
Czasopismo
Rocznik
Tom
122
Numer
3
Strony
289-298
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-12-28
poprawiono
1996-06-24
poprawiono
1996-09-17
Twórcy
  • Chair of Library Management and Scientific Information, Pedagogical University of Bydgoszcz (WSP), Jagiellońska 11, 85-067 Bydgoszcz, Poland
Bibliografia
  • [A] O. N. Ageev, Dynamical systems with a Lebesgue component of even multiplicity in the spectrum, Mat. Sb. 136 (1988), 307-319 (in Russian).
  • [BL] F. Blanchard and M. Lemańczyk, Measure preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal. 1 (1993), 257-294.
  • [C] R. V. Chacon, Approximation and spectral multiplicity, in: Lecture Notes in Math. 160, Springer, 1970, 18-27.
  • [CFS] I. P. Cornfeld, S. W. Fomin and Y. G. Sinai, Ergodic Theory, Springer, 1982.
  • [GKLL] G. R. Goodson, J. Kwiatkowski, M. Lemańczyk and P. Liardet, On the multiplicity function of ergodic group extensions of rotations, Studia Math. 102 (1992), 157-174.
  • [KL] J. Kwiatkowski, Jr., and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, preprint.
  • [L] M. Lemańczyk, Toeplitz $Z_2$-extensions, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 1-43.
  • [MN] J. Mathew and M. G. Nadkarni, A measure preserving transformation whose spectrum has Lebesgue component of finite multiplicity, Bull. London Math. Soc. 16 (1984), 402-406.
  • [O] V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 776-779 (in Russian).
  • [P] W. Parry, Topics in Ergodic Theory, Cambridge Univ. Press, 1981.
  • [Q] M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1294, Springer, 1987.
  • [Ro] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral multiplicities, Invent. Math. 72 (1983), 299-314.
  • [Ru] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York, 1962.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv122i3p289bwm
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