Abstract
We show that for a\(\mathbb{Z}^2 \)-action Ψ being the Kronecker sum of a symbolic strictly ergodic\(\mathbb{Z}\)-actionT and a Chacon\(\mathbb{Z}\)-actionS, the rank (covering number) of Ψ is the same as that forT. Using this result we construct, for a given natural numberr≥2 and a real numberb∈(0,1) withr\b≥1, a\(\mathbb{Z}^d \)-action with rankr, covering numberb and a simple spectrum. On the other hand, for any positive integersr, m with 1≤m≤r≤∞ we construct a\(\mathbb{Z}^d \)-action with rankr and spectral multiplicitym.
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[C] R. V. Chacon,A geometric construction of measure preserving transformations, inProc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Part 2, Univ. of California Press, 1965, pp. 335–360.
[E] E. Eberlein,On topological entropy of semigroups of commuting transformations, inCollection: International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), Astérisque40 (1976), 17–62.
[Fe] S. Ferenczi,Systems of finite rank, Colloq. Math.73 (1997), 35–65.
[Fi] I. Filipowicz, Product\(\mathbb{Z}^d \)-action on a Lebesgue space and their applications, Studia Math.122 (1997), 289–298.
[FK] I. Filipowicz and J. Kwiatkowski,Rank, covering number and a simple spectrum, J. Analyse Math.66 (1995), 185–215.
[J] A. del Junco,A transformation with simple spectrum which is not rank one, Canad. J. Math.29 (1977), 655–663.
[K] J. Kwiatkowski,Inverse limit of M-cocycles and applications, Fund. Math.157 (1998), 261–276.
[KL] J. Kwiatkowski and Y. Lacroix,Multiplicity, rank pairs, J. Analyse Math.71 (1997), 205–235.
[KW] Y. Katznelson and B. Weiss,Commuting measure-preserving transformations, Israel J. Math.12 (1972), 161–173.
[M] M. Mentzen,Some examples of automorphisms with rank r and simple spectrum, Bull. Polish Acad. Sci. Math.35 (1978), 417–424.
[W] P. Walters,An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.
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Filipowicz, I., Kamiński, B. & Kwiatkowski, J. Topological and metric product\(\mathbb{Z}^d \)-actions and their applications-actions and their applications. J. Anal. Math. 83, 21–39 (2001). https://doi.org/10.1007/BF02790255
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DOI: https://doi.org/10.1007/BF02790255