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Mixing with staircase multiplicity functions

Published online by Cambridge University Press:  10 July 2008

OLEG AGEEV
OLEG AGEEV*
Affiliation:
Department of Mathematics, Moscow State Technical University, 2nd Baumanscaya St. 5, 105005 Moscow, Russia School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia (email: ageev@mx.bmstu.ru, ageev@maths.unsw.edu.au)
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Abstract

Every subgroup of the symmetric group defines a natural factor of the Cartesian power of a transformation. We calculate the set of values of the spectral multiplicity function of such factors (under certain conditions on the transformation) in terms of the number of orbits of diagonal actions of these subgroups. An analogous statement also holds on the unitary level for operators that preserve 1. In particular, we prove that for every positive integer n, there exists a transformation which is mixing of all orders and has a staircase multiplicity function of length n; that is, the essential values of the spectral multiplicity function are {1,2,…,n}.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1687 - 1700
Copyright
Copyright © 2008 Cambridge University Press

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References

[1] Adams, T.. Smorodinsky’s conjecture on rank-one mixing. Proc. Amer. Math. Soc. 126 (1998), 739–744.CrossRefGoogle Scholar
[2] Ageev, O.. On ergodic transformations with homogeneous spectrum. J. Dyn. Control Syst. 5 (1999), 149–152.CrossRefGoogle Scholar
[3] Ageev, O.. On the spectrum of Cartesian powers of classical automorphisms. Mat. Zametki 68 (2000), 643–647 (Engl. Transl. Math. Notes 68 (2000), 547–551).Google Scholar
[4] Ageev, O.. The homogeneous spectrum problem in ergodic theory. Invent. Math. 160 (2005), 417–446.CrossRefGoogle Scholar
[5] Baxter, J. R.. A class of ergodic transformations having simple spectrum. Proc. Amer. Math. Soc. 27 (1971), 275–279.CrossRefGoogle Scholar
[6] Bunimovich, L. A. et al. Dynamical Systems, Ergodic Theory and Applications. Springer, Berlin, 2000.Google Scholar
[7] Champernowne, D. G.. The construction of decimals normal in the scale ten. J. London Math. Soc. 8 (1933), 254–260.CrossRefGoogle Scholar
[8] Cornfel’d, I. P., Fomin, S. V. and Sinai, Ya. G.. Ergodic Theory. Springer, Berlin, 1980.Google Scholar
[9] Goodson, G. R.. A survey of recent results in the spectral theory of ergodic dynamical systems. J. Dyn. Control Syst. 5 (1999), 173–226.CrossRefGoogle Scholar
[10] Halmos, P. R.. Lectures on Ergodic Theory. Chelsea Publishing Co., New York, 1956.Google Scholar
[11] Kalikow, S. A.. Twofold mixing implies threefold mixing for rank one transformations. Ergod. Th. & Dyn. Sys. 4 (1984), 237–259.CrossRefGoogle Scholar
[12] Katok, A. B.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[13] Katok, A. B. and Thouvenot, J. P.. Spectral Properties and Combinatorial Constructions in Ergodic Theory (Handbook in Dynamical systems, 1B). Elsevier B.V., Amsterdam, 2006, pp. 649–743.Google Scholar
[14] Kerber, A.. Algebraic Combinatorics via Finite Group Actions. BI-Wiss., Mannheim, 1991.Google Scholar
[15] Nadkarni, M. G.. Spectral Theory of Dynamical Systems. Birkhäuser, Basel, 1998.Google Scholar
[16] Robinson, E. Jr. Mixing and spectral multiplicity. Ergod. Th. & Dyn. Sys. 5 (1985), 617–624.CrossRefGoogle Scholar
[17] Ryzhikov, V. V.. Joinings and multiple mixing of the actions of finite rank. Funct. Anal. Appl. 27 (1993), 128–140.CrossRefGoogle Scholar
[18] Ryzhikov, V. V.. Transformations having homogeneous spectra. J. Dyn. Control Syst. 5 (1999), 145–148.CrossRefGoogle Scholar
[19] Ryzhikov, V. V.. Homogeneous spectrum, disjointness of convolutions, and mixing properties of dynamical systems. Selected Russian Mathematics 1 (1999), 13–24.Google Scholar
[20] Ryzhikov, V. V.. Weak limits of powers, simple spectrum of symmetric products, and rank one mixing constructions. Sb. Math. 198 (2007), 733–754.CrossRefGoogle Scholar