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New spectral multiplicities for mixing transformations

Published online by Cambridge University Press:  14 March 2011

ALEXANDRE I. DANILENKO
ALEXANDRE I. DANILENKO*
Affiliation:
Institute for Low Temperature Physics and Engineering of Ukrainian National Academy of Sciences, 47 Lenin Avenue, Kharkov, 61164, Ukraine (email: [email protected])
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Abstract

It is shown that if E is a subset of such that 1∈E or 2∈E then there is a mixing transformation whose set of spectral multiplicities is E.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 517 - 534
Copyright
Copyright © Cambridge University Press 2011

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References

[Ad]Adams, T. M.. Smorodinsky’s conjecture on rank one systems. Proc. Amer. Math. Soc. 126 (1998), 739–744.CrossRefGoogle Scholar
[Ag1]Ageev, O. N.. On ergodic transformations with homogeneous spectrum. J. Dynam. Control Syst. 5 (1999), 149–152.CrossRefGoogle Scholar
[Ag2]Ageev, O. N.. On the multiplicity function of generic group extensions with continuous spectrum. Ergod. Th. & Dynam. Sys. 21 (2001), 321–338.CrossRefGoogle Scholar
[Ag3]Ageev, O. N.. The homogeneous spectrum problem in ergodic theory. Invent. Math. 160 (2005), 417–446.CrossRefGoogle Scholar
[Ag4]Ageev, O. N.. Mixing with staircase multiplicity function. Ergod. Th. & Dynam. Sys. 28 (2008), 1687–1700.CrossRefGoogle Scholar
[Da1]Danilenko, A. I.. On cocycles with values in group extensions. Generic results. Mat. Fiz. Anal. Geom. 7 (2000), 153–171.Google Scholar
[Da2]Danilenko, A. I.. Funny rank one weak mixing for nonsingular Abelian actions. Israel. J. Math. 121 (2001), 29–54.CrossRefGoogle Scholar
[Da3]Danilenko, A. I.. Explicit solution of Rokhlin’s problem on homogeneous spectrum and applications. Ergod. Th. & Dynam. Sys. 26 (2006), 1467–1490.CrossRefGoogle Scholar
[Da4]Danilenko, A. I.. (C,F)-actions in ergodic theory. Geometry and Dynamics of Groups and Spaces (Progress in Mathematics, 265). 2008, pp. 325–351.CrossRefGoogle Scholar
[Da5]Danilenko, A. I.. On new spectral multiplicities for ergodic maps. Studia Math. 197 (2010), 57–68.CrossRefGoogle Scholar
[DaR1]Danilenko, A. I. and Ryzhikov, V. V.. Spectral multiplicities for infinite measure preserving transformations. Funct. Anal. Appl. 44 (2010), 161–170.CrossRefGoogle Scholar
[DaR2]Danilenko, A. I. and Ryzhikov, V. V.. Mixing constructions with infinite invariant measure and spectral multiplicities. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[dJ]del Junco, A.. A simple map with no prime factors. Israel J. Math. 104 (1998), 301–320.CrossRefGoogle Scholar
[FM]Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
[G-Li]Goodson, G. R., Kwiatkowski, J., Lemańczyk, M. and Liardet, P.. On the multiplicity function of ergodic group extensions of rotations. Studia Math. 102 (1992), 157–174.CrossRefGoogle Scholar
[Kal]Kalikow, S. A.. Twofold mixing implies threefold mixing for rank one transformations. Ergod. Th. & Dynam. Sys. 4 (1984), 237–259.CrossRefGoogle Scholar
[Ka]Katok, A. B.. Combinatorial Constructions in Ergodic Theory and Dynamics (University Lecture Series, 30). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[KaL]Katok, A. and Lemańczyk, M.. Some new cases of realization of spectral multiplicity function for ergodic transformations. Fund. Math. 206 (2009), 185–215.CrossRefGoogle Scholar
[KL]Kwiatkowski, J. Jr and Lemańczyk, M.. On the multiplicity function of ergodic group extensions. II. Studia Math. 116 (1995), 207–215.CrossRefGoogle Scholar
[Le]Lemańczyk, M.. Spectral theory of dynamical systems. Encyclopedia of Complexity and Systems Science. Springer, Berlin, 2009.Google Scholar
[Ne]Newton, D.. On Gausssian processes with simple spectrum. Z. Wahrsch. Verv. Geb. 5 (1966), 207–209.CrossRefGoogle Scholar
[Os]Oseledec, V. I.. On the spectrum of ergodic automorphisms. Soviet Math. Dokl. 168 (1966), 776–779.Google Scholar
[Ro1]Robinson, E. A.. Ergodic measure-preserving transformations with arbitrary finite spectral multiplicities. Invent. Math. 72 (1983), 299–314.CrossRefGoogle Scholar
[Ro2]Robinson, E. A.. Mixing and spectral multiplicity. Ergod. Th. & Dynam. Sys. 5 (1985), 617–624.CrossRefGoogle Scholar
[Ro3]Robinson, E. A.. Transformations with highly nonhomogeneous spectrum of finite multiplicity. Israel J. Math. 56 (1986), 75–88.CrossRefGoogle Scholar
[Ru]Rudolph, D.. k-fold mixing lifts to weakly mixing isometric extensions. Ergod. Th. & Dynam. Sys. 5 (1985), 445–447.CrossRefGoogle Scholar
[Ry1]Ryzhikov, V. V.. Mixing, rank and minimal self-joining of actions with invariant measure. Mat. Sb. 183 (1992), 133–160.Google Scholar
[Ry2]Ryzhikov, V. V.. Transformations having homogeneous spectra. J. Dynam. Control Systems 5 (1999), 145–148.CrossRefGoogle Scholar
[Ry3]Ryzhikov, V. V.. Homogeneous spectrum, disjointness of convolutions, and mixing properties of dynamical systems. Selected Russian Math. 1 (1999), 13–24.Google Scholar
[Ry4]Ryzhikov, V. V.. Weak limits of powers, the simple spectrum of symmetric products and mixing constructions of rank 1. Sb. Math. 198 (2007), 733–754.CrossRefGoogle Scholar
[Ry5]Ryzhikov, V. V.. Spectral multiplicities and asymptotic operator properties of actions with invariant measure. Mat. Sb. 200 (2009), 107–120 (in Russian).Google Scholar
[Sc]Schmidt, K.. Cocycles of Ergodic Transformation Groups (Lecture Notes in Mathematics, 1). McMillan Co., India, 1977.Google Scholar
[Ti]Tikhonov, S. V.. A complete metric on the set of mixing transformations. Sb. Math. 198 (2007), 575–596.CrossRefGoogle Scholar