Spectral rigidity of group actions: Applications to the case gr

Author:
Oleg N. Ageev

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1331-1338

MSC (2000):
Primary 37A05, 37A15, 37A25, 37A30, 37A35, 28D05, 28D15; Secondary 47A05, 47A35, 47D03

Published electronically:
October 6, 2005

MathSciNet review:
2199176

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We apply a technique to study the notion of spectral rigidity of group actions to a group gr. As an application, we prove that there exist rank one weakly mixing transformations conjugate to its square, thereby giving a positive answer to a well-known question.

**1.**Oleg Ageev,*The homogeneous spectrum problem in ergodic theory*, Invent. Math.**160**(2005), no. 2, 417–446. MR**2138072**, 10.1007/s00222-004-0422-z**2.**I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ,*Ergodic theory*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR**832433****3.**Eli Glasner and Jonathan L. King,*A zero-one law for dynamical properties*, Topological dynamics and applications (Minneapolis, MN, 1995) Contemp. Math., vol. 215, Amer. Math. Soc., Providence, RI, 1998, pp. 231–242. MR**1603201**, 10.1090/conm/215/02944**4.**G. R. Goodson,*A survey of recent results in the spectral theory of ergodic dynamical systems*, J. Dynam. Control Systems**5**(1999), no. 2, 173–226. MR**1693318**, 10.1023/A:1021726902801**5.**G. R. Goodson,*Ergodic dynamical systems conjugate to their composition squares*, Acta Math. Univ. Comenian. (N.S.)**71**(2002), no. 2, 201–210. MR**1980380****6.**G. Goodson,*Spectral properties of ergodic dynamical systems conjugate to their composition squares*, preprint.**7.**Paul R. Halmos,*Lectures on ergodic theory*, Chelsea Publishing Co., New York, 1960. MR**0111817****8.**Andrés del Junco,*Transformations with discrete spectrum are stacking transformations*, Canad. J. Math.**28**(1976), no. 4, 836–839. MR**0414822****9.**Andrés del Junco,*Disjointness of measure-preserving transformations, minimal self-joinings and category*, Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., vol. 10, Birkhäuser, Boston, Mass., 1981, pp. 81–89. MR**633762****10.**A. A. Kirillov,*Elements of the theory of representations*, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt; Grundlehren der Mathematischen Wissenschaften, Band 220. MR**0412321****11.**Donald S. Ornstein and Benjamin Weiss,*Ergodic theory of amenable group actions. I. The Rohlin lemma*, Bull. Amer. Math. Soc. (N.S.)**2**(1980), no. 1, 161–164. MR**551753**, 10.1090/S0273-0979-1980-14702-3**12.**Donald S. Ornstein and Benjamin Weiss,*Entropy and isomorphism theorems for actions of amenable groups*, J. Analyse Math.**48**(1987), 1–141. MR**910005**, 10.1007/BF02790325

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Additional Information

**Oleg N. Ageev**

Affiliation:
Department of Mathematics, Moscow State Technical University, 2nd Baumanscaya St. 5, 105005 Moscow, Russia

Address at time of publication:
Max Planck Institute of Mathematics, P.O. Box 7280, D-53072 Bonn, Germany

Email:
ageev@mx.bmstu.ru, ageev@mpim-bonn.mpg.de

DOI:
https://doi.org/10.1090/S0002-9939-05-08380-2

Keywords:
Group actions,
ergodic theory,
conjugations to its squares

Received by editor(s):
November 20, 2004

Published electronically:
October 6, 2005

Additional Notes:
The author was supported in part by the Max Planck Institute of Mathematics, Bonn, and the Programme of Support of Leading Scientific Schools of the RF (grant no. NSh-457.2003.1)

Communicated by:
Michael Handel

Article copyright:
© Copyright 2005
American Mathematical Society