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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
Posted:
October 6, 2005
MathSciNet review:
2199176
Full-text PDF Free Access
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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.
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- O. Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math. 160 (2005), 417-446. MR 2138072
- 2.
- I.P. Cornfel'd , S.V. Fomin, Ya. G. Sinai, Ergodic Theory (Springer-Verlag, 1980). MR 0832433 (87f:28019)
- 3.
- E. Glasner, J.L. King, A zero-one law for dynamical properties, Topol. Dyn. and Appl. 215 (1998), 231-242. MR 1603201 (99d:28039)
- 4.
- G. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173-226. MR 1693318 (2000f:28021)
- 5.
- G. Goodson, Ergodic dynamical systems conjugate to their composition squares, Acta Math. Univ. Comenian. (N.S.) 71 (2002), 201-210. MR 1980380 (2004c:37008)
- 6.
- G. Goodson, Spectral properties of ergodic dynamical systems conjugate to their composition squares, preprint.
- 7.
- P.R. Halmos, Lectures on ergodic theory (N.Y.: Chelsea Publ. Comp., 1960). MR 0111817 (22:2677)
- 8.
- A. del Junco, Transformations with discrete spectrum are stacking transformations, Canad. J. Math. 24 (1976), 836-839. MR 0414822 (54:2914)
- 9.
- A. del Junco, Disjointness of measure-preserving transformations, self-joinings and category, Prog. Math. K (1981), 81-89. MR 0633762 (82m:28035)
- 10.
- A.A. Kirillov, Elements of the theory of representations (Springer-Verlag, 1976). MR 0412321 (54:447)
- 11.
- D. Ornstein, B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161-164. MR 0551753 (80j:28031)
- 12.
- D. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1-141. MR 0910005 (88j:28014)
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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:
http://dx.doi.org/10.1090/S0002-9939-05-08380-2
PII:
S 0002-9939(05)08380-2
Keywords:
Group actions,
ergodic theory,
conjugations to its squares
Received by editor(s):
November 20, 2004
Posted:
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
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