Spectral and mixing properties of actions of amenable groups

Author:
Nir Avni

Journal:
Electron. Res. Announc. Amer. Math. Soc. **11** (2005), 57-63

MSC (2000):
Primary 37A15; Secondary 37A20

Published electronically:
June 10, 2005

MathSciNet review:
2150945

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Abstract | References | Similar Articles | Additional Information

Abstract: We generalize two theorems about K-automorphisms from to all amenable groups with good entropy theory (this class includes all unimodular amenable groups which are not an increasing union of compact subgroups). The first theorem is that such actions are uniformly mixing; the second is that their spectrum is Lebesgue with countable multiplicity. For the proof we will develop an entropy theory for discrete amenable equivalence relations.

**[CFW]**A. Connes, J. Feldman, and B. Weiss,*An amenable equivalence relation is generated by a single transformation*, Ergodic Theory Dynamical Systems**1**(1981), no. 4, 431–450 (1982). MR**662736****[DP]**Alexandre I. Danilenko and Kyewon K. Park,*Generators and Bernoullian factors for amenable actions and cocycles on their orbits*, Ergodic Theory Dynam. Systems**22**(2002), no. 6, 1715–1745. MR**1944401**, 10.1017/S014338570200072X**[DG]**A. H. Dooley and V. Ya. Golodets,*The spectrum of completely positive entropy actions of countable amenable groups*, J. Funct. Anal.**196**(2002), no. 1, 1–18. MR**1941988**, 10.1006/jfan.2002.3966**[KL]**B. Kamiński and P. Liardet,*Spectrum of multidimensional dynamical systems with positive entropy*, Studia Math.**108**(1994), no. 1, 77–85. MR**1259025****[L]**Elon Lindenstrauss,*Pointwise theorems for amenable groups*, Invent. Math.**146**(2001), no. 2, 259–295. MR**1865397**, 10.1007/s002220100162**[OW]**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**[RS]**V. A. Rohlin and Ja. G. Sinaĭ,*The structure and properties of invariant measurable partitions*, Dokl. Akad. Nauk SSSR**141**(1961), 1038–1041 (Russian). MR**0152629****[RW]**Daniel J. Rudolph and Benjamin Weiss,*Entropy and mixing for amenable group actions*, Ann. of Math. (2)**151**(2000), no. 3, 1119–1150. MR**1779565**, 10.2307/121130**[S]**Ja. G. Sinaĭ,*Dynamical systems with countable Lebesgue spectrum. I*, Izv. Akad. Nauk SSSR Ser. Mat.**25**(1961), 899–924 (Russian). MR**0148852****[W]**Benjamin Weiss,*Actions of amenable groups*, Topics in dynamics and ergodic theory, London Math. Soc. Lecture Note Ser., vol. 310, Cambridge Univ. Press, Cambridge, 2003, pp. 226–262. MR**2052281**, 10.1017/CBO9780511546716.012

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Additional Information

**Nir Avni**

Affiliation:
Department of Mathematics, Hebrew University of Jerusalem, Israel

Email:
anir@math.huji.ac.il

DOI:
https://doi.org/10.1090/S1079-6762-05-00147-2

Received by editor(s):
May 27, 2004

Published electronically:
June 10, 2005

Communicated by:
Klaus Schmidt

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.