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
DOI:
https://doi.org/10.1090/S1079-6762-05-00147-2
Published electronically:
June 10, 2005
MathSciNet review:
2150945
Full-text PDF Free Access
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Abstract: We generalize two theorems about K-automorphisms from $\mathbb {Z}$ 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.
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- 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, DOI https://doi.org/10.1017/S014338570200072X
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[CFW]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).
[DP]DP A. I. Danilenko and K. K. Park, Generators and Bernoullian factors for amenable actions and cocycles on their orbits, Ergodic Theory and Dynamical Systems 22 (2002), no. 6, 1715–1745.
[DG]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.
[KL]KL B. Kamiński and P. Liardet, Spectrum of multidimensional dynamical systems with positive entropy, Studia Math. 108 (1994), no. 1, 77–85.
[L]L E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146 (2001), no. 2, 259–295.
[OW]OW D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141.
[RS]RS V. A. Rohlin and Ja. G. Sinaĭ, The structure and properties of invariant measurable partitions (Russian), Dokl. Akad. Nauk SSSR 141 (1961), 1038–1041.
[RW]RW D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math. (2) 151 (2000), no. 3, 1119–1150.
[S]S Ja. G. Sinaĭ, Dynamical systems with countable Lebesgue spectrum (Russian), Izv. Akad. Nauk SSSR Ser. Math. 25 (1961), 899–924.
[W]W B. Weiss, Actions of amenable groups, Topics in Dynamics and Ergodic Theory, LMS Lect. Note Series 310, 226–262.
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Additional Information
Nir Avni
Affiliation:
Department of Mathematics, Hebrew University of Jerusalem, Israel
Email:
anir@math.huji.ac.il
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.
