Pointwise theorems for amenable groups
Author:
Elon Lindenstrauss
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82-90
MSC (1991):
Primary 28D15
DOI:
https://doi.org/10.1090/S1079-6762-99-00065-7
Published electronically:
June 30, 1999
MathSciNet review:
1696824
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Abstract: In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups.
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- William R. Emerson, The pointwise ergodic theorem for amenable groups, Amer. J. Math. 96 (1974), 472–487. MR 354926, DOI 10.2307/2373555
- Frederick P. Greenleaf and William R. Emerson, Group structure and the pointwise ergodic theorem for connected amenable groups, Advances in Math. 14 (1974), 153–172. MR 384997, DOI 10.1016/0001-8708(74)90027-9
- Andrés del Junco and Joseph Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), no. 3, 185–197. MR 553340, DOI 10.1007/BF01673506
- Donald Ornstein and Benjamin Weiss, The Shannon-McMillan-Breiman theorem for a class of amenable groups, Israel J. Math. 44 (1983), no. 1, 53–60. MR 693654, DOI 10.1007/BF02763171
- D. Ornstein and B. Weiss, The Shannon-McMillan-Breiman theorem for countable partitions, unpublished, c. 1985, 4 pages.
- Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
- A. Shulman, Maximal ergodic theorems on groups, Dep. Lit. NIINTI, No. 2184, 1988.
- A. A. Tempel′man, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR 176 (1967), 790–793 (Russian). MR 0219700
- Arkady Tempelman, Ergodic theorems for group actions, Mathematics and its Applications, vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Informational and thermodynamical aspects; Translated and revised from the 1986 Russian original. MR 1172319, DOI 10.1007/978-94-017-1460-0
- A. P. Calderon, A general ergodic theorem, Annals of Mathematics 58 (1953), no. 1, 182–191.
- W. R. Emerson, The pointwise ergodic theorem for amenable groups, American Journal of Mathematics 96 (1974), no. 3, 472–478.
- W. R. Emerson and F. P. Greenleaf, Groups structure and the pointwise ergodic theorem for connected amenable groups, Advances in Math. 14 (1974), 153–172.
- A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185–197.
- D. Ornstein and B. Weiss, The Shannon-McMillan-Breiman theorem for a class of amenable groups, Israel Journal of Mathematics 44 (1983), no. 1, 53–60.
- D. Ornstein and B. Weiss, The Shannon-McMillan-Breiman theorem for countable partitions, unpublished, c. 1985, 4 pages.
- D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal D’analyse Mathématique 48 (1987), 1–142.
- A. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, Rhode Island, 1988.
- D. Rudolph, Fundamentals of Measurable Dynamics — Ergodic theory on Lebesgue spaces, Oxford University Press, New York, 1990.
- A. Shulman, Maximal ergodic theorems on groups, Dep. Lit. NIINTI, No. 2184, 1988.
- A. Tempelman, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR 176 (1967), no. 4, 790–793; English translation: Soviet Math. Dokl. 8 (1967), no. 5, 1213–1216.
- A. Tempelman, Ergodic theorems for group actions, informational and thermodynamical aspects, Kluwer Academic Publishers, Dordrecht, 1992.
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Additional Information
Elon Lindenstrauss
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
MR Author ID:
605709
Email:
elon@math.huji.ac.il
Keywords:
Amenable groups,
pointwise convergence,
ergodic theorems
Received by editor(s):
January 18, 1999
Published electronically:
June 30, 1999
Additional Notes:
The author would like to thank the Clore Foundation for its support.
Communicated by:
Yitzhak Katznelson
Article copyright:
© Copyright 1999
American Mathematical Society
