Book Review
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MathSciNet review:
2816388
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Book Information:
Authors:
Alexander Gorodnik and
Amos Nevo
Title:
The ergodic theory of lattice subgroups
Additional book information:
Princeton University Press,
Princeton,
2010,
xiv + 121 pp.,
ISBN 978-0-691-14185-5,
US $29.95/US$60.00
V. Bergelson and A. Gorodnik, Weakly mixing group actions: a brief survey and an example, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 3–25. MR 2090763
M. Cowling, U. Haagerup, and R. Howe, Almost $L^2$ matrix coefficients, J. Reine Angew. Math. 387 (1988), 97–110. MR 946351
W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), no. 1, 143–179. MR 1230289, DOI 10.1215/S0012-7094-93-07107-4
Alex Eskin and Curt McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), no. 1, 181–209. MR 1230290, DOI 10.1215/S0012-7094-93-07108-6
Alex Eskin, Shahar Mozes, and Nimish Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2) 143 (1996), no. 2, 253–299. MR 1381987, DOI 10.2307/2118644
Alex Gorodnik, François Maucourant, and Hee Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 3, 383–435 (English, with English and French summaries). MR 2482443, DOI 10.24033/asens.2071
Roger E. Howe and Calvin C. Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979), no. 1, 72–96. MR 533220, DOI 10.1016/0022-1236(79)90078-8
François Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A561–A563 (French, with English summary). MR 512106
Elon Lindenstrauss, Pointwise theorems for amenable groups, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82–90. MR 1696824, DOI 10.1090/S1079-6762-99-00065-7
Grigoriy A. Margulis, On some aspects of the theory of Anosov systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows; Translated from the Russian by Valentina Vladimirovna Szulikowska. MR 2035655, DOI 10.1007/978-3-662-09070-1
Shahar Mozes, Mixing of all orders of Lie groups actions, Invent. Math. 107 (1992), no. 2, 235–241. MR 1144423, DOI 10.1007/BF01231889
Amos Nevo, Pointwise ergodic theorems for actions of groups, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 871–982. MR 2186253, DOI 10.1016/S1874-575X(06)80038-X
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, DOI 10.1090/S0273-0979-1980-14702-3
Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18. MR 1546100, DOI 10.1215/S0012-7094-39-00501-6
References
- V. Bergelson, A. Gorodnik. Weakly mixing group actions: a brief survey and an example. Modern Dynamical Systems and Applications, 3–25, Cambridge Univ. Press, Cambridge, 2004. MR 2090763 (2005m:37010)
- M. Cowling, U. Haagerup, R. Howe. Almost $L^2$ matrix coefficients. J. Reine Angew. Math.387 (1988), 97–110. MR 946351 (89i:22008)
- W. Duke, Z. Rudnick, P. Sarnak. Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), no. 1, 143–179. MR 1230289 (94k:11072)
- A. Eskin, C. McMullen. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209. MR 1230290 (95b:22025)
- A. Eskin, S. Mozes, N. Shah. Unipotent flows and counting lattice points on homogeneous varieties. Ann. of Math. (2) 143 (1996), no. 2, 253–299. MR 1381987 (97d:22012)
- A. Gorodnik, F. Maucourant, H. Oh. Manin’s and Peyre’s conjectures on rational points and adelic mixing. Ann. Sci. Éc. Norm. Sup. (4) 41 (2008), no. 3, 383–435. MR 2482443 (2010a:14047)
- R. Howe, C. Moore. Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979), no. 1, 72–96. MR 533220 (80g:22017)
- F. Ledrappier. Un champ markovien peut être d’entropie nulle et mélangeant. (French) C. R. Acad. Sci. Paris Ser. A-B 287 (1978), no. 7, A561–A563. MR 512106 (80b:28030)
- E. Lindenstrauss. Pointwise theorems for amenable groups. Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 82–90 (electronic). MR 1696824 (2000g:28042)
- G.A. Margulis. On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004. vi+139 pp. ISBN: 3-540-40121-0 MR 2035655 (2004m:37049)
- S. Mozes. Mixing of all orders of Lie groups actions. Invent. Math. 107 (1992), no. 2, 235–241. MR 1144423 (93e:22011)
- A. Nevo. Pointwise ergodic theorems for actions of groups. Handbook of Dynamical Systems. Vol. 1B, 871–982, Elsevier B. V., Amsterdam, 2006 . MR 2186253 (2006k:37006)
- D. Ornstein, B. 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 (80j:28031)
- N. Wiener. The ergodic theorem. Duke Math. J. 5 (1939), no. 1, 1–18. MR 1546100
Review Information:
Reviewer:
Manfred Einsiedler
Affiliation:
ETH Zürich, Departement Mathematik
Email:
manfred.einsiedler@math.ethz.ch
Journal:
Bull. Amer. Math. Soc.
48 (2011), 475-480
DOI:
https://doi.org/10.1090/S0273-0979-2011-01335-0
Published electronically:
March 9, 2011
Review copyright:
© Copyright 2011
American Mathematical Society
