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The rate of mixing for geodesic and horocycle flows

Published online by Cambridge University Press:  19 September 2008

Marina Ratner
Marina Ratner
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA
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Abstract

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Using elementary methods we show that for every unitary representation π7 of G = SL(2, ℝ) with no non-zero invariant vectors the matrix coefficients 〈 π(a(t))v, w 〉 of

decay exponentially fast for any vectors v, w Holder continuous in the direction of the rotation subgroup of G.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 2 , June 1987 , pp. 267 - 288
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Borel, A. & Wallach, N.. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Annals. Annals of Math. Studies 94 (1980).Google Scholar
[2]Collet, P., Epstein, H. & Gallavotti, G.. Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Commun. Math. Phys. 95 61–112 (1984).CrossRefGoogle Scholar
[3]Gelbart, S.. Automorphic Forms on Adele Groups. Annals of Math. Studies 83 (1975).Google Scholar
[4]Gelfand, I., Graev, M. & Pyatetskii-Shapiro, I.. Representation Theory and Automorphic Functions. W. B. Saunders Co., Philadelphia, 1969.Google Scholar
[5]Guillemin, V. & Kazhdan, D.. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 301–312 (1980).CrossRefGoogle Scholar
[6]Hopf, E.. Statistik der geodatischen Linien in Mänigfaltigkeiten negativer Krümung. Ber. Voch. Sachs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
[7]Howe, R. & Moore, C.. Asymptotic properties of unitary representations. J. Fund. Anal. 32 (1979), 727ndash;96.CrossRefGoogle Scholar
[8]Katznelson, Y.. An Introduction to Harmonic Analysis. Dover Publications, 1976.Google Scholar
[9]Kubota, T.. Elementary Theory of Eisenstein Series. Halsted Press, 1973.Google Scholar
[10]Lang, S.. Real Analysis. Addison-Wesley, 1969.Google Scholar
[11]Lang, S.. SL2(R). Springer-Verlag, 1985.CrossRefGoogle Scholar
[12]Lax, P. & Phillips, R.. The asymptotic distribution of lattice points in Euclidean and non Euclidean spaces. J. Fund. Anal. 46 (1983), #3.Google Scholar
[13]Mackey, G.. The Theory of Unitary Group Representations. Univ. of Chicago Press, 1976.Google Scholar
[14]Milicic, D.. Asymptotic behavior of matrix coefficients of the discrete series. Duke Math. J. 44 (1977), 59–88.CrossRefGoogle Scholar
[15]Patterson, S.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.CrossRefGoogle Scholar
[16]Trombi, P.. Asymptotic expansions of matrix coefficients: The real rank one case. J. Fund. Anal. 30 (1978), 83–105.CrossRefGoogle Scholar