zbMATH — the first resource for mathematics

The rate of mixing for geodesic and horocycle flows. (English) Zbl 0623.22008
Author’s summary: “Using elementary methods we show that for every unitary representation \(\pi\) of \(G=SL(2, {\mathbb{R}})\) with no non-zero invariant vectors the matrix coefficients \(<\pi (a(t))v,w>\) of \[ a(t)=\left( \begin{matrix} e^ t\\ 0\end{matrix} \quad \begin{matrix} 0\\ e^{- t}\end{matrix} \right) \] decay exponentially fast for any vectors v, w Hölder continuous in the direction of the rotation subgroup of G.”
Reviewer: T.Jørgensen

22D40 Ergodic theory on groups
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
22D10 Unitary representations of locally compact groups
22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations
Full Text: DOI
[1] Gelfand, Representation Theory and Automorphic Functions (1969)
[2] DOI: 10.1016/0040-9383(80)90015-4 · Zbl 0465.58027 · doi:10.1016/0040-9383(80)90015-4
[3] DOI: 10.1007/BF01215756 · Zbl 0585.58022 · doi:10.1007/BF01215756
[4] Borel, Annals of Math. Studies 94 pp none– (1980)
[5] DOI: 10.1016/0022-1236(78)90057-5 · Zbl 0392.43011 · doi:10.1016/0022-1236(78)90057-5
[6] DOI: 10.1007/BF02392046 · Zbl 0336.30005 · doi:10.1007/BF02392046
[7] DOI: 10.1215/S0012-7094-77-04403-9 · Zbl 0398.22022 · doi:10.1215/S0012-7094-77-04403-9
[8] Mackey, The Theory of Unitary Group Representations (1976) · Zbl 0344.22002
[9] Lax, J. Fund. Anal. 46 pp none– (1983)
[10] Lang, SL2(R) (1985) · doi:10.1007/978-1-4612-5142-2
[11] Kubota, Elementary Theory of Eisenstein Series (1973)
[12] Lang, Real Analysis (1969)
[13] Katznelson, An Introduction to Harmonic Analysis (1976)
[14] Howe, J. Fund. Anal. 32 pp 727ndash;96– (1979)
[15] Hopf, Ber. Voch. Sachs. Akad. Wiss. 91 pp 261– (1939)
[16] Gelbart, Annals of Math. Studies 83 pp none– (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.