On pairs of closed geodesics on hyperbolic surfaces

Nigel J. E. Pitt

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 1, page 1-25
  • ISSN: 0373-0956

Abstract

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In this article we prove a trace formula for double sums over totally hyperbolic Fuchsian groups Γ . This links the intersection angles and common perpendiculars of pairs of closed geodesics on Γ / H with the inner products of squares of absolute values of eigenfunctions of the hyperbolic laplacian Δ . We then extract quantitative results about the intersection angles and common perpendiculars of these geodesics both on average and individually.

How to cite

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Pitt, Nigel J. E.. "On pairs of closed geodesics on hyperbolic surfaces." Annales de l'institut Fourier 49.1 (1999): 1-25. <http://eudml.org/doc/75333>.

@article{Pitt1999,
abstract = {In this article we prove a trace formula for double sums over totally hyperbolic Fuchsian groups $\Gamma $. This links the intersection angles and common perpendiculars of pairs of closed geodesics on $\Gamma /H$ with the inner products of squares of absolute values of eigenfunctions of the hyperbolic laplacian $\Delta $. We then extract quantitative results about the intersection angles and common perpendiculars of these geodesics both on average and individually.},
author = {Pitt, Nigel J. E.},
journal = {Annales de l'institut Fourier},
keywords = {hyperbolic surfaces; closed geodesics; eigenfunctions of Laplacian; trace formula; Petersson inner product},
language = {eng},
number = {1},
pages = {1-25},
publisher = {Association des Annales de l'Institut Fourier},
title = {On pairs of closed geodesics on hyperbolic surfaces},
url = {http://eudml.org/doc/75333},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Pitt, Nigel J. E.
TI - On pairs of closed geodesics on hyperbolic surfaces
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 1
SP - 1
EP - 25
AB - In this article we prove a trace formula for double sums over totally hyperbolic Fuchsian groups $\Gamma $. This links the intersection angles and common perpendiculars of pairs of closed geodesics on $\Gamma /H$ with the inner products of squares of absolute values of eigenfunctions of the hyperbolic laplacian $\Delta $. We then extract quantitative results about the intersection angles and common perpendiculars of these geodesics both on average and individually.
LA - eng
KW - hyperbolic surfaces; closed geodesics; eigenfunctions of Laplacian; trace formula; Petersson inner product
UR - http://eudml.org/doc/75333
ER -

References

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  1. [1] A. BEARDON, The Geometry of Discrete Groups, Springer, 1983. Zbl0528.30001MR85d:22026
  2. [2] H. IWANIEC, Prime geodesic theorem, J. reine. angew. Math., 349 (1984), 136-159. Zbl0527.10021MR85h:11025
  3. [3] H. IWANIEC, Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la Revista Matemática Iberoamericana (1995). Zbl0847.11028MR96f:11078
  4. [4] J. LEHNER, Discontinuous Groups and Automorphic functions, Amer. Math. Soc. (1964). Zbl0178.42902MR29 #1332
  5. [5] W. LUO and P. SARNAK, Quantum ergodicity of eigenfunctions on PSL2(ℤ), IHES Publ., 81 (1995 207-237). Zbl0852.11024MR97f:11037
  6. [6] N. PITT, Talk given at the XIV Escola de Algebra, IMPA, Rio de Janeiro, Aug. 1996. 
  7. [7] P. SARNAK, Arithmetic Quantum Chaos, Israel Mathematical Conference Proceedings, 8 (1995). Zbl0831.58045MR96d:11059
  8. [8] P. SARNAK, Class numbers of indefinite binary quadratic forms, J. Number Theory, 15 (1982), 229-247. Zbl0499.10021
  9. [9] A. SEGER and C. SOGGE, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J., 38 (1989), 669-682. Zbl0703.35133MR91f:58097
  10. [10] A. SELBERG, Collected Papers, Vol. I, Springer, 1989. Zbl0675.10001MR92h:01083
  11. [11] S. ZELDITCH, Selberg Trace Formulae, Pseudodifferential operators and geodesic periods of automorphic forms, Duke Math. J. (1988), 295-344. Zbl0646.10024

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