no. 1
On pairs of closed geodesics on hyperbolic surfaces
Annales de l'Institut Fourier, Tome 49 (1999) no. 1, pp. 1-25.

Dans cet article nous démontrons une formule de trace pour les doubles sommes sur les groupes fuchsiens totalement hyperboliques Γ. Ceci relie les angles d’intersection et les perpendiculaires communes des paires de géodésiques fermées sur Γ/H avec les produits scalaires des carrés de la valeur absolue des fonctions propres du laplacien hyperbolique Δ. Nous arrivons donc à des résultats quantitatifs sur les angles d’intersection et les perpendiculaires communes de ces géodésiques, en moyenne et individuellement.

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.

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     title = {On pairs of closed geodesics on hyperbolic surfaces},
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     volume = {49},
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Pitt, Nigel J. E. On pairs of closed geodesics on hyperbolic surfaces. Annales de l'Institut Fourier, Tome 49 (1999) no. 1, pp. 1-25. doi : 10.5802/aif.1667. https://aif.centre-mersenne.org/articles/10.5802/aif.1667/

[1] A. Beardon, The Geometry of Discrete Groups, Springer, 1983. | MR | Zbl

[2] H. Iwaniec, Prime geodesic theorem, J. reine. angew. Math., 349 (1984), 136-159. | EuDML | MR | Zbl

[3] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Biblioteca de la Revista Matemática Iberoamericana (1995). | MR | Zbl

[4] J. Lehner, Discontinuous Groups and Automorphic functions, Amer. Math. Soc. (1964). | MR | Zbl

[5] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL2(ℤ)\ℍ, IHES Publ., 81 (1995 207-237). | EuDML | Numdam | MR | Zbl

[6] N. Pitt, Talk given at the XIV Escola de Algebra, IMPA, Rio de Janeiro, Aug. 1996.

[7] P. Sarnak, Arithmetic Quantum Chaos, Israel Mathematical Conference Proceedings, 8 (1995). | MR | Zbl

[8] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory, 15 (1982), 229-247. | MR | Zbl

[9] A. Seger and C. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J., 38 (1989), 669-682. | MR | Zbl

[10] A. Selberg, Collected Papers, Vol. I, Springer, 1989. | MR | Zbl

[11] S. Zelditch, Selberg Trace Formulae, Pseudodifferential operators and geodesic periods of automorphic forms, Duke Math. J. (1988), 295-344. | Zbl

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