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Remarks on the length spectrum, and counting. (Remarques sur le spectre des longueurs d’une surface et comptages.) (French) Zbl 1058.53063
Summary: This paper deals with the length spectrum associated to a negatively curved manifold. In particular we prove that the length spectrum of a surface is not included in a direct subgroup of \(\mathbb R\). We also compare the length spectrum for different Riemannian structures.

MSC:
53D25 Geodesic flows in symplectic geometry and contact geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53C22 Geodesics in global differential geometry
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[1] [BE] Y. Benoist:Propriétés asymptotiques des groupes linéaires, Geometric and functional Analysis,7: (1997), 1-47. · Zbl 0947.22003
[2] [Bo] M. Bourdon:Structure conforme au bord et flot géodésique d’un CAT (?1)-espace, L’enseignement mathématique,41: (1995), 63-102.
[3] [Bu] M. Burger:Intersection, Manhattan curve and Patterson-Sullivan theory in rank 2, Intern. Math. Math. res. notices, (1993), no. 7, 217-225. · Zbl 0829.57023
[4] [D] F. Dal’bo:Famille de groupes agissant sur le produit de deux variétés de Hadamard, Sém. de Grenoble, 1997.
[5] [D-P] F. Dal’bo & M. Peigné:Some negatively curved manifold with cusps, mixing and counting, J. Reigne Angew. Math.497: (1998), 141-169. · Zbl 0890.53043
[6] [G-H] Y. Guivarc’h & J. Hardy:Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d’Anosov, Ann. I.H.P. no. 1, (1998), 73-98.
[7] [K] I. Kim:Rigidity of rank one symmetric spaces and their product, (1997), (prépublication).
[8] [L] F. Ledrappier:Structure au bord des variétés à courbure négative, Sém. de Grenoble, (1994-1995), 97-122.
[9] [O1] J.-P. Otal:Sur la géométrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Revista matematica Ibero americana,8: (1992), no. 3.
[10] [O2] J.-P. Otal:Le spectre marqué des longueurs des surfaces à courbure négative, Annales of Math.13: (1990), 151-162. · Zbl 0699.58018
[11] [P] M. Peigné:Aspects stochastiques d’actions de groupes et de semi-groupes, (habilitation Rennes 1998).
[12] [P-P] W. Parry & M. Pollicott:An analogue of the prime number theorem for closed orbits of axiome A flows, Annales of Math.118: (1983), 573-591. · Zbl 0537.58038
[13] [P-S] M. Pollicott & S. Sharp:The circle problem on surfaces of variable negative curvature, Mh. Math.123: (1997), 61-70. · Zbl 0876.58040
[14] [R] J. Rudolph:Ergodic behavior of Sullivan’s geometric measure on a geometicaly finite hyperbolic manifold, Ergod. Th. Dym. Syst.2: (1982), 491-512. · Zbl 0525.58025
[15] [S-S] R. Schwartz & R. Sharp:The correlation of length spectra of two hyperbolic surfaces, Comm. Math. Phys.153: (1993), 423-430. · Zbl 0772.58045
[16] [T] P. Tukia:On homomorphism of geometricaly finite möbius groups, Publ. Math. I.H.E.S.,61: (1985), 171-214. · Zbl 0572.30036
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