Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux

Babillot, Martine; Peigné, Marc

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 33 (2000) no. 1, pp. 81-120.

[1] T. Akaza, 3/2-dimensional measure of singular sets of some Kleinian groups, J. Math. Soc. Japan 24 (1972) 448-464. | MR | Zbl

[2] E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, in : Collected Papers, Addison-Wesley, 1965, pp. 499-501.

[3] T. Adachi and T. Sunada, Homology of closed geodesics in a negatively curved manifold, J. Differential Geom. 26 (1987) 81-99. | MR | Zbl

[4] A. Broise, F. Dal'Bo and M. Peigné, Méthode des opérateurs de transfert : transformations dilatantes de l'intervalle et dénombrement de géodésiques fermées, Soc. Math. France, Astérisque 238, 1996. | Zbl

[5] A.F. Beardon, The exponent of convergence of Poincaré series, Proc. London Math. Soc. 3 (18) (1968) 461-483. | MR | Zbl

[6] V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990) 459-477. | MR | Zbl

[7] M. Babillot and F. Ledrappier, Lalley's theorem on periodic orbits of hyperbolic flows, Ergodic Theory Dynamical Systems 18 (1998) 17-39. | MR | Zbl

[8] R. Bowen, The equidistribution of closed geodesics, Amer. J. Math. 94 (1972) 413-423. | MR | Zbl

[9] R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1979). | Numdam | MR | Zbl

[10] M. Babillot and M. Peigné, Closed geodesics in homology classes on hyperbolic manifolds with cusps, C. R. Acad. Sci. Paris Série I 324 (1997) 901-906. | MR | Zbl

[11] L. Breiman, Probability, Addison-Wesley, 1968. | MR | Zbl

[12] X. Bressaud, Opérateurs de transfert sur le décalage à alphabet infini et applications, Thèse de l'Université Paris 6, 1996.

[13] R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math. IHES 50 (1979) 153-170. | Numdam | MR | Zbl

[14] W. Doeblin and R. Fortet, Sur les chaines à liaison complètes, Bull. Soc. Math. France 65 (1937) 132-148. | JFM | Numdam | Zbl

[15] P.G. Doyle, On the bass note of a Schottky group, Acta Math. 160 (1988) 249-284. | MR | Zbl

[16] F. Dal'Bo and M. Peigné, Groupes du ping-pong et géodésiques fermées en courbure -1, Ann. Inst. Fourier 46 (3) (1996) 755-799. | Numdam | MR | Zbl

[17] N. Enriquez, J. Franchi and Y. Le Jan, Stable windings for geodesics under Patterson-Sullivan measure, Prépublication 431 du Laboratoire de Probabilités de l'Université Paris VI, 1998.

[18] N. Enriquez and Y. Le Jan, Statistic of the winding of geodesics on a Riemann surface with finite area and constant negative curvature, Rev. Mat. Iber. 13 (1997) 377-401. | MR | Zbl

[19] C. Epstein, Asymptotics for closed geodesics in a homology class-the finite volume case, Duke Math. J. 55 (1987) 717-757. | MR | Zbl

[20] J. Franchi, Asymptotic singular homology of a complete hyperbolic 3-manifold of finite volume, Proc. London Math. Soc. 79 (1999) 451-480. | MR | Zbl

[21] Y. Guivarc'H and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. ENS 26 (1993) 23-50. | Numdam | MR | Zbl

[22] L. Guillopé, Fonctions Zêta de Selberg et surfaces de géométrie finie, Adv. Studies in Pure Math. 21 (1992) 33-72. | MR | Zbl

[23] Y. Guivarc'H and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov, et applications aux difféomorphismes d'Anosov, Ann. IHP 24 73-98. | Numdam | MR | Zbl

[24] N.T.A. Haydn, Meromorphic extensions of the Zeta function for Axiom A flows, Ergodic Theory Dynamical Systems 10 (1990) 347-360. | MR | Zbl

[25] D. Heijhal, The Selberg trace formula and the Riemann Zeta function, Duke Math. J. 43 (1976) 441-482. | MR | Zbl

[26] H. Hennion, Sur un théorème spectral et son application aux noyaux Lipschitziens, Proc. AMS 118 (1993) 637-634. | MR | Zbl

[27] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen (I), Math. Ann. 138 (1959) 1-26 ; (II), Math. Ann. 142 (1961) 385-398 ; (III), Math. Ann. 143 (1961) 463-464. | MR | Zbl

[28] S. Isola, On the rate of convergence to equilibrium of countable ergodic Markov chains, Prépublication, Università degli Studi di Bologna, 1997.

[29] S. Isola, Dynamical Zeta functions and correlation functions for intermittent interval maps, Prépublication, Università degli Studi di Bologna, 1997.

[30] S. Isola, Renewal sequences and intermittency, Prépublication, Università degli Studi di Bologna, 1997.

[31] A. Katsuda and T. Sunada, Closed orbits in homology classes, Publ. Math. IHES 71 (1990) 5-32. | Numdam | MR | Zbl

[32] S. Lalley, Renewal theorems in symbolic dynamics with applications to geodesic flows, non euclidean tesselations and their fractal limits, Acta Math. 163 (1989) 1-55. | MR | Zbl

[33] S. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J. 58 (1989) 795-821. | MR | Zbl

[34] S. Lalley, Probabilistic methods in certain counting problems in ergodic theory, in : T. Bedford, M. Keane and C. Series, eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford, 1991. | MR | Zbl

[35] B. Maskitt, Kleinian Groups, Springer, Berlin, 1988. | Zbl

[36] G.A. Margulis, Applications of ergodic theory for the investigation of manifolds of negative curvature, Func. Anal. Appl. 3 (1969) 335-336. | MR | Zbl

[37] C. Mcmullen, Hausdorff dimension and conformal dynamics, I : Kleinian Groups and strong limits, à paraître J. Diff. Geom. III : Computation of dimension, Amer. J. Math. 120 (1998) 691-721. | MR | Zbl

[38] R. Miatello and N.R. Wallach, The resolvant of the Laplacian on locally symmetric spaces, J. Differential Geom. 36 (1992) 663-698. | MR | Zbl

[39] W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions, Israel J. Math. 45 (1983) 573-591. | MR | Zbl

[40] S.J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976) 241-273. | MR | Zbl

[41] M. Pollicott, Homology and closed geodesics in a compact negatively curved surface, Amer. J. Math. 113 (1991) 379-385. | MR | Zbl

[42] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188, Soc. Math. France, 1990. | Numdam | MR | Zbl

[43] R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J. 55 (1987) 287-297. | MR | Zbl

[44] T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points : thermodynamic formalism and phase transitions, J. Stat. Phys. 66 (1992) 503-514. | MR | Zbl

[45] J.G. Ratcliffe, Foundations of Hyperbolic Geometry, Springer, New York, 1994.

[46] T. Roblin, Sur la théorie ergodique des groupes discrets en géométrie hyperbolique, Thèse de doctorat de l'Université Paris XI, 1999.

[47] J. Rousseau-Egele, Un théorème de la limite locale pour une classe de fonctions dilatantes et monotones par morceaux, Ann. Probab. 11 (1983) 772-788. | MR | Zbl

[48] W. Rudin, Analyse Réelle et Complexe, Masson, Paris, 1975. | MR | Zbl

[49] H.H. Rugh, Intermittency and regularized Fredholm determinants, Invent. Math. 135 (1999) 1-24. | MR | Zbl

[50] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956) 47-87. | MR | Zbl

[51] C. Series, The modular surface and continued fractions, J. London Math. Soc. 31 (1985) 69-80. | MR | Zbl

[52] R. Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dynamical Systems 13 (1993) 387-408. | MR | Zbl

[53] F. Spitzer, Principles of Random Walks, Van Nostrand, Princeton, 1964. | MR | Zbl

[54] D. Sullivan, The density at infinity of a discrete group of hyperbolic isometries, Publ. Math. IHES 50 (1979) 171-209. | Numdam | MR | Zbl

[55] D. Sullivan, Entropy, Hausdorff measure old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984) 259-277. | MR | Zbl

[56] L.S. Young, Recurrence times and rates of mixing, à paraître, Israel J. Math. (1999). | MR | Zbl

[57] S. Zelditch, Trace formula for compact Γ\PSL2(R) and the equidistribution theory of closed geodesics, Duke Math. J. 59 (1989) 27-81. | MR | Zbl