Expanding maps on Cantor sets and analytic continuation of zeta functions
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 1, pp. 116-153.
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     title = {Expanding maps on {Cantor} sets and analytic continuation of zeta functions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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Naud, Frédéric. Expanding maps on Cantor sets and analytic continuation of zeta functions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 1, pp. 116-153. doi : 10.1016/j.ansens.2004.11.002. http://www.numdam.org/articles/10.1016/j.ansens.2004.11.002/

[1] Anantharaman N., Géodésiques fermées d'une surface sous contraintes homologiques, Thèse de doctorat, Université Paris 6, 2000.

[2] Anantharaman N., Precise counting results for closed orbits of Anosov flows, Ann. Sci. École Norm. Sup. (4) 33 (1) (2000) 33-56. | Numdam | MR | Zbl

[3] Anosov D.V., Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967) 209. | MR | Zbl

[4] Baladi V., Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific, Singapore, 2000. | MR | Zbl

[5] Baladi V., Vallée B., Euclidian algorithms are Gaussian, J. Number Theory (2004), submitted for publication. | Zbl

[6] Beardon A.F., The Geometry of Discrete Groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995, Corrected reprint of the 1983 original. | MR | Zbl

[7] Bowen R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 1975. | MR | Zbl

[8] Bowen R., Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 11-25. | Numdam | MR | Zbl

[9] Bowen R., Series C., Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 153-170. | Numdam | MR | Zbl

[10] Button J., All Fuchsian Schottky groups are classical Schottky groups, in: The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 117-125, (electronic). | MR | Zbl

[11] Dolgopyat D., On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (2) (1998) 357-390. | MR | Zbl

[12] Dolgopyat D., Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (5) (1998) 1097-1114. | MR | Zbl

[13] Dolgopyat D., Prevalence of rapid mixing. II. Topological prevalence, Ergodic Theory Dynam. Systems 20 (4) (2000) 1045-1059. | MR | Zbl

[14] Guillopé L., Zworski M., The wave trace for Riemann surfaces, Geom. Funct. Anal. 9 (6) (1999) 1156-1168. | MR | Zbl

[15] Guillopé L., Sur la distribution des longueurs des géodésiques fermées d'une surface compacte à bord totalement géodésique, Duke Math. J. 53 (3) (1986) 827-848. | MR | Zbl

[16] Guillopé L., Lin K.K., Zworski M., The Selberg zeta function for convex co-compact Schottky groups, Comm. Math. Phys. 245 (1) (2004) 149-176. | MR | Zbl

[17] Hejhal D.A., The Selberg Trace Formula for PSL(2,R), vol. I, Lecture Notes in Mathematics, vol. 548, Springer-Verlag, Berlin, 1976. | MR | Zbl

[18] Huber H., Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 142 (1960/1961) 385-398. | MR | Zbl

[19] Iwaniec H., Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[20] Jenkinson O., Pollicott M., Calculating Hausdorff dimensions of Julia sets and Kleinian limit sets, Amer. J. Math. 124 (3) (2002) 495-545. | MR | Zbl

[21] Lalley S.P., Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163 (1-2) (1989) 1-55. | MR | Zbl

[22] Liverani C., Decay of correlations, Ann. of Math. (2) 142 (2) (1995) 239-301. | MR | Zbl

[23] Mazzeo R.R., Melrose R.B., Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (2) (1987) 260-310. | MR | Zbl

[24] Mcmullen C.T., Hausdorff dimension and conformal dynamics. III. Computation of dimension, Amer. J. Math. 120 (4) (1998) 691-721. | MR | Zbl

[25] Morita T., Markov systems and transfer operators associated with cofinite Fuchsian groups, Ergodic Theory Dynam. Systems 17 (5) (1997) 1147-1181. | MR | Zbl

[26] Naud F., Precise asymptotics of the length spectrum for finite geometry Riemann surfaces, IMRN (2004), submitted for publication. | MR | Zbl

[27] Nicholls P.J., The Ergodic Theory of Discrete Groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. | MR | Zbl

[28] Parry W., Pollicott M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990) 268. | Numdam | MR | Zbl

[29] Patterson S.J., Perry P.A., The divisor of Selberg's zeta function for Kleinian groups, Duke Math. J. 106 (2) (2001) 321-390, Appendix A by Charles Epstein. | MR | Zbl

[30] Pollicott M., Error terms in “prime orbit theorems” for locally constant suspended flows, Quart. J. Math. Oxford Ser. (2) 41 (163) (1990) 313-323. | Zbl

[31] Pollicott M., Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. Math. 85 (2) (1991) 161-192. | MR | Zbl

[32] Pollicott M., Rocha A.C., A remarkable formula for the determinant of the Laplacian, Invent. Math. 130 (2) (1997) 399-414. | MR | Zbl

[33] Pollicott M., Sharp R., Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math. 120 (5) (1998) 1019-1042. | MR | Zbl

[34] Pollicott M., Sharp R., Error terms for closed orbits of hyperbolic flows, Ergodic Theory Dynam. Systems 21 (2) (2001) 545-562. | MR | Zbl

[35] Randol B., On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977) 241-247. | MR | Zbl

[36] Ratcliffe J.G., Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. | MR | Zbl

[37] Ruelle D., Flots qui ne mélangent pas exponentiellement, C. R. Acad. Sci. Paris Sér. I Math. 296 (4) (1983) 191-193. | MR | Zbl

[38] Ruelle D., An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math. 72 (1991) 175-193, 1990. | Numdam | MR | Zbl

[39] Sarnak P., Prime geodesic theorems, PhD thesis, Stanford University, 1980.

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

[41] Stoyanov L., Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math. 123 (4) (2001) 715-759. | MR | Zbl

[42] Strain J., Zworski M., Growth of the zeta functions for a quadratic map and the dimension of the julia set, Nonlinearity 17 (5) (2004) 1607-1622. | MR | Zbl

[43] Sullivan D., The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 171-202. | Numdam | MR | Zbl

[44] Sullivan D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (3-4) (1984) 259-277. | MR | Zbl

[45] Walters P., An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. | MR | Zbl

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