Zeta functions and the periodic orbit structure of hyperbolic dynamics
Astérisque, no. 187-188 (1990) , 272 p.
@book{AST_1990__187-188__1_0,
     author = {Parry, William and Pollicott, Mark},
     title = {Zeta functions and the periodic orbit structure of hyperbolic dynamics},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {187-188},
     year = {1990},
     zbl = {0726.58003},
     language = {en},
     url = {http://www.numdam.org/item/AST_1990__187-188__1_0/}
}
TY  - BOOK
AU  - Parry, William
AU  - Pollicott, Mark
TI  - Zeta functions and the periodic orbit structure of hyperbolic dynamics
T3  - Astérisque
PY  - 1990
IS  - 187-188
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_1990__187-188__1_0/
LA  - en
ID  - AST_1990__187-188__1_0
ER  - 
%0 Book
%A Parry, William
%A Pollicott, Mark
%T Zeta functions and the periodic orbit structure of hyperbolic dynamics
%S Astérisque
%D 1990
%N 187-188
%I Société mathématique de France
%U http://www.numdam.org/item/AST_1990__187-188__1_0/
%G en
%F AST_1990__187-188__1_0
Parry, William; Pollicott, Mark. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque, no. 187-188 (1990), 272 p. http://numdam.org/item/AST_1990__187-188__1_0/

[1] L. M. Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk. SSSR, 128 (1959), 647-650. | Zbl

[2] L. M. Abramov, On the entropy of a flow, Dokl. Akad. Nauk. SSSR, 128 (1959), 873-875. | Zbl

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

[4] R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Research report, I.B.M., Yorktown Heights. | Zbl | DOI

[5] D. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math., 90 (1967), 1-235. | MR | Zbl

[6] D. Anosov and Y. G. Sinai, Some smooth ergodic systems, Russian Math. Surv., 22 (1967), 103-167. | Zbl | MR | DOI

[7] E. Artin and B. Mazur, On periodic points, Annals of Math., 81 (1965), 82-99. | MR | Zbl | DOI

[8] R. Bhatia and K. R. Parthasarathy, Lectures on functional analysis, I, MacMillan, Delhi, 1977. | MR | Zbl

[9] R. Bowen and O. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Symp. Pure Math., 14 (1970), 43-50. | MR | Zbl | DOI

[10] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747. | MR | Zbl | DOI

[11] R. Bowen, One-dimensional hyperbolic sets for flows, J. Diff. Equations, 12 (1972), 173-179. | MR | Zbl | DOI

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

[13] R. Bowen and P. Walters, Expansive one-parameter flows, J. Diff. Eq., 12 (1972), 180-193. | MR | Zbl | DOI

[14] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30. | MR | Zbl | DOI

[15] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math., 95 (1973), 429-460. | MR | Zbl | DOI

[16] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, SLN 470, Springer, Berlin, 1975. | MR | Zbl

[17] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202. | MR | Zbl | EuDML | DOI

[18] R. Bowen, On Axiom A diffeomorphisms, Reg. Conf. Series 35, Amer. Math. Soc., 1977. | Zbl | MR

[19] R. Bowen and C. Series, Markov maps associated to Fuchsian groups, Pub. Math. (IHES), 50 (1979), 153-170. | Zbl | MR | EuDML | Numdam | DOI

[20] M. Brin, Ergodic theory of frame flows. In Ergodic theory and dynamical systems, II (ed. A. Katok), Birkhauser Basel, 1982. | MR | Zbl | DOI

[21] F. Browder, On the spectral theory of elliptic differential operators, I, Math. Ann., 142 (1961), 22-130. | EuDML | Zbl | MR | DOI

[22] J. W. S. Cassels and A. Fröhlich, Algebraic number theory, Academic, London, 1967. | Zbl | MR

[23] Z. Coelho and W. Parry, Central limit asymptotics for subshifts of finite type, Israel J. Math., 69, no. 2 (1990), 235-249. | MR | Zbl | DOI

[24] C. Corduneau, Almost periodic functions, Interscience, New York, 1968. | MR | Zbl

[25] I. P. Cornfield, S. V. Fomin and Y. G. Sinai, Ergodic theory, Springer, Berlin, 1983. | Zbl | MR

[26] H. Delange, Généralisation du théorème de Ikehara, Ann. Sci. E.N.S., 71 (1954), 213-242. | MR | Zbl | EuDML | Numdam

[27] M. Denker and W. Phillip, Approximation by Brownian motion for Gibbs measures and flows under a function, Ergod. Th. and Dynam. Sys., 4 (1984), 541-552. | MR | Zbl | DOI

[28] M. Denker, C. Grillenburger and K. Sigmund, Ergodic theory on compact spaces, SLN 527, Springer, Berlin, 1976. | MR | Zbl

[29] P. Eberlein, Geodesic flows on negatively curved manifolds, I, Annals of Math., 95(1972), 492-510. | MR | Zbl | DOI

[30] W. &Amp; F. Ellison, Prime numbers, Herman, Paris, 1975. | Zbl

[31] W. Feller, An introduction to probability theory and its applications, vol. 2, New York, J. Wiley & Sons, 1971. | Zbl

[32] J. Franks, Homology and dynamical systems, Reg. Conf. Series, 49, Amer. Math. Soc., 1982. | Zbl | MR

[33] D. Fried, Rationality for isolated expansive sets, Adv. Math., 65 (1987), 35-38. | MR | Zbl | DOI

[34] G. Gallavotti, Funzioni zeta ed insiemi basilas, Accad. Lincei, Rend. Sc. fismat. e nat., 61 (1976), 309-317. | MR | Zbl

[35] F. Gantmacher, The theory of matrices, vol. II, Chelsea, New York, 1974. | MR | Zbl

[35*] Y. Guivarc'H, Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés, Ergod. Th. and Dynam. Sys., 9 (1989), 433-453. | MR | Zbl | DOI

[35**] Y. Guivarc'H and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphisms d'Anosov, Annales de l'Inst. H. Poincaré, 24 (1) (1988), 73-98. | MR | Zbl | EuDML | Numdam

[36] N. T. A. Haydn, Meromorphic extension of the zeta function for Axiom A flows, Ergod. Th. and Dynam. Sys., 10 (1990), 347-360. | MR | Zbl

[37] G. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc., 45 (1939), 241-246. | JFM | MR | DOI

[38] D. Hejhal, The Selberg trace formula for PSL(2,), SLN 548, Springer, Berlin, 1976. | MR | Zbl

[39] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, SLN 583, Springer, Berlin, 1977. | MR | Zbl

[40] F. Hofbauer and G. Keller, Zeta-functions and transfer operators for piecewise linear transformaltions, J. Reine Angew. Math., 352 (1984), 100-113. | MR | Zbl | EuDML

[41] E. Hopf, Ergodentheorie, Springer, Berlin, 1937. | JFM | Zbl

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

[43] C. Ionescu-Tulcea and G. Marinescu, Théorie ergodique pour les classes d'opérations non completement continues, Annals Math., 52 (1950), 140-147. | MR | Zbl | DOI

[44] T. Kato, Perturbation theory of linear operators, Springer, Berlin, 1966. | MR | Zbl

[45] A. Katsuda and T. Sunada, Dynamical L-functions and homology of closed orbits, Bull. Amer. Math. Soc., 20, no. 1 (1989), 73-77. | MR | Zbl | DOI

[46] A. Katsuda and T. Sunada, Closed orbits in homology classes (Preprint). | MR | Zbl | Numdam | DOI

[47] A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math., 109 (1987), 145-156. | Zbl | MR

[48] Y. Katznelson, An introduction to harmonic analysis, J. Wiley & Sons, New York, 1968. | MR | Zbl

[49] M. Keane, Strongly mixing g-measures, Invent. Math., 16 (1972), 309-324. | MR | Zbl | EuDML | DOI

[50] G. Keller, Un théorème de la limite centrale pour une classes de transformations monotone par morceaux, CRAS, A291 (1980), 155-158. | MR | Zbl

[51] G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193. | MR | Zbl | DOI

[52] S. Lalley, Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math., 8 (1987), 154-193. | MR | Zbl | DOI

[53] O. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Commun. Math. Phys., 13 (1969), 194-215. | MR | DOI

[54] S. Lang, Algebraic number theory, Addison-Wesley, 1970. | MR | Zbl

[55] F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Z. Wahr. Verw. Geb., 30 (1974), 185-202. | MR | Zbl | DOI

[56] A. Livsic, Cohomology properties of dynamical systems, Math. USSR-Izv., 6 (1972), 1278-1301. | Zbl | MR | DOI

[57] G. Margulis, On some applications of ergodic theory to the study of manifolds of negative curvature, Func. Anal. App., 3 (1969), 89-90. | Zbl | MR

[58] M. Misiurewicz, A short proof of the variational principle for a Z + N action on a compact space, Asterisque, 40 (1976), 147-187. | Numdam | Zbl | MR

[59] M. Morse, Symbolic dynamics, Mimeographed notes, Princeton, 1966. | Zbl

[60] R. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. | MR | Zbl | DOI

[61] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-65. | MR | Zbl | DOI

[62] W. Parry, Topics in ergodic theory, C.U.P., Cambridge, 1981. | MR | Zbl

[63] W. Parry and S. Tuncel, Classification problems in ergodic theory, C.U.P., Cambridge, 1982. | MR | Zbl | DOI

[64] W. Parry and S. Tuncel, On the stochastic and topological structure of Markov chains, Bull. London Math. Soc., 14 (1982), 16-27. | MR | Zbl | DOI

[65] 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), 41-52. | MR | Zbl | DOI

[66] W. Parry and M. Pollicott, An analogue of the prime number theorem and closed orbits of Axiom A flows, Annals of Math., 118 (1983), 573-591. | MR | Zbl | DOI

[67] W. Parry, Bowen's equidistribution theory and the Dirichlet density theorem, Ergod. Th. and Dynam. Sys., 4 (1984), 117-134. | MR | Zbl | DOI

[68] W. Parry and M. Pollicott, The Chebotarev theorem for Galois coverings of Axiom A flows, Ergod. Th. and Dynam. Sys., 6 (1986), 133-148. | MR | Zbl | DOI

[69] W. Parry, Sychronisation of canonical measures for hyperbolic attractors, Commun. Math. Phys., 106 (1986), 267-275. | MR | Zbl | DOI

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

[71] M. Pollicott, A complex Ruelle operator theorem and two counter examples, Ergod. Th.and Dynam. Sys., 4 (1984), 135-146. | Zbl | MR | DOI

[72] M. Pollicott, Meromorphic extensions for generalised zeta functions, Invent Math., 85 (1986). | MR | Zbl | EuDML | DOI

[73] M. Ratner, Markov partitions for Anosov flows on 3-dimensional manifolds, Mat. Zam., 6 (1969), 693-704. | Zbl | MR

[74] M. Ratner, The central limit theorem for y-flows on three dimensional manifolds, Dokl. Akad. Nauk. SSSR, 186 (1969), 519-521. | MR | Zbl

[75] M. Ratner, The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math., 16 (1973), 181-197. | MR | Zbl | DOI

[76] S. M. Rees, Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergod. Th. and Dynam. Sys., 1 (1981), 107-133. | MR | Zbl

[76*] J. Rousseau-Egelé, Un théorème de la limite locale pour une classe de transformations dilatantes, Ann. of Probability, 11, no. 3 (1983), 772-788. | Zbl | MR | DOI

[77] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Commun. Math. Phys., 9 (1968), 267-278. | MR | Zbl | DOI

[78] D. Ruelle, Statistical mechanics on a compact set with action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 185 (1973), 237-253. | MR | Zbl | DOI

[79] D. Ruelle, Zeta functions for expanding maps and Anosov flows, Invent. Math., 34 (1978), 231-242. | MR | EuDML | Zbl | DOI

[80] D. Ruelle, Generalised zeta functions for Axiom A basic sets, Bull. Amer. Math. Soc., 82 (1976), 153-156. | MR | Zbl | DOI

[81] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654. | MR | Zbl | DOI

[82] D. Ruelle, Thermodynamic formalism, Addison-Wesley, 1978. | Zbl | MR

[83] D. Ruelle, Flows which do not exponentially mix, C.R.A.S., 296 (1983), 191-194. | MR | Zbl

[84] P. Sarnak, Prime geodesic theorems, Ph.D. thesis, Stanford, 1980. | MR

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

[86] E. Seneta, Non-negative matrices, George Allen and Unwin, London, 1973. | MR | Zbl

[87] C. Series, Symbolic dynamics for geodesic flows, Acta Math., 146 (1981), 103-128. | MR | Zbl | DOI

[88] C. Series, The infinite word problem and limit sets of Fuchsian groups, Ergod. Th. and Dynam. Sys., 1 (1981), 337-360. | MR | Zbl | DOI

[89] C. Series, Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Erg. Th. and Dynam. Sys., 6 (1986), 601-625. | MR | Zbl

[90] M. Shub, Global stability of dynamical systems, Springer, Berlin, 1989. | Zbl | MR

[91] Y. G. Sinai, The central limit theorem for geodesic flows on manifolds of constant negative curvature, Dokl. Akad. Nauk. SSSR, 133 (1960), 1303-1306. | MR | Zbl

[92] Y. G. Sinai, The construction of Markov partitions, Fun. Anal. Appl., 2 (1968), 70-80. | MR | Zbl

[93] Y. G. Sinai, The asymptotic behaviour of the number of closed geodesics on a compact manifold of negative curvature, Transl. A.M.S., 73 (1968), 227-250. | Zbl | MR

[94] Y. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surv., 27 (1972), 21-70. | Zbl | MR | DOI

[95] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1976), 747-817. | MR | Zbl | DOI

[96] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Pub. Math. (IHES), 50 (1979), 171-209. | MR | Zbl | EuDML | Numdam | DOI

[97] D. Sullivan, Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc., 6 (1982), 57-73. | MR | Zbl | DOI

[98] T. Sunada, Geodesic flows and random walks, Adv. Stud. Pure Math., 3 (1984), 47-85. | MR | Zbl | DOI

[99] F. Tangerman, Ph. D. thesis, Boston University, 1988.

[100] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. | MR | Zbl

[101] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1976), 937-971. | MR | Zbl | DOI

[102] P. Walters, An introduction to ergodic theory, G.T.M. 79, Springer, Berlin, 1982. | MR | Zbl

[103] N. Wiener, The Fourier integral and certain of its applications, C.U.P., Cambridge, 1967. | Zbl | MR | JFM