The thermodynamic formalism approach to Selberg’s zeta function for 𝑃𝑆𝐿(2,𝐙)

Mayer, Dieter H.

doi:10.1090/S0273-0979-1991-16023-4

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-9485(online) ISSN 0273-0979(print)

The thermodynamic formalism approach to Selberg's zeta function for ${\text{PSL}}\left( {2,\mathbf{Z}} \right)$

Author: Dieter H. Mayer
Journal: Bull. Amer. Math. Soc. 25 (1991), 55-60
MSC (1985): Primary 58F20, 58F25; Secondary 11F72, 11M26
MathSciNet review: 1080004
Full-text PDF

Roy L. Adler and Leopold Flatto, Cross section map for the geodesic flow on the modular surface, Conference in modern analysis and probability (New Haven, Conn., 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 9–24. MR 737384 (85j:58128), http://dx.doi.org/10.1090/conm/026/737384
Emil Artin, The collected papers of Emil Artin, Edited by Serge Lang and John T. Tate, Addison–Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. MR 0176888 (31 #1159)
Rufus Bowen and Caroline Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 153–170. MR 556585 (81b:58026)
David Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 491–517. MR 875085 (88k:58134)
A. Grothendieck, La théorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319–384 (French). MR 0088665 (19,558d)
Dieter H. Mayer, The Ruelle-Araki transfer operator in classical statistical mechanics, Lecture Notes in Physics, vol. 123, Springer-Verlag, Berlin, 1980. MR 576928 (84m:82006)
Dieter H. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys. 130 (1990), no. 2, 311–333. MR 1059321 (91g:58216)
Dieter H. Mayer, Continued fractions and related transformations, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 175–222. MR 1130177
William Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions, Israel J. Math. 45 (1983), no. 1, 41–52. MR 710244 (85c:58089), http://dx.doi.org/10.1007/BF02760669
Mark Pollicott, Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France 114 (1986), no. 4, 431–446. MR 882589 (88j:58102)
Mark Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. Math. 85 (1991), no. 2, 161–192. MR 1093004 (92f:58136), http://dx.doi.org/10.1016/0001-8708(91)90054-B
D. Ruelle, Zeta functions and statistical mechanics, (Rennes, 1975) Soc. Math. France, Paris, 1976, pp. 167–176. Astérisque, No. 40. MR 0458497 (56 #16697)
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655 (80g:82017)
Peter Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), no. 1, 113–120. MR 885573 (89e:58116)
A. Selberg, 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 0088511 (19,531g)
Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563 (87c:58094), http://dx.doi.org/10.1112/jlms/s2-31.1.69
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014 (37 #3598), http://dx.doi.org/10.1090/S0002-9904-1967-11798-1
A. B. Venkov, Spectral theory of automorphic functions, the Selberg zeta function and some problems of analytic number theory and mathematical physics, Uspekhi Mat. Nauk 34 (1979), no. 3(207), 69–135, 248 (Russian). MR 542238 (81c:10034)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|