Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Mayer's transfer operator approach to Selberg's zeta function

Authors: A. Momeni and A. B. Venkov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 529-553
MSC (2010): Primary 11M36, 11M41
Published electronically: May 24, 2013
MathSciNet review: 3088005
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: These notes are based on three lectures given by the second author at Copenhagen University (October 2009) and at Aarhus University, Denmark (December 2009). Mostly, a survey of the results of Dieter Mayer on relationships between Selberg and Smale-Ruelle dynamical zeta functions is presented. In a special situation, the dynamical zeta function is defined for a geodesic flow on a hyperbolic plane quotient by an arithmetic cofinite discrete group. More precisely, the flow is defined for the corresponding unit tangent bundle. It turns out that the Selberg zeta function for this group can be expressed in terms of a Fredholm determinant of a classical transfer operator of the flow. The transfer operator is defined in a certain space of holomorphic functions, and its matrix representation in a natural basis is given in terms of the Riemann zeta function and the Euler gamma function.

References [Enhancements On Off] (What's this?)

  • 1. R. L. Adler and L. Flatto, Cross section map for the geodesic flow on the modular surface, Conference in Modern Analysis and Probability (New Haven, 1982), Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 9-24. MR 0737384 (85j:58128)
  • 2. R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. No. 50 (1979), 153-170. MR 0556585 (81b:58026)
  • 3. C.-H. Chang and D. Mayer, Thermodynamic formalism and Selberg's zeta function for modular groups, Regul. Chaotic Dyn. 5 (2000), 281-312. MR 1789478 (2001k:37045)
  • 4. I. Efrat, Dynamics of the continued fraction map and the spectral theory of $ \mathrm {SL}(2,\mathbb{Z})$, Invent. Math. 114 (1993), 207-218. MR 1235024 (94h:11052)
  • 5. I. M. Gel'fand and N. Ya. Vilenkin, Generalized functions. Vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961; English transl., Acad. Press, New York-London, 1964. MR 0146653 (26:4173); MR 0435834 (55:8786d)
  • 6. I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, sums, series and products, Fizmatgiz, Moscow, 1963; English transl., Acad. Press, New York-London, 1965. MR 0161996 (28:5198); MR 0197789 (33:5952)
  • 7. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 pp. MR 0075539 (17:763c)
  • 8. J. Lewis and D. Zagier, Period functions and the Selberg zeta function for the modular group, The Mathematical Beauty of Physics (Saclay, 1996), Adv. Ser. Math. Phys., vol. 24, World Sci., Singapore, 1997, pp. 83-97. MR 1490850 (99c:11108)
  • 9. D. Mayer, The thermodynamic formalism approach to Selberg's zeta function for $ \mathrm {PSL}(2,\mathbb{Z})$, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 55-60. MR 1080004 (91j:58130)
  • 10. -, On a zeta function related to the continued fraction transformation, Bull. Soc. Math. France 104 (1976), 195-203. MR 0418168 (54:6210)
  • 11. -, Continued fractions and related transformations, Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, 1989), Oxford Univ. Press, New York, 1991, pp. 175-222. MR 1130177
  • 12. -, Thermodynamics formalism and quantum mechanics on the modular surface, From Phase Transitions to Chaos, World Sci. Publ., River Edge, NJ, 1992, pp. 521-529. MR 1172552 (93h:58121)
  • 13. -, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys. 130 (1990), 311-333. MR 1059321 (91g:58216)
  • 14. D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc. 49 (2002), 887-895. MR 1920859 (2003d:37026)
  • 15. -, Dynamical zeta functions for piecewise monotone maps of the interval, CRM Monogr. Ser., vol. 4, Amer. Math. Soc., Providence, RI, 1994. MR 1274046 (95m:58101)
  • 16. H. H. Schaefer, Topological vector spaces, The Macmillan Co., New York, 1966. MR 0193469 (33:1689)
  • 17. 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)
  • 18. Ya. G. Sinaĭ, Gibbs measures in ergodic theory, Uspekhi Mat. Nauk 27 (1972), no. 4, 21-64; English transl., Russian Math. Surveys 27 (1972), no. 4, 21-69. MR 0399421 (53:3265)
  • 19. A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172 pp.; English transl., Math. Appl. (Soviet Ser.), vol. 51, Kluwer Acad. Publ. Group, Dordrecht, 1990. MR 0665585 (85j:11060a); MR 1135112 (93a:11046)
  • 20. -, The automorphic scattering matrix for the Hecke group $ \Gamma [2\cos (\pi /q)]$, Internat. Conf. on Analytical Methods in Number Theory and Analysis (Moscow, 1981), Trudy Mat. Inst. Steklov. 163 (1984), 32-36; English transl. in Proc. Steklov Inst. Math. 1985, no. 4 (163). MR 0769866 (86e:11041)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 11M36, 11M41

Retrieve articles in all journals with MSC (2010): 11M36, 11M41

Additional Information

A. Momeni
Affiliation: Department of Statistical Physics and Nonlinear Dynamics, Institute of Theoretical Physics, Clausthal University of Technology, 38678, Clausthal-Zellerfeld, Germany

A. B. Venkov
Affiliation: Institute for Mathematics and Centre of Quantum Geometry QGM, University of Aarhus, 8000, Aarhus C, Denmark

Keywords: Mayer's transfer operator, Selberg's zeta function
Received by editor(s): September 22, 2011
Published electronically: May 24, 2013
Additional Notes: The authors would like to thank Dieter Mayer for several important remarks and we would like to say also that all possible mistakes in the text belong to us but not to Mayer’s theory we presented in this paper. This work was supported by DAAD, the International Center of TU Clausthal, and the Danish National Research Foundation Center of Excellence, Center for Quantum Geometry of Moduli Spaces(QGM)
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society