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Selberg Zeta Functions and Transfer Operators

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2139))

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Abstract

In recent years the application of the transfer operator method in the study of Selberg zeta functions and the spectral theory of hyperbolic spaces has made significant progress, in both analytical investigations and numerical investigations. We consider transfer operators for the geodesic flow on surfaces of constant negative curvature, therefore systems where a particle is moving freely on such a surface with constant velocity.

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References

  1. Adler, R., Flatto, L.: Cross section map for the geodesic flow on the modular surface. Contemp. Math. 26, 9–24 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Artin, E.: Ein mechanisches System mit quasi-ergodischen Bahnen. Abh. Math. Sem. d. Hamburgischen Universität 3, 170–175 (1924) (Also in Collected Papers, Addison-Wesley, pp. 499–501 (1965))

    Google Scholar 

  3. Atkin, A.O.L., Lehner, J.: Hecke operators on Γ 0(m). Math. Ann. 185 (2), 134–160 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Avelin, H.: Deformation of Γ 0(5)-cusp forms. Math. Comput. 76, 361–384 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Avelin, H.: Computations of Eisenstein series on Fuchsian groups. Math. Comput. 77, 1779–1800 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baladi, V.: Positive Transfer Operators and Decay of Correlations. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  7. Banach, S.: Théorie des opérations linéaires. Ed. S. Banach, B. Knaster, K. Kuratowski, S. Mazurkiewicz, W. Sierpiński i H. Steinhaus, Monografie Matematyczne, vol. 1. Instytut Matematyczny Polskiej Akademi Nauk (1932)

    Google Scholar 

  8. Booker, A., Strömbergsson, A., Venkatesh, A.: Effective computation of Maass cusp forms. IMRN 2006, 1–34 (2006)

    MATH  Google Scholar 

  9. Borthwick, D.: Distribution of resonances for hyperbolic surfaces. Exp. Math. 23 (1), 25–45 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bowen, R.: Symbolic dynamics for hyperbolic flows. Am. J. Math. 95 (2), 429–460 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bowen, R., Chazottes, J. (eds.) Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd edn. Springer, Berlin/Heidelberg (2008)

    MATH  Google Scholar 

  12. Bowen, R., Ruelle, D.: The Ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. Publications Mathématiques de L’IHÉS 50 (1), 153–170 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bruggeman, R., Fraczek, M., Mayer, D.: Perturbation of zeros of the Selberg zeta-function for Γ 0(4). Exp. Math. 22 (3), 217–252 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bruggeman, R., Lewis, J., Zagier, D.: Function theory related to the group \(\mathrm{PSL}\!\left (2, \mathbb{R}\right )\). Dev. Math. 28, 107–201 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Bruggeman, R., Lewis, J., Zagier, D.: Period Functions for Maass Forms and Cohomology. Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  17. Chang, C.H., Mayer, D.: The transfer operator approach to Selberg’s zeta function and modular and Maass wave forms for \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). In: Hejhal, D., Gutzwiller, M., et al. (eds.) Emerging Applications of Number Theory, pp. 72–142. Springer, New York (1999)

    Google Scholar 

  18. Chang, C.H., Mayer, D.: An extension of the thermodynamic formalism approach to Selberg’s zeta function for general modular groups. In: Fiedler, B. (eds.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 523–562. Springer, Berlin/New York (2001)

    Chapter  Google Scholar 

  19. Cornfeld, I., Fomin, S., Sinai, Y.: Ergodic Theory. Grundlehren der mathematischen Wissenschaften, vol. 245. Springer, New York (1982)

    Google Scholar 

  20. Deshouillers, J., Iwaniec, H., Phillips, R., Sarnak, P.: Maass cusp forms. Proc. Natl. Acad. Sci. USA 82, 3533–3534 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Devaney, R.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Westview Press, Cambridge/Boulder (2003)

    MATH  Google Scholar 

  22. Diu, B., Guthmann, C., Lederer, D., Roulet, B.: Grundlagen der Statistischen Physik. de Gruyter (1994)

    Google Scholar 

  23. Eckmann, J., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Efrat, I.: Dynamics of the continued fraction map and the spectral theory of \(\mathrm{SL}\!\left (2, \mathbb{Z}\right )\). Invent. math. 114, 207–218 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Farmer, D., Lemurell, S.: Deformations of Maass forms. Math. Comput. 74 (252), 1967–1982 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fraczek, M.: Spezielle Eigenfunktionen des Transfer-Operators für Hecke Kongruenz Untergruppen. Diploma Thesis, Clausthal University (2006)

    Google Scholar 

  27. Fraczek, M., Mayer, D.: Symmetries of the transfer operator for Γ 0(n) and a character deformation of the Selberg zeta function for Γ 0(4). Algebra Number Theory 6 (3), 587–610 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fraczek, M., Mayer, D., Mühlenbruch, T.: A realization of the Hecke algebra on the space of period functions for Γ 0(n). J. reine angew. Math. 603, 133–163 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Fried, D.: The zeta functions of Ruelle and Selberg. Ann. Sci. Ec. Norm. Sup. 19, 491–517 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fried, D.: Symbolic dynamics for triangle groups. Invent. Math. 125, 487–521 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gottlieb, D., Shu, C.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39 (4), 644–668 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Guillope, L., Lin, K., Zworski, M.: The Selberg zeta function for convex co-compact Schottky groups. Commun. Math. Phys. 245 (1), 149–175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hadamard, J.: Les surfaces à courbures opposées et leurs lignes géodésique. J. Math. Pures Appl. 4, 27–73 (1898)

    MATH  Google Scholar 

  34. Hejhal, D.: Eigenvalues of the Laplacian for Hecke triangle groups. Mem. Am. Math. Soc. 97 (496) (1992). http://www.ams.org/books/memo/0469/

  35. Hejhal, D.: On the calculation of Maass cusp forms. In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 175–186. Cambridge University Press, Cambridge, UK (2011)

    Chapter  Google Scholar 

  36. Hilgert, J., Mayer, D., Movasati, H.: Transfer operators for Γ 0(n) and the Hecke operators for period functions of \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). Math. Proc. Camb. Philos. Soc. 139, 81–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hove, L.V.: Sur L’intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension. Physica 16 (2), 137–143 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31 (1), 253–258 (1925)

    Google Scholar 

  39. Iwaniec, H.: Spectral Methods of Automorphic Forms. American Mathematical Society, Providence (2002)

    Book  MATH  Google Scholar 

  40. Judge, C.: On the existence of maass cusp forms on hyperbolic surfaces with cone points. J. Am. Math. Soc. 8 (3), 715–759 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kac, M.: On the partition function of a one-dimensinal lattice gas. Phys. Fluids 2, 8–12 (1959)

    Article  MATH  Google Scholar 

  42. Lasota, A., Mackey, M.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Springer, New York (1994)

    Book  MATH  Google Scholar 

  43. Lenz, W.: Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. Physikalische Zeitschrift 21, 613–615 (1920)

    Google Scholar 

  44. Lewis, J., Zagier, D.: Period functions for Maass wave forms, I. Ann. Math. 153, 191–258 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lieb, E., Mattis, D.: Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles. Perspectives in Physics: A Series of Reprint Collections. Academic Press, New York (1966)

    Google Scholar 

  46. Manin, Y., Marcolli, M.: Continued fractions, modular symbols and non commutative geometry. Selecta Math. (N.S.) 8, 475–520 (2002)

    Google Scholar 

  47. Margulis, G.: On Some Aspects of the Theory of Anosov Systems. With a Survey by Richard Sharp: Periodic Orbits of Hyperbolic Flows. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg (2004)

    Google Scholar 

  48. Matthies, C., Steiner, F.: Selberg’s ζ function and the quantization of chaos. Phys. Rev. A 44 (12), R7877–R7880 (1991)

    Article  MathSciNet  Google Scholar 

  49. Mayer, D.: The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Physics, vol. 123. Springer, Berlin/Heidelberg (1980)

    Google Scholar 

  50. Mayer, D.: On the thermodynamic formalism for the Gauss map. Commun. Math. Phys. 130 (2), 311–333 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mayer, D.: Continued fractions and related transformations, chap. 7 In: Bedford, T., et al. (eds.) Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, pp. 175–222. Oxford University Press, Oxford/New York (1991)

    Google Scholar 

  52. Mayer, D.: The thermodynamic formalism approach to Selberg’s zeta function for \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). Bull. Am. Math. Soc. (N.S.) 25 (1), 55–60 (1991)

    Google Scholar 

  53. Mayer, D.: Transfer operators, the Selberg zeta function and the Lewis-Zagier theory of period functions. In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 143–174. Cambridge University Press, Cambridge, UK (2011)

    Chapter  Google Scholar 

  54. Mayer, D., Mühlenbruch, T., Strömberg, F.: The transfer operator for Heck triangle groups. In: Discrete and Continuous Dynamical Systems - Series A, vol. 32, number 7, pp. 2453–2484 (2012)

    Google Scholar 

  55. Mayer, D., Strömberg, F.: Symbolic dynamics for the geodesic flow on Hecke surfaces. J. Mod. Dyn. 2 (4), 581–627 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Möller, M., Pohl, A.: Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant. Ergod. Theory Dyn. Syst. 33 (1), 247–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Morita, T.: Markov systems and transfer operators associated with cofinite Fuchsian groups. Ergod. Theory Dyn. Syst. 17 (5), 1147–1181 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. Mühlenbruch, T.: Hecke operators on period functions for Γ 0(n). J. Number Theory 118, 208–235 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  59. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)

    MathSciNet  MATH  Google Scholar 

  60. Petridis, Y., Risager, M.: Dissolving cusp forms: higher order Fermi’s golden rules. Mathematika 59, 269–301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  61. Phillips, R., Sarnak, P.: On cusp forms for co-finite subgroups of \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). Invent. Math. 80, 339–364 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  62. Phillips, R., Sarnak, P.: The spectrum of fermat curves. Geom. Funct. Anal. 1, 80–146 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  63. Phillips, R., Sarnak, P.: Cusp forms for character varieties. Geom. Funct. Anal. 4, 93–118 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  64. Pollicott, M.: Some applications of thermodynamic formalism to manifolds with constant negative curvature. Adv. Math. 85 (2), 161–192 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ratner, M.: Markov partitions for Anosov flows on 3-dimensional manifolds. Mat. Zam. 6, 693–704 (1969)

    Google Scholar 

  66. Ratner, M.: Markov partitions for Anosov flows on n-dimensional manifolds. Israel J. Math. 6, 92–114 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  67. Ruelle, D.: Generalized zeta-functions for Axiom A basic sets. Bull. Am. Math. Soc. 82 (1), 153–156 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  68. Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Inventiones Mathematicae 34 (3), 231–242 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  69. Ruelle, D.: Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge/New York (2004)

    Book  MATH  Google Scholar 

  70. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956)

    MathSciNet  MATH  Google Scholar 

  71. Selberg, A.: Atle Selberg Collected Papers, vol. 1, chap. Harmonic Analysis (Göttingen lectures), 1954, pp. 626–674. Springer, Springer, Berlin/Heidelberg (1989)

    Google Scholar 

  72. Selberg, A.: Remarks on the distribution of poles of Eisenstein series. In: Festschrift in Honor of I.I. Piatetski-Shapiro, vol. 2, pp. 251–278 (1990) (Also in Collected Papers, vol. 2, pp. 15–45. Springer, Springer, Berlin/Heidelberg (1991))

    Google Scholar 

  73. Series, C.: The modular surface and continued fractions. J. Lond. Math. Soc. (2) 31, 69–80 (1985)

    Google Scholar 

  74. Sinai, Y.: Classical dynamical systems with countably-multiple Lebesque spectrum. Izv. Akad. Nauk SSSR Ser. Mat. 30, 15–68 (1966)

    MathSciNet  Google Scholar 

  75. Sinai, Y.: Introduction to Ergodic Theory. Princeton University Press, Princeton (1977)

    Google Scholar 

  76. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73 (6), 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  77. Strömberg, F.: Computation of Selberg Zeta Functions on Hecke Triangle Groups. arXiv:0804.4837v1 (2008, preprint)

    Google Scholar 

  78. Strömberg, F.: Maass Waveforms on (Γ 0(N), χ) (Computational Aspects). In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 187–228. Cambridge University Press, Cambridge, UK (2011)

    Chapter  Google Scholar 

  79. Then, H.: Maass cusp forms for large eigenvalues. Math. Comput. 74 (249), 363–381 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  80. Winkler, A.: Cusp forms and Hecke groups. Journal für die reine und angewandte Mathematik (Crelles Journal) (386), 187–204 (1988)

    Google Scholar 

  81. Wolpert, S.: Disappearance of cusp forms in special families. Ann. Math. 139 (2), 239–291 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Mathematics Institute, University of Warwick, Coventry, UK

    Markus Szymon Fraczek

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Fraczek, M.S. (2017). Introduction. In: Selberg Zeta Functions and Transfer Operators. Lecture Notes in Mathematics, vol 2139. Springer, Cham. https://doi.org/10.1007/978-3-319-51296-9_1

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