Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-669899f699-b58lm Total loading time: 0 Render date: 2025-04-30T08:50:11.203Z Has data issue: false hasContentIssue false

IV - Transfer Operators, the Selberg Zeta Function and the Lewis-Zagier Theory of Period Functions

Published online by Cambridge University Press:  05 January 2012

By
Dieter H. Mayer
Edited by
Jens Bolte  and
Frank Steiner
Dieter H. Mayer
Affiliation:
Institut für Theoretische Physik, TU-Clausthal
Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
Get access

Summary

Summary. In these lectures we discuss the transfer operator approach to the Selberg zeta function for the geodesic flow on the unit tangent bundle of a modular surface Γ\ℍ. Thereby Γ denotes a subgroup of the full modular group SL(2,ℤ) of finite index and ℍ is the hyperbolic plane. It turns out that this function can be expressed in terms of the Fredholm determinant of the classical transfer operator ℒβ of this flow when appropriately extended to the complex “temperature”-planeβ.

The work of J. Lewis and D. Zagier, respectively our own work for the full modular group SL(2,ℤ) has shown that the eigenfunctions of this transfer operator at those β-values which belong to the zeroes of the Selberg function are closely related to automorphic forms for this group, both holomorphic and real analytic. For negative integer values of β they coincide with the period polynomials of Eichler, Shimura and Manin which justifies to call them quite generally period functions.

Of special interest are the eigenfunctions of the transfer operator ℒβ for β-values on the critical line ℜβ = ½. They are related to the Maass wave forms, that means the eigenfunctions of the hyperbolic Laplacian on the surface. In the language of quantum mechanics these functions are the eigenstates of the Schrödinger operator for a free particle moving on such a surface with constant negative curvature.

Type
Chapter
Information
Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology , pp. 143 - 174
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

1. AtkinA. O., L., Lehner, J.: Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134–160.CrossRefGoogle Scholar
2. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16. World Scientific, River Edge, NJ, (2000).Google Scholar
3. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470 (1975).CrossRefGoogle Scholar
4. Chang, C.-H., Mayer, D.: Thermodynamic formalism and Selberg's zeta function for modular groups. Regul. Chaotic Dyn. 5 (2000), no. 3, 281–312.CrossRefGoogle Scholar
5. Chang, C.-H., Mayer, D.: Eigenfunctions of the transfer operators and the period functions for modular groups. In “Dynamical, spectral, and arithmetic zeta functions” (San Antonio, TX, 1999). Contemp. Math. 290, pp. 1–40, Amer. Math. Soc., Providence, RI, 2001.Google Scholar
6. Chang, C.H., Mayer, D.: The period function of the nonholomorphic Eisenstein series for PSL(2, ℝ). Math. Phys. Elec. J. 4(6) (1998).Google Scholar
7. Chang, C.H., Mayer, D.: The transfer operator approach to Selberg's zeta function and modular Maass wave forms for PSL(2, ℝ). In Emerging Applications of Number Theory, eds. D., Hejhal, M., Gutzwiller et al., IMA Volumes 109 (1999) 72–142.CrossRefGoogle Scholar
8. Choie, Y., Zagier, D.: Rational period functions for PSL(2, ℝ). In “A tribute to Emil Grosswald: number theory and related analysis”. Contemp. Math. 143, pp. 89–108 Amer. Math. Soc., Providence, RI, 1993.Google Scholar
9. Connes, A.: Formule de trace en geométrié non-commutative et hypothèse de Riemann. C. R. Acad. Sci. Paris 323 Serie I (1996) 1231–1236.Google Scholar
10. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer VerlagBerlin (1993).CrossRefGoogle Scholar
11. Deitmar, A., Hilgert, J.: A Lewis correspondence for submodular groups. Forum Math. 19 (2007) 1075–1099.CrossRefGoogle Scholar
12. Eichler, M.: Eine Verallgemeinerung der Abelschen Integrale. Math. Zeitschrift 67 (1957) 267–298.CrossRefGoogle Scholar
13. Fraczek, M., Mayer, D., Mühlenbruch, T.: A realization of the Hecke algebra on the space of period functions for Γ0(n). Crelle's J. Reine Angew. Math. 603 (2007) 133–166.Google Scholar
14. Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16(1955).Google Scholar
15. Hilgert, J., Mayer, D., Movasati, H.: Transfer operators for Γ0(n) and the Hecke operators for the period functions of PSL(2, ℝ). Math. Proc. Camb. Phil. Soc. 139 (2005) 81–116.CrossRefGoogle Scholar
16. Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31 (1925) 253–258.CrossRefGoogle Scholar
17. Lewis, J.B.: Spaces of holomorphic functions equivalent to the even Maass cusp forms. Invent. Math. 127 (1997), no. 2, 271–306.CrossRefGoogle Scholar
18. Lewis, J.B.; Zagier, D.: Period functions and the Selberg zeta function for the modular group. In “The mathematical beauty of physics” (Saclay, 1996), Adv. Ser. Math. Phys. 24, pp. 83–97. World Sci. Publishing, River Edge, NJ, 1997.Google Scholar
19. Lewis, J.B.; Zagier, D.: Period functions for Maass wave forms. I. Ann. of Math. (2) 153 (2001), no. 1, 191–258.Google Scholar
20. Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Annalen 121 (1949) 141–183.CrossRefGoogle Scholar
21. Manin, Yu.I., Marcolli, M.: Continued fractions, modular symbols, and non-commutative geometry. Selecta Mathematica (New Series). 8, no. 3 (2002) 475–520.CrossRefGoogle Scholar
22. Martin, F.: Périodes de formes modulaires de poids 1. Thèse, Université Paris 7 (2001).
23. Mayer, D.: The thermodynamic formalism approach to Selberg's zeta function for PSL(2, ℝ). Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 55–60.CrossRefGoogle Scholar
24. Mayer, D.: The Ruelle-Araki transfer operator in classical statistical mechanics. Lecture Notes in Physics, 123. Springer-Verlag, Berlin-New York, 1980.Google Scholar
25. Mayer, D.: Continued fractions and related transformations. In: “Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces”, pp. 175–222. Eds. T., Bedford et al., Oxford University Press, Oxford, 1991.Google Scholar
27. Merel, L.: Universal Fourier expansions of modular forms. In “On Artins conjecture for odd 2-dimensional representations”, ch. IV. Ed.: G., Frey, Lecture Notes in Math. 1585, Springer Verlag, Berlin, 1994.Google Scholar
28. Mühlenbruch, T.: Systems of automorphic forms and period functions. Proefschrift Universiteit Utrecht, (2003).Google Scholar
29. Pollicott, M.: Distribution of closed geodesics on the modular surface and quadratic irrationals. Bull. Soc. Math. France 114 (1986) 431–446.CrossRefGoogle Scholar
30. Ruelle, D.: Thermodynamic formalism. Addison-Wesley Publishing Co., Reading, Mass., 1978.Google Scholar
31. Ruelle, D.: Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. CRM Monograph Series 4 (1994) Am. Math. Soc..Google Scholar
32. Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. Math. 34 (1976) 231–242.CrossRefGoogle Scholar
33. Sarnak, P.: Arithmetic chaos. Israel Math. Conf. Proc. 8 (1995) 183–236.Google Scholar
34. Series, C.: The modular surface and continued fractions. J. London Math. Soc. (2) 31 (1985) 69–80.CrossRefGoogle Scholar
35. Venkov, A.B., Zograf, P.G.: Analogues of Artin's factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1150–1158, 1343. Engl. tran.: Math. USSR-Izv. 21 (1983), no. 3, 435–443.Google Scholar
36. Zagier, D.: Periods of modular forms, traces of Hecke operators, and multiple zeta values. In “Research into automorphic forms and L functions” (Kyoto, 1992). Sūrikaisekikenkyūsho Kōkyūroku No. 843, (1993), 162–170.Google Scholar
37. Zagier, D.: Hecke operators and periods of modular forms. Israel Math. Conf. Proc. 3 (1990) 321–336.Google Scholar
38. Zagier, D.: Periods of modular forms and Jacobi theta functions. Invent. math. 104 (1991) 449–465.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×