# American Institute of Mathematical Sciences

July  2012, 32(7): 2453-2484. doi: 10.3934/dcds.2012.32.2453

## The transfer operator for the Hecke triangle groups

 1 Lower Saxony Professorship, Institute for Theoretical Physics, TU Clausthal, D-38678 Clausthal-Zellerfeld, Germany 2 Department of Mathematics and Computer Science, FernUniversität in Hagen, D-58084 Hagen, Germany 3 Department of Mathematics, TU Darmstadt, D-64289 Darmstadt, Germany

Received  December 2009 Revised  March 2010 Published  March 2012

In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for $q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the so-called Hurwitz-Nakada continued fractions. We also derive functional equations for the eigenfunctions of the transfer operator which for eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.
Citation: Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
##### References:
 [1] R. W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maaß wave forms. II: Cohomology, preprint. [2] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology, Journal of Number Theory, 129 (2009), 158-181. doi: 10.1016/j.jnt.2008.08.003. [3] C.-H. Chang and D. Mayer, Thermodynamic formalism and Selberg's zeta function for modular groups, Regul. Chaotic Dyn., 5 (2000), 281-312. doi: 10.1070/rd2000v005n03ABEH000150. [4] C.-H. Chang and D. Mayer, Eigenfunctions of the transfer operators and the period functions for modular groups, in "Dynamical, Spectral, and Arithmetic Zeta Functions" (eds. M. L. Lapidus and M. Van Frankenhuysen) (San Antonio, TX, 1999), Contemp. Math., 290, Amer. Math. Soc., Providence, RI, (2001), 1-40. [5] M. Fraczek, D. Mayer and T. Mühlenbruch, A realization of the Hecke algebra on the space of period functions for $\Gamma_0(n)$, J. Reine Angew. Math., 603 (2007), 133-163. doi: 10.1515/CRELLE.2007.014. [6] D. Hejhal, "The Selberg Trace Formula for $\PSL(2,\mathbbR)$," Vol. 2, Lecture Notes in Mathematics, 1001, Springer-Verlag, Berlin, 1983. [7] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $\PSL(2,\mathbbZ)$, Math. Proc. Camb. Phil. Soc., 139 (2005), 81-116. doi: 10.1017/S0305004105008480. [8] A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwickelung reeller Grössen, Acta Math., 12 (1889), 367-405. doi: 10.1007/BF02391885. [9] J. Lewis and D. Zagier, Period functions for Maass wave forms. I., Ann. of Math., 153 (2001), 191-258. doi: 10.2307/2661374. [10] D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. doi: 10.1007/BF02473355. [11] D. Mayer, On composition operators on Banach spaces of holomorphic functions, Journal of Functional Analysis, 35 (1980), 191-206. doi: 10.1016/0022-1236(80)90004-X. [12] D. Mayer and T. Mühlenbruch, Nearest $\lambda_q$-multiple fractions, in "Spectrum and Dynamics" (eds. D. Jakobson, S. Nonnenmacher and I. Polterovich), CRM Proceedings and Lecture Notes, 52, AMS, Providence, RI, (2010), 147-184. Available from: arXiv:0902.3953. [13] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, Journal of Modern Dynamics, 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581. [14] H. Nakada, Continued fractions, geodesic flows and Ford circles, in "Algorithms, Fractals, and Dynamics" (ed. T. Takahashi) (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191. [15] R. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbbR)$, Invent. Math., 80 (1985), 339-364. doi: 10.1007/BF01388610. [16] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563. doi: 10.1215/S0012-7094-54-02154-7. [17] D. Rosen and T. A. Schmidt, Hecke groups and continued fractions, Bull. Austral. Math. Soc., 46 (1992), 459-474. doi: 10.1017/S0004972700012120. [18] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, 4, AMS, Providence, R.I., 1994. [19] T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Mathematische Zeitschrift, 220 (1995), 369-397. doi: 10.1007/BF02572621. [20] A. Selberg, Remarks on the distribution of poles of Eisenstein series, in "Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday," Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc., 3, Weizmann, Jerusalem, (1990), 251-278. [21] F. Strömberg, Computation of Selberg's zeta functions on Hecke triangle groups, arXiv:0804.4837.

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##### References:
 [1] R. W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maaß wave forms. II: Cohomology, preprint. [2] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology, Journal of Number Theory, 129 (2009), 158-181. doi: 10.1016/j.jnt.2008.08.003. [3] C.-H. Chang and D. Mayer, Thermodynamic formalism and Selberg's zeta function for modular groups, Regul. Chaotic Dyn., 5 (2000), 281-312. doi: 10.1070/rd2000v005n03ABEH000150. [4] C.-H. Chang and D. Mayer, Eigenfunctions of the transfer operators and the period functions for modular groups, in "Dynamical, Spectral, and Arithmetic Zeta Functions" (eds. M. L. Lapidus and M. Van Frankenhuysen) (San Antonio, TX, 1999), Contemp. Math., 290, Amer. Math. Soc., Providence, RI, (2001), 1-40. [5] M. Fraczek, D. Mayer and T. Mühlenbruch, A realization of the Hecke algebra on the space of period functions for $\Gamma_0(n)$, J. Reine Angew. Math., 603 (2007), 133-163. doi: 10.1515/CRELLE.2007.014. [6] D. Hejhal, "The Selberg Trace Formula for $\PSL(2,\mathbbR)$," Vol. 2, Lecture Notes in Mathematics, 1001, Springer-Verlag, Berlin, 1983. [7] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $\PSL(2,\mathbbZ)$, Math. Proc. Camb. Phil. Soc., 139 (2005), 81-116. doi: 10.1017/S0305004105008480. [8] A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwickelung reeller Grössen, Acta Math., 12 (1889), 367-405. doi: 10.1007/BF02391885. [9] J. Lewis and D. Zagier, Period functions for Maass wave forms. I., Ann. of Math., 153 (2001), 191-258. doi: 10.2307/2661374. [10] D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. doi: 10.1007/BF02473355. [11] D. Mayer, On composition operators on Banach spaces of holomorphic functions, Journal of Functional Analysis, 35 (1980), 191-206. doi: 10.1016/0022-1236(80)90004-X. [12] D. Mayer and T. Mühlenbruch, Nearest $\lambda_q$-multiple fractions, in "Spectrum and Dynamics" (eds. D. Jakobson, S. Nonnenmacher and I. Polterovich), CRM Proceedings and Lecture Notes, 52, AMS, Providence, RI, (2010), 147-184. Available from: arXiv:0902.3953. [13] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, Journal of Modern Dynamics, 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581. [14] H. Nakada, Continued fractions, geodesic flows and Ford circles, in "Algorithms, Fractals, and Dynamics" (ed. T. Takahashi) (Okayama/Kyoto, 1992), Plenum, New York, (1995), 179-191. [15] R. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbbR)$, Invent. Math., 80 (1985), 339-364. doi: 10.1007/BF01388610. [16] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563. doi: 10.1215/S0012-7094-54-02154-7. [17] D. Rosen and T. A. Schmidt, Hecke groups and continued fractions, Bull. Austral. Math. Soc., 46 (1992), 459-474. doi: 10.1017/S0004972700012120. [18] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, 4, AMS, Providence, R.I., 1994. [19] T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups, Mathematische Zeitschrift, 220 (1995), 369-397. doi: 10.1007/BF02572621. [20] A. Selberg, Remarks on the distribution of poles of Eisenstein series, in "Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday," Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc., 3, Weizmann, Jerusalem, (1990), 251-278. [21] F. Strömberg, Computation of Selberg's zeta functions on Hecke triangle groups, arXiv:0804.4837.
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