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 & Continuous Dynamical Systems - A, 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., (). Google Scholar

[2]

R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, Journal of Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar

[3]

C.-H. Chang and D. Mayer, Thermodynamic formalism and Selberg's zeta function for modular groups,, Regul. Chaotic Dyn., 5 (2000), 281. doi: 10.1070/rd2000v005n03ABEH000150. Google Scholar

[4]

C.-H. Chang and D. Mayer, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar

[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. doi: 10.1515/CRELLE.2007.014. Google Scholar

[6]

D. Hejhal, "The Selberg Trace Formula for $\PSL(2,\mathbbR)$," Vol. 2,, Lecture Notes in Mathematics, 1001 (1983). Google Scholar

[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. doi: 10.1017/S0305004105008480. Google Scholar

[8]

A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwickelung reeller Grössen,, Acta Math., 12 (1889), 367. doi: 10.1007/BF02391885. Google Scholar

[9]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I.,, Ann. of Math., 153 (2001), 191. doi: 10.2307/2661374. Google Scholar

[10]

D. Mayer, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. doi: 10.1007/BF02473355. Google Scholar

[11]

D. Mayer, On composition operators on Banach spaces of holomorphic functions,, Journal of Functional Analysis, 35 (1980), 191. doi: 10.1016/0022-1236(80)90004-X. Google Scholar

[12]

D. Mayer and T. Mühlenbruch, Nearest $\lambda_q$-multiple fractions,, in, 52 (2010), 147. Google Scholar

[13]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, Journal of Modern Dynamics, 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar

[14]

H. Nakada, Continued fractions, geodesic flows and Ford circles,, in, (1995), 179. Google Scholar

[15]

R. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbbR)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar

[16]

D. Rosen, A class of continued fractions associated with certain properly discontinuous groups,, Duke Math. J., 21 (1954), 549. doi: 10.1215/S0012-7094-54-02154-7. Google Scholar

[17]

D. Rosen and T. A. Schmidt, Hecke groups and continued fractions,, Bull. Austral. Math. Soc., 46 (1992), 459. doi: 10.1017/S0004972700012120. Google Scholar

[18]

D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar

[19]

T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups,, Mathematische Zeitschrift, 220 (1995), 369. doi: 10.1007/BF02572621. Google Scholar

[20]

A. Selberg, Remarks on the distribution of poles of Eisenstein series,, in, 3 (1990), 251. Google Scholar

[21]

F. Strömberg, Computation of Selberg's zeta functions on Hecke triangle groups,, \arXiv{0804.4837}., (). Google Scholar

show all references

References:
[1]

R. W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maaß wave forms. II: Cohomology,, preprint., (). Google Scholar

[2]

R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, Journal of Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar

[3]

C.-H. Chang and D. Mayer, Thermodynamic formalism and Selberg's zeta function for modular groups,, Regul. Chaotic Dyn., 5 (2000), 281. doi: 10.1070/rd2000v005n03ABEH000150. Google Scholar

[4]

C.-H. Chang and D. Mayer, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar

[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. doi: 10.1515/CRELLE.2007.014. Google Scholar

[6]

D. Hejhal, "The Selberg Trace Formula for $\PSL(2,\mathbbR)$," Vol. 2,, Lecture Notes in Mathematics, 1001 (1983). Google Scholar

[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. doi: 10.1017/S0305004105008480. Google Scholar

[8]

A. Hurwitz, Über eine besondere Art der Kettenbruch-Entwickelung reeller Grössen,, Acta Math., 12 (1889), 367. doi: 10.1007/BF02391885. Google Scholar

[9]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I.,, Ann. of Math., 153 (2001), 191. doi: 10.2307/2661374. Google Scholar

[10]

D. Mayer, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. doi: 10.1007/BF02473355. Google Scholar

[11]

D. Mayer, On composition operators on Banach spaces of holomorphic functions,, Journal of Functional Analysis, 35 (1980), 191. doi: 10.1016/0022-1236(80)90004-X. Google Scholar

[12]

D. Mayer and T. Mühlenbruch, Nearest $\lambda_q$-multiple fractions,, in, 52 (2010), 147. Google Scholar

[13]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, Journal of Modern Dynamics, 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar

[14]

H. Nakada, Continued fractions, geodesic flows and Ford circles,, in, (1995), 179. Google Scholar

[15]

R. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbbR)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar

[16]

D. Rosen, A class of continued fractions associated with certain properly discontinuous groups,, Duke Math. J., 21 (1954), 549. doi: 10.1215/S0012-7094-54-02154-7. Google Scholar

[17]

D. Rosen and T. A. Schmidt, Hecke groups and continued fractions,, Bull. Austral. Math. Soc., 46 (1992), 459. doi: 10.1017/S0004972700012120. Google Scholar

[18]

D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar

[19]

T. A. Schmidt and M. Sheingorn, Length spectra of the Hecke triangle groups,, Mathematische Zeitschrift, 220 (1995), 369. doi: 10.1007/BF02572621. Google Scholar

[20]

A. Selberg, Remarks on the distribution of poles of Eisenstein series,, in, 3 (1990), 251. Google Scholar

[21]

F. Strömberg, Computation of Selberg's zeta functions on Hecke triangle groups,, \arXiv{0804.4837}., (). Google Scholar

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