# American Institute of Mathematical Sciences

October  2012, 6(4): 563-596. doi: 10.3934/jmd.2012.6.563

## A dynamical approach to Maass cusp forms

 1 Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3–5, 37073 Göttingen, Germany

Received  September 2012 Published  January 2013

For nonuniform cofinite Fuchsian groups $\Gamma$ that satisfy a certain additional geometric condition, we show that the Maass cusp forms for $\Gamma$ are isomorphic to $1$-eigenfunctions of a finite-term transfer operator. The isomorphism is constructive.
Citation: Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563
##### References:
 [1] E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. Google Scholar [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. Reine Angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1. Google Scholar [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2012). Google Scholar [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, J. Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar [5] C.-H. Chang and D. Mayer, The period function of the nonholomorphic Eisenstein series for $PSL(2,\mathbf Z)$,, Math. Phys. Electron. J., 4 (1998). Google Scholar [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$,, in, 109 (1999), 73. Google Scholar [7] _____, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups,, in, (2001), 523. Google Scholar [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042. Google Scholar [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667. Google Scholar [11] 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. Google Scholar [12] D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084. Google Scholar [13] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,\mathbf Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480. Google Scholar [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one,, in, (2009), 97. doi: 10.1142/9789812832825_0006. Google Scholar [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120. Google Scholar [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math. (2), 153 (2001), 191. doi: 10.2307/2661374. Google Scholar [17] B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. Google Scholar [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195. Google Scholar [19] _____, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. Google Scholar [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. Google Scholar [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. Google Scholar [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, J. Mod. Dyn., 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar [23] M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory and Dynamical Systems, (2011). Google Scholar [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X. Google Scholar [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar [26] _____, The Weyl theorem and the deformation of discrete groups,, Comm. Pure Appl. Math., 38 (1985), 853. Google Scholar [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow,, in preparation., (). Google Scholar [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds,, \arXiv{1008.0367}, (2010). Google Scholar [29] _____, Period functions for Maass cusp forms for $\Gamma_0(p)$: A transfer operator approach,, International Mathematics Research Notices, (2012). doi: 10.1093/imrn/rns146. Google Scholar [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature,, Adv. in Math., 85 (1991), 161. doi: 10.1016/0001-8708(91)90054-B. Google Scholar [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [32] _____, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887. Google Scholar [33] C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69. Google Scholar

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##### References:
 [1] E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. Google Scholar [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. Reine Angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1. Google Scholar [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2012). Google Scholar [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology,, J. Number Theory, 129 (2009), 158. doi: 10.1016/j.jnt.2008.08.003. Google Scholar [5] C.-H. Chang and D. Mayer, The period function of the nonholomorphic Eisenstein series for $PSL(2,\mathbf Z)$,, Math. Phys. Electron. J., 4 (1998). Google Scholar [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$,, in, 109 (1999), 73. Google Scholar [7] _____, Eigenfunctions of the transfer operators and the period functions for modular groups,, in, 290 (2001), 1. Google Scholar [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups,, in, (2001), 523. Google Scholar [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042. Google Scholar [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667. Google Scholar [11] 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. Google Scholar [12] D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084. Google Scholar [13] J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,\mathbf Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480. Google Scholar [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one,, in, (2009), 97. doi: 10.1142/9789812832825_0006. Google Scholar [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120. Google Scholar [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math. (2), 153 (2001), 191. doi: 10.2307/2661374. Google Scholar [17] B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. Google Scholar [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195. Google Scholar [19] _____, On the thermodynamic formalism for the Gauss map,, Comm. Math. Phys., 130 (1990), 311. Google Scholar [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. Google Scholar [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. Google Scholar [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces,, J. Mod. Dyn., 2 (2008), 581. doi: 10.3934/jmd.2008.2.581. Google Scholar [23] M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory and Dynamical Systems, (2011). Google Scholar [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X. Google Scholar [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$,, Invent. Math., 80 (1985), 339. doi: 10.1007/BF01388610. Google Scholar [26] _____, The Weyl theorem and the deformation of discrete groups,, Comm. Pure Appl. Math., 38 (1985), 853. Google Scholar [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow,, in preparation., (). Google Scholar [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds,, \arXiv{1008.0367}, (2010). Google Scholar [29] _____, Period functions for Maass cusp forms for $\Gamma_0(p)$: A transfer operator approach,, International Mathematics Research Notices, (2012). doi: 10.1093/imrn/rns146. Google Scholar [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature,, Adv. in Math., 85 (1991), 161. doi: 10.1016/0001-8708(91)90054-B. Google Scholar [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval,", CRM Monograph Series, 4 (1994). Google Scholar [32] _____, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887. Google Scholar [33] C. Series, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69. Google Scholar
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