# 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), 170-175. [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. Reine Angew. Math., 492 (1997), 1-39. doi: 10.1515/crll.1997.492.1. [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology, preprint, 2012. Available from: http://www.staff.science.uu.nl/~brugg103/algemeen/prpr.html. [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology, J. Number Theory, 129 (2009), 158-181. doi: 10.1016/j.jnt.2008.08.003. [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), Paper 6, 8 pp. [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$, in "Emerging Applications of Number Theory" (Minneapolis, MN, 1996), IMA Vol. Math. Appl., 109, Springer, New York, (1999), 73-141. [7] _____, 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., Amer. Math. Soc., 290, Providence, RI, (2001), 1-40. [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 523-562. [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. doi: 10.1515/FORUM.2007.042. [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$, Invent. Math., 114 (1993), 207-218. doi: 10.1007/BF01232667. [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), 133-163. [12] D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084. [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), 81-116. doi: 10.1017/S0305004105008480. [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one, in "Infinite Dimensional Harmonic Analysis IV," World Scientific Publ., Hackensack, NJ, (2009), 97-111. doi: 10.1142/9789812832825_0006. [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127 (1997), 271-306. doi: 10.1007/s002220050120. [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2), 153 (2001), 191-258. doi: 10.2307/2661374. [17] B. Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230. [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203. [19] _____, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55-60. [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., 32 (2012), 2453-2484. [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581. [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, to appear, published as First View, arXiv:1103.5235, (2011). [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups, Ergodic Theory Dynam. Systems, 17 (1997), 1147-1181. doi: 10.1017/S014338579708632X. [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$, Invent. Math., 80 (1985), 339-364. doi: 10.1007/BF01388610. [26] _____, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math., 38 (1985), 853-866. [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow, in preparation. [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds, arXiv:1008.0367, 2010. [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. [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math., 85 (1991), 161-192. doi: 10.1016/0001-8708(91)90054-B. [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, 4, American Mathematical Society, Providence, RI, 1994. [32] _____, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895. [33] C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.

show all references

##### References:
 [1] E. Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170-175. [2] R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. Reine Angew. Math., 492 (1997), 1-39. doi: 10.1515/crll.1997.492.1. [3] R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology, preprint, 2012. Available from: http://www.staff.science.uu.nl/~brugg103/algemeen/prpr.html. [4] R. W. Bruggeman and T. Mühlenbruch, Eigenfunctions of transfer operators and cohomology, J. Number Theory, 129 (2009), 158-181. doi: 10.1016/j.jnt.2008.08.003. [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), Paper 6, 8 pp. [6] _____, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $PSL(2,\mathbf Z)$, in "Emerging Applications of Number Theory" (Minneapolis, MN, 1996), IMA Vol. Math. Appl., 109, Springer, New York, (1999), 73-141. [7] _____, 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., Amer. Math. Soc., 290, Providence, RI, (2001), 1-40. [8] _____, An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 523-562. [9] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. doi: 10.1515/FORUM.2007.042. [10] I. Efrat, Dynamics of the continued fraction map and the spectral theory of $SL(2,\mathbf Z)$, Invent. Math., 114 (1993), 207-218. doi: 10.1007/BF01232667. [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), 133-163. [12] D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084. [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), 81-116. doi: 10.1017/S0305004105008480. [14] J. Hilgert and A. Pohl, Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one, in "Infinite Dimensional Harmonic Analysis IV," World Scientific Publ., Hackensack, NJ, (2009), 97-111. doi: 10.1142/9789812832825_0006. [15] J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127 (1997), 271-306. doi: 10.1007/s002220050120. [16] J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2), 153 (2001), 191-258. doi: 10.2307/2661374. [17] B. Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230. [18] D. Mayer, On a $\zeta$ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203. [19] _____, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. [20] _____, The thermodynamic formalism approach to Selberg's zeta function for $PSL(2,\mathbf Z)$, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55-60. [21] D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., 32 (2012), 2453-2484. [22] D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. doi: 10.3934/jmd.2008.2.581. [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, to appear, published as First View, arXiv:1103.5235, (2011). [24] T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups, Ergodic Theory Dynam. Systems, 17 (1997), 1147-1181. doi: 10.1017/S014338579708632X. [25] R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of $PSL(2,\mathbf R)$, Invent. Math., 80 (1985), 339-364. doi: 10.1007/BF01388610. [26] _____, The Weyl theorem and the deformation of discrete groups, Comm. Pure Appl. Math., 38 (1985), 853-866. [27] A. Pohl, Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow, in preparation. [28] _____, Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds, arXiv:1008.0367, 2010. [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. [30] M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math., 85 (1991), 161-192. doi: 10.1016/0001-8708(91)90054-B. [31] D. Ruelle, "Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval," CRM Monograph Series, 4, American Mathematical Society, Providence, RI, 1994. [32] _____, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895. [33] C. Series, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.
 [1] Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581 [2] Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173 [3] Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581 [4] Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48 [5] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 [6] 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 [7] Feng Luo. Geodesic length functions and Teichmuller spaces. Electronic Research Announcements, 1996, 2: 34-41. [8] Jim Wiseman. Symbolic dynamics from signed matrices. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621 [9] George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43 [10] Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301 [11] Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665 [12] Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158 [13] Jose S. Cánovas, Tönu Puu, Manuel Ruiz Marín. Detecting chaos in a duopoly model via symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 269-278. doi: 10.3934/dcdsb.2010.13.269 [14] Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245 [15] Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487 [16] David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287 [17] Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243 [18] Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198 [19] Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835 [20] Mike Boyle. The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics. Journal of Modern Dynamics, 2019, 15: 427-435. doi: 10.3934/jmd.2019026

2021 Impact Factor: 0.641