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.

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