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

show all references

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

[1]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[2]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106

[3]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288

[4]

Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166

[5]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[6]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117

[7]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130

[8]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[9]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[10]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[11]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006

[12]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[13]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069

[14]

Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314

[15]

Laure Cardoulis, Michel Cristofol, Morgan Morancey. A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020054

[16]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391

[17]

Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001

[18]

Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020034

[19]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[20]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389

2019 Impact Factor: 0.465

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]