2014, 34(5): 2173-2241. doi: 10.3934/dcds.2014.34.2173

Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds

1. 

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

Received  May 2013 Revised  August 2013 Published  October 2013

We construct cross sections for the geodesic flow on the orbifolds $\Gamma $\$ \mathbb{H}$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $\mathbb{H}$ denotes the hyperbolic plane and $\Gamma$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.
Citation: Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173
References:
[1]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Am. Math. Soc., 25 (1991), 229. doi: 10.1090/S0273-0979-1991-16076-3.

[2]

P. Arnoux, Le codage du flot géodésique sur la surface modulaire,, (French) [Coding of the geodesic flow on the modular surface] Enseign. Math. (2), 40 (1994), 29.

[3]

E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. doi: 10.1007/BF02954622.

[4]

R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2013).

[5]

U. Bunke and M. Olbrich, Gamma-cohomology and the selberg zeta function,, J. reine angew. Math., 467 (1995), 199.

[6]

_____, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology,, J. reine angew. Math., 497 (1998), 47.

[7]

R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. reine angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1.

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups,, Publ. Math., 50 (1979), 153.

[9]

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum,, Niels Bohr Institute, (2008). doi: 10.1016/0167-2789(91)90227-Z.

[10]

C.-H. Chang and D. Mayer, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $ PSL(2, Z)$,, Emerging applications of number theory (Minneapolis, 109 (1999), 73. doi: 10.1007/978-1-4612-1544-8_3.

[11]

______, Eigenfunctions of the transfer operators and the period functions for modular groups, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999),, Amer. Math. Soc., 290 (2001), 1. doi: 10.1090/conm/290/04571.

[12]

______, An Extension of the Thermodynamic Formalism Approach to Selberg's Zeta Function for General Modular Groups,, Ergodic theory, (2001), 523.

[13]

A. Deitmar and J. Hilgert, Cohomology of arithmetic groups with infinite dimensional coefficient spaces,, Doc. Math., 10 (2005), 199.

[14]

______, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042.

[15]

I. Efrat, Dynamics of the continued fraction map and the spectral theory of $ SL(2,Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667.

[16]

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.

[17]

L. Ford, Automorphic Functions,, Chelsea publishing company, (1972).

[18]

D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084.

[19]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques,, J. Math. Pures et Appl., 4 (1898), 27.

[20]

J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $ PSL(2,\mathbb Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480.

[21]

J. Hilgert and A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Orbifolds of Rank One,, Infinite dimensional harmonic analysis IV, (2009). doi: 10.1142/9789812832825_0006.

[22]

S. Katok, Fuchsian groups,, Chicago Lectures in Mathematics, (1992).

[23]

J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120.

[24]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math.(2), 153 (2001), 191. doi: 10.2307/2661374.

[25]

B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. doi: 10.1016/S0001-8708(71)80003-8.

[26]

D. Mayer, On a $\zeta $ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195.

[27]

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

[28]

______, The thermodynamic formalism approach to Selberg's zeta function for $ PSL(2,Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. doi: 10.1090/S0273-0979-1991-16023-4.

[29]

D. Mayer, Transfer Operators, the Selberg Zeta Function and the Lewis-Zagier Theory of Periodic Functions,, Hyperbolic geometry and applications in quantum chaos and cosmology, (2012), 146.

[30]

D. Mayer, T. Mühlenbruch, and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. doi: 10.3934/dcds.2012.32.2453.

[31]

T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X.

[32]

M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory Dynam. Systems, 33 (2013), 247. doi: 10.1017/S0143385711000794.

[33]

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.

[34]

A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Good Orbifolds of Rank One,, 2009, ().

[35]

______, Ford fundamental domains in symmetric spaces of rank one,, Geom. Dedicata, 147 (2010), 219. doi: 10.1007/s10711-009-9453-3.

[36]

______, A dynamical approach to Maass cusp forms,, J. Mod. Dyn., 6 (2012), 563.

[37]

______, Odd and Even Maass Cusp Forms for Hecke Triangle Groups, and the Billiard Flow,, , (2013).

[38]

______, Period Functions for Maass Cusp Forms for $\Gamma_0(p)$: A transfer Operator Approach,, International Mathematics Research Notices, 14 (2013), 3250. doi: 10.1093/imrn/rns146.

[39]

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.

[40]

J. Ratcliffe, Foundations of Hyperbolic Manifolds, second ed.,, Graduate Texts in Mathematics, (2006).

[41]

D. Ruelle, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887.

[42]

C. Series, Symbolic dynamics for geodesic flows,, Acta Math., 146 (1981), 103. doi: 10.1007/BF02392459.

[43]

______, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69.

[44]

L. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups,, Trans. Amer. Math. Soc., 351 (1999), 2295. doi: 10.1090/S0002-9947-99-02151-0.

show all references

References:
[1]

R. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics,, Bull. Am. Math. Soc., 25 (1991), 229. doi: 10.1090/S0273-0979-1991-16076-3.

[2]

P. Arnoux, Le codage du flot géodésique sur la surface modulaire,, (French) [Coding of the geodesic flow on the modular surface] Enseign. Math. (2), 40 (1994), 29.

[3]

E. Artin, Ein mechanisches system mit quasiergodischen bahnen,, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170. doi: 10.1007/BF02954622.

[4]

R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology,, preprint, (2013).

[5]

U. Bunke and M. Olbrich, Gamma-cohomology and the selberg zeta function,, J. reine angew. Math., 467 (1995), 199.

[6]

_____, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology,, J. reine angew. Math., 497 (1998), 47.

[7]

R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions,, J. reine angew. Math., 492 (1997), 1. doi: 10.1515/crll.1997.492.1.

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups,, Publ. Math., 50 (1979), 153.

[9]

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum,, Niels Bohr Institute, (2008). doi: 10.1016/0167-2789(91)90227-Z.

[10]

C.-H. Chang and D. Mayer, The transfer operator approach to Selberg's zeta function and modular and Maass wave forms for $ PSL(2, Z)$,, Emerging applications of number theory (Minneapolis, 109 (1999), 73. doi: 10.1007/978-1-4612-1544-8_3.

[11]

______, Eigenfunctions of the transfer operators and the period functions for modular groups, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999),, Amer. Math. Soc., 290 (2001), 1. doi: 10.1090/conm/290/04571.

[12]

______, An Extension of the Thermodynamic Formalism Approach to Selberg's Zeta Function for General Modular Groups,, Ergodic theory, (2001), 523.

[13]

A. Deitmar and J. Hilgert, Cohomology of arithmetic groups with infinite dimensional coefficient spaces,, Doc. Math., 10 (2005), 199.

[14]

______, A Lewis correspondence for submodular groups,, Forum Math., 19 (2007), 1075. doi: 10.1515/FORUM.2007.042.

[15]

I. Efrat, Dynamics of the continued fraction map and the spectral theory of $ SL(2,Z)$,, Invent. Math., 114 (1993), 207. doi: 10.1007/BF01232667.

[16]

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.

[17]

L. Ford, Automorphic Functions,, Chelsea publishing company, (1972).

[18]

D. Fried, Symbolic dynamics for triangle groups,, Invent. Math., 125 (1996), 487. doi: 10.1007/s002220050084.

[19]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques,, J. Math. Pures et Appl., 4 (1898), 27.

[20]

J. Hilgert, D. Mayer and H. Movasati, Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $ PSL(2,\mathbb Z)$,, Math. Proc. Cambridge Philos. Soc., 139 (2005), 81. doi: 10.1017/S0305004105008480.

[21]

J. Hilgert and A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Orbifolds of Rank One,, Infinite dimensional harmonic analysis IV, (2009). doi: 10.1142/9789812832825_0006.

[22]

S. Katok, Fuchsian groups,, Chicago Lectures in Mathematics, (1992).

[23]

J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms,, Invent. Math., 127 (1997), 271. doi: 10.1007/s002220050120.

[24]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I,, Ann. of Math.(2), 153 (2001), 191. doi: 10.2307/2661374.

[25]

B. Maskit, On Poincaré's theorem for fundamental polygons,, Advances in Math., 7 (1971), 219. doi: 10.1016/S0001-8708(71)80003-8.

[26]

D. Mayer, On a $\zeta $ function related to the continued fraction transformation,, Bull. Soc. Math. France, 104 (1976), 195.

[27]

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

[28]

______, The thermodynamic formalism approach to Selberg's zeta function for $ PSL(2,Z)$,, Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55. doi: 10.1090/S0273-0979-1991-16023-4.

[29]

D. Mayer, Transfer Operators, the Selberg Zeta Function and the Lewis-Zagier Theory of Periodic Functions,, Hyperbolic geometry and applications in quantum chaos and cosmology, (2012), 146.

[30]

D. Mayer, T. Mühlenbruch, and F. Strömberg, The transfer operator for the Hecke triangle groups,, Discrete Contin. Dyn. Syst., 32 (2012), 2453. doi: 10.3934/dcds.2012.32.2453.

[31]

T. Morita, Markov systems and transfer operators associated with cofinite Fuchsian groups,, Ergodic Theory Dynam. Systems, 17 (1997), 1147. doi: 10.1017/S014338579708632X.

[32]

M. Möller and A. Pohl, Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant,, Ergodic Theory Dynam. Systems, 33 (2013), 247. doi: 10.1017/S0143385711000794.

[33]

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.

[34]

A. Pohl, Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Good Orbifolds of Rank One,, 2009, ().

[35]

______, Ford fundamental domains in symmetric spaces of rank one,, Geom. Dedicata, 147 (2010), 219. doi: 10.1007/s10711-009-9453-3.

[36]

______, A dynamical approach to Maass cusp forms,, J. Mod. Dyn., 6 (2012), 563.

[37]

______, Odd and Even Maass Cusp Forms for Hecke Triangle Groups, and the Billiard Flow,, , (2013).

[38]

______, Period Functions for Maass Cusp Forms for $\Gamma_0(p)$: A transfer Operator Approach,, International Mathematics Research Notices, 14 (2013), 3250. doi: 10.1093/imrn/rns146.

[39]

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.

[40]

J. Ratcliffe, Foundations of Hyperbolic Manifolds, second ed.,, Graduate Texts in Mathematics, (2006).

[41]

D. Ruelle, Dynamical zeta functions and transfer operators,, Notices Amer. Math. Soc., 49 (2002), 887.

[42]

C. Series, Symbolic dynamics for geodesic flows,, Acta Math., 146 (1981), 103. doi: 10.1007/BF02392459.

[43]

______, The modular surface and continued fractions,, J. London Math. Soc. (2), 31 (1985), 69. doi: 10.1112/jlms/s2-31.1.69.

[44]

L. Vulakh, Farey polytopes and continued fractions associated with discrete hyperbolic groups,, Trans. Amer. Math. Soc., 351 (1999), 2295. doi: 10.1090/S0002-9947-99-02151-0.

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