May  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 and Continuous Dynamical Systems, 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., New Ser., 25 (1991), 229-334. 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), 29-48.

[3]

E. Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170-175. doi: 10.1007/BF02954622.

[4]

R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology, preprint, available on http://www.staff.science.uu.nl/ brugg103/algemeen/prpr.html, 2013.

[5]

U. Bunke and M. Olbrich, Gamma-cohomology and the selberg zeta function, J. reine angew. Math., 467 (1995), 199-219.

[6]

_____, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. reine angew. Math., 497 (1998), 47-81.

[7]

R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. reine angew. Math., 492 (1997), 1-39. doi: 10.1515/crll.1997.492.1.

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci., 50 (1979), 153-170.

[9]

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, 2008, http://ChaosBook.org. 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, MN, 1996), Springer, New York, IMA Vol. Math. Appl., 109 (1999), 73-141. 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., Providence, RI, Contemp. Math., 290 (2001), 1-40. 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, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 523-562.

[13]

A. Deitmar and J. Hilgert, Cohomology of arithmetic groups with infinite dimensional coefficient spaces, Doc. Math., 10 (2005), 199-216 (electronic).

[14]

______, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. 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), 207-218. 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), 133-163.

[17]

L. Ford, Automorphic Functions, Chelsea publishing company, New York, 1972.

[18]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.

[19]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures et Appl., 4 (1898), 27-73; see also: Oeuvres Complétes de Jacques Hadamard, 2 (1698), 729-775.

[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), 81-116. 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, 97–111, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789812832825_0006.

[22]

S. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.

[23]

J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127 (1997), 271-306. doi: 10.1007/s002220050120.

[24]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math.(2), 153 (2001), 191-258. doi: 10.2307/2661374.

[25]

B. Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230. 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), 195-203.

[27]

______, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. 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), 55-60. 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, 146-174, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, Cambridge, 2012.

[30]

D. Mayer, T. Mühlenbruch, and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., 32 (2012), 2453-2484. 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), 1147-1181. 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), 247-283. doi: 10.1017/S0143385711000794.

[33]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. 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), 219-276. doi: 10.1007/s10711-009-9453-3.

[36]

______, A dynamical approach to Maass cusp forms, J. Mod. Dyn., 6 (2012), 563-596.

[37]

______, Odd and Even Maass Cusp Forms for Hecke Triangle Groups, and the Billiard Flow, arXiv:1303.0528, 2013.

[38]

______, Period Functions for Maass Cusp Forms for $\Gamma_0(p)$: A transfer Operator Approach, International Mathematics Research Notices, 14 (2013), 3250-3273. doi: 10.1093/imrn/rns146.

[39]

M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math., 85 (1991), 161-192. doi: 10.1016/0001-8708(91)90054-B.

[40]

J. Ratcliffe, Foundations of Hyperbolic Manifolds, second ed., Graduate Texts in Mathematics, 149. Springer, New York, 2006.

[41]

D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895.

[42]

C. Series, Symbolic dynamics for geodesic flows, Acta Math., 146 (1981), 103-128. doi: 10.1007/BF02392459.

[43]

______, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. 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), 2295-2323. 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., New Ser., 25 (1991), 229-334. 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), 29-48.

[3]

E. Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg, 3 (1924), 170-175. doi: 10.1007/BF02954622.

[4]

R. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms. II: Cohomology, preprint, available on http://www.staff.science.uu.nl/ brugg103/algemeen/prpr.html, 2013.

[5]

U. Bunke and M. Olbrich, Gamma-cohomology and the selberg zeta function, J. reine angew. Math., 467 (1995), 199-219.

[6]

_____, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. reine angew. Math., 497 (1998), 47-81.

[7]

R. Bruggeman, Automorphic forms, hyperfunction cohomology, and period functions, J. reine angew. Math., 492 (1997), 1-39. doi: 10.1515/crll.1997.492.1.

[8]

R. Bowen and C. Series, Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci., 50 (1979), 153-170.

[9]

P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, 2008, http://ChaosBook.org. 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, MN, 1996), Springer, New York, IMA Vol. Math. Appl., 109 (1999), 73-141. 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., Providence, RI, Contemp. Math., 290 (2001), 1-40. 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, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 523-562.

[13]

A. Deitmar and J. Hilgert, Cohomology of arithmetic groups with infinite dimensional coefficient spaces, Doc. Math., 10 (2005), 199-216 (electronic).

[14]

______, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. 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), 207-218. 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), 133-163.

[17]

L. Ford, Automorphic Functions, Chelsea publishing company, New York, 1972.

[18]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521. doi: 10.1007/s002220050084.

[19]

J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures et Appl., 4 (1898), 27-73; see also: Oeuvres Complétes de Jacques Hadamard, 2 (1698), 729-775.

[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), 81-116. 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, 97–111, World Sci. Publ., Hackensack, NJ, 2009. doi: 10.1142/9789812832825_0006.

[22]

S. Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.

[23]

J. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127 (1997), 271-306. doi: 10.1007/s002220050120.

[24]

J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math.(2), 153 (2001), 191-258. doi: 10.2307/2661374.

[25]

B. Maskit, On Poincaré's theorem for fundamental polygons, Advances in Math., 7 (1971), 219-230. 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), 195-203.

[27]

______, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. 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), 55-60. 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, 146-174, London Math. Soc. Lecture Note Ser., 397, Cambridge Univ. Press, Cambridge, 2012.

[30]

D. Mayer, T. Mühlenbruch, and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., 32 (2012), 2453-2484. 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), 1147-1181. 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), 247-283. doi: 10.1017/S0143385711000794.

[33]

D. Mayer and F. Strömberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, J. Mod. Dyn., 2 (2008), 581-627. 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), 219-276. doi: 10.1007/s10711-009-9453-3.

[36]

______, A dynamical approach to Maass cusp forms, J. Mod. Dyn., 6 (2012), 563-596.

[37]

______, Odd and Even Maass Cusp Forms for Hecke Triangle Groups, and the Billiard Flow, arXiv:1303.0528, 2013.

[38]

______, Period Functions for Maass Cusp Forms for $\Gamma_0(p)$: A transfer Operator Approach, International Mathematics Research Notices, 14 (2013), 3250-3273. doi: 10.1093/imrn/rns146.

[39]

M. Pollicott, Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math., 85 (1991), 161-192. doi: 10.1016/0001-8708(91)90054-B.

[40]

J. Ratcliffe, Foundations of Hyperbolic Manifolds, second ed., Graduate Texts in Mathematics, 149. Springer, New York, 2006.

[41]

D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895.

[42]

C. Series, Symbolic dynamics for geodesic flows, Acta Math., 146 (1981), 103-128. doi: 10.1007/BF02392459.

[43]

______, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. 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), 2295-2323. doi: 10.1090/S0002-9947-99-02151-0.

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