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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

______, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. doi: 10.1515/FORUM.2007.042.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

D. Mayer, On a $\zeta $ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203.  Google Scholar

[27]

______, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. doi: 10.1007/BF02473355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

______, Ford fundamental domains in symmetric spaces of rank one, Geom. Dedicata, 147 (2010), 219-276. doi: 10.1007/s10711-009-9453-3.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

______, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

______, A Lewis correspondence for submodular groups, Forum Math., 19 (2007), 1075-1099. doi: 10.1515/FORUM.2007.042.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[26]

D. Mayer, On a $\zeta $ function related to the continued fraction transformation, Bull. Soc. Math. France, 104 (1976), 195-203.  Google Scholar

[27]

______, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311-333. doi: 10.1007/BF02473355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

______, Ford fundamental domains in symmetric spaces of rank one, Geom. Dedicata, 147 (2010), 219-276. doi: 10.1007/s10711-009-9453-3.  Google Scholar

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[41]

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

[42]

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

[43]

______, The modular surface and continued fractions, J. London Math. Soc. (2), 31 (1985), 69-80. doi: 10.1112/jlms/s2-31.1.69.  Google Scholar

[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.  Google Scholar

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