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February  2019, 26: 24-35. doi: 10.3934/era.2019.26.003

Fractal Weyl bounds and Hecke triangle groups

Frédéric Naud 1, , Anke Pohl 2, and  Louis Soares 3,

1. 

Laboratoire de Mathématiques D'avignon, Université d'Avignon, 301 rue Baruch de Spinoza, 84916 Avignon Cedex, France
2. 

University of Bremen, Department 3 - Mathematics, Bibliothekstr. 5, 28359 Bremen, Germany
3. 

Institute for Mathematics, University of Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany

Received  November 10, 2018 Revised  June 07, 2019 Published  July 2019

Fund Project: Naud is supported by Institut Universitaire de France. Pohl acknowledges support by the DFG grants PO 1483/2-1 and PO 1483/2-2

Let $ \Gamma_w $ be a non-cofinite Hecke triangle group with cusp width $ w>2 $ and let $ \varrho\colon\Gamma_w\to U(V) $ be a finite-dimensional unitary representation of $ \Gamma_w $. In this note we announce a new fractal upper bound for the Selberg zeta function of $ \Gamma_w $ twisted by $ \varrho $. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $ \exp\left( C_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \right) $, where $ \delta = \delta_w $ denotes the Hausdorff dimension of the limit set of $ \Gamma_w. $ This bound implies fractal Weyl bounds on the resonances of the Laplacian for any geometrically finite surface $ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $ whose fundamental group $ \widetilde{\Gamma} $ is a finite index, torsion-free subgroup of $ \Gamma_w $.

Keywords: Fractal Weyl bound, resonances, hyperbolic surfaces of infinite area, Hecke triangle groups, Selberg zeta function, transfer operator.
Mathematics Subject Classification: Primary: 58J50, Secondary: 37C30, 37D35, 11M36.
Citation: Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003
References:
[1]

T. Apostol, On the Lerch zeta function, Pacific J. Math., (1951), 161–167. doi: 10.2140/pjm.1951.1.161. Google Scholar

[2]

O. Bandtlow and O. Jenkinson, Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., 218 (2008), 902–925. doi: 10.1016/j.aim.2008.02.005. Google Scholar

[3]

D. Borthwick, C. Judge and P. Perry, Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv., 80 (2005), 483–515. doi: 10.4171/CMH/23. Google Scholar

[4]

C. Chang and D. Mayer, 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, 2001,523–562. Google Scholar

[5]

K. Datchev and S. Dyatlov, Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal., 23 (2013), 1145–1206. doi: 10.1007/s00039-013-0225-8. Google Scholar

[6]

S. Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, To appear in JEMS.Google Scholar

[7]

S. Dyatlov and M. Zworski, Fractal Uncertainty for Transfer Operators , International Mathematics Research Notices, 2018. doi: 10.1093/imrn/rny026. Google Scholar

[8]

F. Faure and M. Tsuiji, Fractal Weyl law for the Ruelle spectrum of Anosov flows, arXiv: 1706.09307.Google Scholar

[9]

K. Fedosova and A. Pohl, Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, arXiv: 1709.00760.Google Scholar

[10]

D. Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. Éc. Norm. Supér. (4), 19 (1986), 491–517. doi: 10.24033/asens.1515. Google Scholar

[11]

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

[12]

L. Guillopé, K. Lin and M. Zworski, The Selberg zeta function for convex co-compact Schottky groups, Commun. Math. Phys., 245 (2004), 149–176. doi: 10.1007/s00220-003-1007-1. Google Scholar

[13]

L. Guillopé and M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2), 145 (1997), 597–660. doi: 10.2307/2951846. Google Scholar

[14]

E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen, Math. Ann., 112 (1936), 664–699. doi: 10.1007/BF01565437. Google Scholar

[15]

D. Jakobson and F. Naud, On the critical line of convex co-compact hyperbolic surfaces, Geom. Funct. Anal., 22 (2012), 352–368. doi: 10.1007/s00039-012-0154-y. Google Scholar

[16]

D. Jakobson and F. Naud, Resonances and density bounds for convex co-compact congruence subgroups of $SL_2(\Bbb{Z})$, Israel J. Math., 213 (2016), 443–473. doi: 10.1007/s11856-016-1332-7. Google Scholar

[17]

M. Katsurada, On an asymptotic formula of Ramanujan for a certain theta-type series, Acta Arith., 97 (2001), 157–172. doi: 10.4064/aa97-2-4. Google Scholar

[18]

P. Lax and R. Phillips, Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Comm. Pure Appl. Math., 37 (1984), 303–328. doi: 10.1002/cpa.3160370304. Google Scholar

[19]

W. Lu, S. Sridhar and M. Zworski, Fractal Weyl laws for chaotic open systems, Phys. Rev. Lett., 91 (2003), 154101. doi: 10.1103/PhysRevLett.91.154101. Google Scholar

[20]

Y. Manin and M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Sel. Math., New Ser., 8 (2002), 475–521. doi: 10.1007/s00029-002-8113-3. Google Scholar

[21]

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

[22]

D. Mayer, The thermodynamic formalism approach to Selberg's zeta function for $ \rm{PSL} $(2, Z), Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55–60. doi: 10.1090/S0273-0979-1991-16023-4. Google Scholar

[23]

D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., Ser. A, 32 (2012), 2453–2484. doi: 10.3934/dcds.2012.32.2453. Google Scholar

[24]

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

[25]

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

[26]

F. Naud, Density and location of resonances for convex co-compact hyperbolic surfaces, Invent. Math., 195 (2014), 723–750. doi: 10.1007/s00222-013-0463-2. Google Scholar

[27]

S. Nonnenmacher, J. Sjöstrand and M. Zworski, Fractal Weyl law for open quantum chaotic maps, Ann. of Math. (2), 179 (2014), 179–251. doi: 10.4007/annals.2014.179.1.3. Google Scholar

[28]

S. Nonnenmacher and M. Zworski, Fractal Weyl laws in discrete models of chaotic scattering, J. Phys. A, 38 (2005), 10683–10702. Google Scholar

[29]

S. Patterson and P. Perry, The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein, Duke Math. J., 106 (2001), 321–390. doi: 10.1215/S0012-7094-01-10624-8. Google Scholar

[30]

A. Pohl, A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area, Commun. Math. Phys., 337 (2015), 103–126. doi: 10.1007/s00220-015-2304-1. Google Scholar

[31]

A. Pohl, Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations, Contemp. Math., 669 (2016), 205–236. doi: 10.1090/conm/669/13430. Google Scholar

[32]

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

[33]

D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231–242. doi: 10.1007/BF01403069. Google Scholar

[34]

E. Seiler and B. Simon, An inequality among determinants, Proc. Nat. Acad. Sci. USA, 72 (1975), 3277–3278. doi: 10.1073/pnas.72.9.3277. Google Scholar

[35]

J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J., 60 (1990), 1–57. doi: 10.1215/S0012-7094-90-06001-6. Google Scholar

[36]

J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J., 137 (2007), 381–459. doi: 10.1215/S0012-7094-07-13731-1. Google Scholar

[37]

B. Stratmann and M. Urbański, The box-counting dimension for geometrically finite Kleinian groups, Fundam. Math., 149 (1996), 83–93. Google Scholar

[38]

A. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math., 153 (1982); A translation of Trudy Mat. Inst. Steklov., 153 (1981), 172pp. Google Scholar

[39]

A. Venkov and P. Zograf, On analogues of the Artin factorization formulas in the spectral theory of automorphic functions connected with induced representations of Fuchsian groups, Math. USSR, Izv., 21 (1983), 435-443. Google Scholar

[40]

M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Invent. Math., 136 (1999), 353–409. doi: 10.1007/s002220050313. Google Scholar

show all references

References:
[1]

T. Apostol, On the Lerch zeta function, Pacific J. Math., (1951), 161–167. doi: 10.2140/pjm.1951.1.161. Google Scholar

[2]

O. Bandtlow and O. Jenkinson, Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions, Adv. Math., 218 (2008), 902–925. doi: 10.1016/j.aim.2008.02.005. Google Scholar

[3]

D. Borthwick, C. Judge and P. Perry, Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces, Comment. Math. Helv., 80 (2005), 483–515. doi: 10.4171/CMH/23. Google Scholar

[4]

C. Chang and D. Mayer, 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, 2001,523–562. Google Scholar

[5]

K. Datchev and S. Dyatlov, Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal., 23 (2013), 1145–1206. doi: 10.1007/s00039-013-0225-8. Google Scholar

[6]

S. Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, To appear in JEMS.Google Scholar

[7]

S. Dyatlov and M. Zworski, Fractal Uncertainty for Transfer Operators , International Mathematics Research Notices, 2018. doi: 10.1093/imrn/rny026. Google Scholar

[8]

F. Faure and M. Tsuiji, Fractal Weyl law for the Ruelle spectrum of Anosov flows, arXiv: 1706.09307.Google Scholar

[9]

K. Fedosova and A. Pohl, Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, arXiv: 1709.00760.Google Scholar

[10]

D. Fried, The zeta functions of Ruelle and Selberg. I, Ann. Sci. Éc. Norm. Supér. (4), 19 (1986), 491–517. doi: 10.24033/asens.1515. Google Scholar

[11]

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

[12]

L. Guillopé, K. Lin and M. Zworski, The Selberg zeta function for convex co-compact Schottky groups, Commun. Math. Phys., 245 (2004), 149–176. doi: 10.1007/s00220-003-1007-1. Google Scholar

[13]

L. Guillopé and M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2), 145 (1997), 597–660. doi: 10.2307/2951846. Google Scholar

[14]

E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen, Math. Ann., 112 (1936), 664–699. doi: 10.1007/BF01565437. Google Scholar

[15]

D. Jakobson and F. Naud, On the critical line of convex co-compact hyperbolic surfaces, Geom. Funct. Anal., 22 (2012), 352–368. doi: 10.1007/s00039-012-0154-y. Google Scholar

[16]

D. Jakobson and F. Naud, Resonances and density bounds for convex co-compact congruence subgroups of $SL_2(\Bbb{Z})$, Israel J. Math., 213 (2016), 443–473. doi: 10.1007/s11856-016-1332-7. Google Scholar

[17]

M. Katsurada, On an asymptotic formula of Ramanujan for a certain theta-type series, Acta Arith., 97 (2001), 157–172. doi: 10.4064/aa97-2-4. Google Scholar

[18]

P. Lax and R. Phillips, Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces. I, Comm. Pure Appl. Math., 37 (1984), 303–328. doi: 10.1002/cpa.3160370304. Google Scholar

[19]

W. Lu, S. Sridhar and M. Zworski, Fractal Weyl laws for chaotic open systems, Phys. Rev. Lett., 91 (2003), 154101. doi: 10.1103/PhysRevLett.91.154101. Google Scholar

[20]

Y. Manin and M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Sel. Math., New Ser., 8 (2002), 475–521. doi: 10.1007/s00029-002-8113-3. Google Scholar

[21]

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

[22]

D. Mayer, The thermodynamic formalism approach to Selberg's zeta function for $ \rm{PSL} $(2, Z), Bull. Amer. Math. Soc. (N.S.), 25 (1991), 55–60. doi: 10.1090/S0273-0979-1991-16023-4. Google Scholar

[23]

D. Mayer, T. Mühlenbruch and F. Strömberg, The transfer operator for the Hecke triangle groups, Discrete Contin. Dyn. Syst., Ser. A, 32 (2012), 2453–2484. doi: 10.3934/dcds.2012.32.2453. Google Scholar

[24]

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

[25]

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

[26]

F. Naud, Density and location of resonances for convex co-compact hyperbolic surfaces, Invent. Math., 195 (2014), 723–750. doi: 10.1007/s00222-013-0463-2. Google Scholar

[27]

S. Nonnenmacher, J. Sjöstrand and M. Zworski, Fractal Weyl law for open quantum chaotic maps, Ann. of Math. (2), 179 (2014), 179–251. doi: 10.4007/annals.2014.179.1.3. Google Scholar

[28]

S. Nonnenmacher and M. Zworski, Fractal Weyl laws in discrete models of chaotic scattering, J. Phys. A, 38 (2005), 10683–10702. Google Scholar

[29]

S. Patterson and P. Perry, The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein, Duke Math. J., 106 (2001), 321–390. doi: 10.1215/S0012-7094-01-10624-8. Google Scholar

[30]

A. Pohl, A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area, Commun. Math. Phys., 337 (2015), 103–126. doi: 10.1007/s00220-015-2304-1. Google Scholar

[31]

A. Pohl, Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations, Contemp. Math., 669 (2016), 205–236. doi: 10.1090/conm/669/13430. Google Scholar

[32]

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

[33]

D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231–242. doi: 10.1007/BF01403069. Google Scholar

[34]

E. Seiler and B. Simon, An inequality among determinants, Proc. Nat. Acad. Sci. USA, 72 (1975), 3277–3278. doi: 10.1073/pnas.72.9.3277. Google Scholar

[35]

J. Sjöstrand, Geometric bounds on the density of resonances for semiclassical problems, Duke Math. J., 60 (1990), 1–57. doi: 10.1215/S0012-7094-90-06001-6. Google Scholar

[36]

J. Sjöstrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J., 137 (2007), 381–459. doi: 10.1215/S0012-7094-07-13731-1. Google Scholar

[37]

B. Stratmann and M. Urbański, The box-counting dimension for geometrically finite Kleinian groups, Fundam. Math., 149 (1996), 83–93. Google Scholar

[38]

A. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math., 153 (1982); A translation of Trudy Mat. Inst. Steklov., 153 (1981), 172pp. Google Scholar

[39]

A. Venkov and P. Zograf, On analogues of the Artin factorization formulas in the spectral theory of automorphic functions connected with induced representations of Fuchsian groups, Math. USSR, Izv., 21 (1983), 435-443. Google Scholar

[40]

M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Invent. Math., 136 (1999), 353–409. doi: 10.1007/s002220050313. Google Scholar

Figure 1.  Fundamental domain for the Hecke triangle group $ \Gamma_w $ with cusp width $ w>2 $
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