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2019, 26: 24-35.
doi: 10.3934/era.2019.26.003
Fractal Weyl bounds and Hecke triangle groups
| 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 |
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
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References:
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T. Apostol, On the Lerch zeta function, Pacific J. Math., (1951), 161–167.
doi: 10.2140/pjm.1951.1.161. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
S. Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, To appear in JEMS. |
| [7] |
S. Dyatlov and M. Zworski, Fractal Uncertainty for Transfer Operators , International Mathematics Research Notices, 2018.
doi: 10.1093/imrn/rny026. |
| [8] |
F. Faure and M. Tsuiji, Fractal Weyl law for the Ruelle spectrum of Anosov flows, arXiv: 1706.09307. |
| [9] |
K. Fedosova and A. Pohl, Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, arXiv: 1709.00760. |
| [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. |
| [11] |
D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487–521.
doi: 10.1007/s002220050084. |
| [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. |
| [13] |
L. Guillopé and M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2), 145 (1997), 597–660.
doi: 10.2307/2951846. |
| [14] |
E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen, Math. Ann., 112 (1936), 664–699.
doi: 10.1007/BF01565437. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311–333.
doi: 10.1007/BF02473355. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
S. Nonnenmacher and M. Zworski, Fractal Weyl laws in discrete models of chaotic scattering, J. Phys. A, 38 (2005), 10683–10702. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231–242.
doi: 10.1007/BF01403069. |
| [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. |
| [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. |
| [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. |
| [37] |
B. Stratmann and M. Urbański, The box-counting dimension for geometrically finite Kleinian groups, Fundam. Math., 149 (1996), 83–93. |
| [38] |
A. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math., 153 (1982); A translation of Trudy Mat. Inst. Steklov., 153 (1981), 172pp. |
| [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.
|
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
S. Dyatlov, Improved fractal Weyl bounds for hyperbolic manifolds, To appear in JEMS. |
| [7] |
S. Dyatlov and M. Zworski, Fractal Uncertainty for Transfer Operators , International Mathematics Research Notices, 2018.
doi: 10.1093/imrn/rny026. |
| [8] |
F. Faure and M. Tsuiji, Fractal Weyl law for the Ruelle spectrum of Anosov flows, arXiv: 1706.09307. |
| [9] |
K. Fedosova and A. Pohl, Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, arXiv: 1709.00760. |
| [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. |
| [11] |
D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487–521.
doi: 10.1007/s002220050084. |
| [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. |
| [13] |
L. Guillopé and M. Zworski, Scattering asymptotics for Riemann surfaces, Ann. of Math. (2), 145 (1997), 597–660.
doi: 10.2307/2951846. |
| [14] |
E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen, Math. Ann., 112 (1936), 664–699.
doi: 10.1007/BF01565437. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys., 130 (1990), 311–333.
doi: 10.1007/BF02473355. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [28] |
S. Nonnenmacher and M. Zworski, Fractal Weyl laws in discrete models of chaotic scattering, J. Phys. A, 38 (2005), 10683–10702. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), 231–242.
doi: 10.1007/BF01403069. |
| [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. |
| [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. |
| [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. |
| [37] |
B. Stratmann and M. Urbański, The box-counting dimension for geometrically finite Kleinian groups, Fundam. Math., 149 (1996), 83–93. |
| [38] |
A. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. Math., 153 (1982); A translation of Trudy Mat. Inst. Steklov., 153 (1981), 172pp. |
| [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.
|
| [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. |
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