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The Selberg Zeta Function for Convex Co-Compact Schottky Groups

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Abstract

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space ℍn +1: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|n +1) , and it gives new bounds on the number of resonances (scattering poles) of Γ\ℍn +1 . The proof of this result is based on the application of holomorphic L 2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\ℍn +1 as the simplest model of quantum chaotic scattering.

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References

  1. Anderson, J.W., Rocha, A.C.: Analyticity of Hausdorff dimension of limit sets of Kleinian groups. Ann. Acad. Sci. Fennicae 22, 349–364 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Bunke, U., Olbrich, M.: Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. Math. 149, 627–689 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Button, J.: All Fuchsian Schottky groups are classical Schottky groups. Geometry and Topology Monographs, 1, 1998, The Epstein Birthday Schrift, pp. 117–125

    Google Scholar 

  4. Eckhardt, B., Russberg, G., Cvitanović, P., Rosenqvist, P.E., Scherer, P.: Pinball Scattering. In: Quantum Chaos, Cambridge: Cambridge Univ. Press, 1995, pp. 405–433

  5. Fried, D.: The Zeta functions of Ruelle and Selberg. Ann. Sc. Éc. Norm. Sup 19, 491–517 (1986)

    MathSciNet  MATH  Google Scholar 

  6. Gohberg, I.C., Krein, M.G.: An introduction to the theory of linear non-self-adjoint operators. Translation of Mathematical Monographs, 18, Providence, RI: A.M.S., 1969

  7. Guillopé, L.: Sur la distribution des longueurs des géodésiques fermées d’une surface compacte à bord totalement géodésique. Duke Math. J. 53, 827–848 (1986)

    MathSciNet  Google Scholar 

  8. Guillopé, L., Zworski, M.: Scattering asymptotics for Riemann surfaces. Ann. Math. 145, 597–660 (1997)

    MathSciNet  Google Scholar 

  9. Jenkinson, O., Pollicott, M.: Calculating Hausdorff dimension of Julia sets and Kleinian limit sets. Am. J. Math. 124, 495–545 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Lalley, S.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits. Acta Math. 163, 1–55 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, 1998

  12. Lin, K.: Numerical study of quantum resonances in chaotic scattering. J. Comp. Phys. 176, 295–329 (2002)

    Article  MATH  Google Scholar 

  13. Lin, K., Zworski, M.: Quantum resonances in chaotic scattering. Chem. Phys. Lett. 355, 201–205 (2002)

    Article  Google Scholar 

  14. Lu, W., Sridhar, S., Zworski, M.: Fractal Weyl laws for chaotic open systems. Phys. Rev. Lett. 91, 154101 (2003)

    Article  Google Scholar 

  15. Maskit, B.: A characterization of Schottky groups. J. Analyse Math. 19, 227–230 (1967)

    MATH  Google Scholar 

  16. Maskit, B.: Kleinian groups. Grundlehren Math. Wiss. 287, Berlin: Springer-Verlag, 1988

  17. Mather, J.N.: Characterization of Anosov diffeomorphisms. Nederl. Akad. Wet. Proc., Ser. A 71, 479–483 (1968)

    Google Scholar 

  18. McMullen, C.: The classification of conformal dynamical systems. In: Current Developments in Mathematics, 1995, Cambridge: International Press, 1995, pp. 323–360

  19. McMullen, C.: Hausdorff dimension and conformal dynamics III: Computation of dimension. Am. J. Math. 120, 691–721 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Melrose, R.B.: Polynomial bounds on the number of scattering poles. J. Funct. Anal. 53, 287–303 (1983)

    MathSciNet  MATH  Google Scholar 

  21. Naud, F.: Expanding maps on Cantor sets, analytic continuation of zeta functions with applications to convex co-compact surfaces. Preprint, 2003

  22. Patterson, S.J., Perry, P.: Divisor of the Selberg zeta function for Kleinian groups in even dimensions, with an appendix by C. Epstein. Duke. Math. J. 326, 321–390 (2001)

    Google Scholar 

  23. Perry, P.: Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Geom. Funct. Anal. 11, 132–141 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Perry, P.: A Poisson formula and lower bounds on the number of resonances for hyperbolic manifolds. Int. Math. Res. Notices 34, 1837–1851 (2003)

    Article  Google Scholar 

  25. Petkov, V., Zworski, M.: Breit-Wigner approximation and the distribution of resonances. Commun. Math. Phys. 204, 329–351 (1999); Erratum, Commun. Math. Phys. 214, 733–735 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Poincaré, H.: Mémoire sur les groupes Kleinéens. Acta Math. 3, 49–92 (1883)

    Google Scholar 

  27. Pollicott, M.: Kleinian groups, Laplacian on forms and currents at infinity. Proc. Am. Math. Soc. 110, 269–279 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Pollicott, M.: Some applications of thermodynamic formalism to manifolds with constant negative curvature. Adv. Math. 85, 161–192 (1991)

    MathSciNet  MATH  Google Scholar 

  29. Pollicott, M., Rocha, A.C.: A remarkable formula for the determinant of the Laplacian. Inv. Math. 130, 399–414 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ratcliffe, J.G.: Foundations of hyperbolic manifolds. Graduate Texts in Mathematics. 149, Berlin: Springer-Verlag, 1994

  31. Ruelle, D.: Zeta-functions for expanding Anosov maps and flows. Inv. Math. 34, 231–242 (1976)

    MATH  Google Scholar 

  32. Rugh, H.H.: The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5, 1237–1263 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sjöstrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60, 1–57 (1990)

    MathSciNet  Google Scholar 

  34. Strain, J., Zworski, M.: Growth of the zeta function for a quadratic map and the dimension of the Julia set. Preprint, 2003

  35. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Publ. Math. IHÉS 50, 259–277 (1979)

    Google Scholar 

  36. Wunsch, J., Zworski, M.: Distribution of resonances for asymptotically euclidean manifolds. J. Diff. Geom. 55, 43–82 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Zworski, M.: Sharp polynomial bounds on the number of scattering poles. Duke Math. J. 59, 311–323 (1989)

    MathSciNet  MATH  Google Scholar 

  38. Zworski, M.: Dimension of the limit set and the distribution of resonances for convex co-compact hyperbolic surfaces. Inv. Math. 136, 353–409 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zworski, M.: Resonances in Physics and Geometry. Notices Am. Math. Soc. 46, 319–328 (1999)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jean Leray (UMR CNRS-UN 6629), Département de Mathématiques, Faculté des Sciences et des Techniques, 2, rue de la Houssinière, 44322, Nantes Cedex 3, France

    Laurent Guillopé

  2. University of California, Mathematics Department, Evans Hall, Berkeley, CA 94720, USA

    Kevin K. Lin & Maciej Zworski

Authors
  1. Laurent Guillopé
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  2. Kevin K. Lin
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  3. Maciej Zworski
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Correspondence to Maciej Zworski.

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Communicated by P. Sarnak

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Guillopé, L., Lin, K. & Zworski, M. The Selberg Zeta Function for Convex Co-Compact Schottky Groups. Commun. Math. Phys. 245, 149–176 (2004). https://doi.org/10.1007/s00220-003-1007-1

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