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Geometric and Functional Analysis

, Volume 27, Issue 4, pp 744–771 | Cite as

Fourier dimension and spectral gaps for hyperbolic surfaces

  • Jean Bourgain
  • Semyon Dyatlov
Article

Abstract

We obtain an essential spectral gap for a convex co-compact hyperbolic surface \({M=\Gamma\backslash\mathbb H^2}\) which depends only on the dimension \({\delta}\) of the limit set. More precisely, we show that when \({\delta > 0}\) there exists \({\varepsilon_0=\varepsilon_0(\delta) > 0}\) such that the Selberg zeta function has only finitely many zeroes s with \({{\rm Re} s > \delta-\varepsilon_0}\). The proof uses the fractal uncertainty principle approach developed in Dyatlov and Zahl (Geom Funct Anal 26:1011–1094, 2016). The key new component is a Fourier decay bound for the Patterson–Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group \({\Gamma}\) are nonlinear, together with estimates on exponential sums due to Bourgain (J Anal Math 112:193–236, 2010) which follow from the discretized sum-product theorem in \({\mathbb{R}}\).

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References

  1. Bor16.
    D. Borthwick. Spectral Theory of Infinite-Area Hyperbolic Surfaces, second edition. Birkhäuser, Basel (2016).Google Scholar
  2. Bou03.
    Bourgain J.: On the Erdős–Volkmann and Katz–Tao ring conjectures. Geometric and Functional Analysis 13, 334–365 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. Bou10.
    Bourgain J.: The discretized sum-product and projection theorems. Journal d’Analyse Mathématique 112, 193–236 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. BD16.
    J. Bourgain and S. Dyatlov. Spectral Gaps Without the Pressure Condition, preprint, arXiv:1612.09040.
  5. BG12.
    Bourgain J., Gamburd A.: A spectral gap theorem in SU(d). Journal of the European Mathematical Society 14, 1455–1511 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. BGK06.
    Bourgain J., Glibichuk A., Konyagin S.: Estimates for the number of sums and products and for exponential sums in fields of prime order. Journal of the London Mathematical Society 73(2), 380–398 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. Dol98.
    Dolgopyat D.: On decay of correlations in Anosov flows. Annals of Mathematics 147(2), 357–390 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. DJ16.
    S. Dyatlov and L. Jin. Resonances for open quantum maps and a fractal uncertainty principle. Communications in Mathematical Physics, published online.Google Scholar
  9. DJ17.
    S. Dyatlov and L. Jin. Dolgopyat’s Method and Fractal Uncertainty Principle, preprint, arXiv:1702.03619.
  10. DZ16.
    Dyatlov S., Zahl J.: Spectral gaps, additive energy, and a fractal uncertainty principle. Geometric and Functional Analysis 26, 1011–1094 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. GR89.
    Gaspard P., Rice S.: Scattering from a classically chaotic repeller. The Journal of chemical physics 90, 2225–2241 (1989)MathSciNetCrossRefGoogle Scholar
  12. Gre09.
    B. Green. Sum-Product Phenomena in \({\mathbb{F}_p}\): A Brief Introduction, arXiv:0904.2075.
  13. Ika88.
    Ikawa M.: Decay of solutions of the wave equation in the exterior of several convex bodies. Annales de l’institut Fourier 38, 113–146 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. JS13.
    T. Jordan and T. Sahlsten, Fourier Transforms of Gibbs Measures for the Gauss Map, to appear in Math. Ann., arXiv:1312.3619.
  15. Mat95.
    Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  16. Nau05.
    Naud F.: Expanding maps on Cantor sets and analytic continuation of zeta functions. Annales de l’ENS, 38(4), 116–153 (2005)MathSciNetMATHGoogle Scholar
  17. Non11.
    Nonnenmacher S.: Spectral problems in open quantum chaos. Nonlinearity 24, R123 (2011)CrossRefMATHGoogle Scholar
  18. NZ09.
    Nonnenmacher S., Zworski M.: Quantum decay rates in chaotic scattering. Acta Mathematica 203, 149–233 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. OW16.
    Oh H., Winter D.: Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of \({{\rm SL}_2(\mathbb{Z})}\). Journal of the American Mathematical Society 29, 1069–1115 (2016)MathSciNetCrossRefMATHGoogle Scholar
  20. Pat76.
    Patterson S.J.: The limit set of a Fuchsian group. Acta Mathematica 136, 241–273 (1976)MathSciNetCrossRefMATHGoogle Scholar
  21. PS10.
    Petkov V., Stoyanov L.: Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. Analysis and PDE 3, 427–489 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. Sto11.
    Stoyanov L.: Spectra of Ruelle transfer operators for axiom A flows. Nonlinearity 24, 1089–1120 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. Sul79.
    Sullivan D.: The density at infinity of a discrete group of hyperbolic motions. Publications Mathématiques de l’IHÉS 50, 171–202 (1979)MathSciNetCrossRefMATHGoogle Scholar
  24. Zwo17.
    Zworski M.: Mathematical study of scattering resonances. Bulletin of Mathematical Sciences 7, 1–85 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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