Abstract
We obtain an essential spectral gap for n-dimensional convex co-compact hyperbolic manifolds with the dimension \({\delta}\) of the limit set close to \({{n-1\over 2}}\). The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors–David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.
Similar content being viewed by others
References
J. Anderson and A. Rocha. Analyticity of Hausdorff dimension of limit sets of Kleinian groups. Annales Academiæ Scientiarum Fennicæ Mathematica 22 (1997), 349–364
S. Barkhofen, T. Weich, A. Potzuweit, H.-J. Stöckmann, U. Kuhl, and M. Zworski. Experimental observation of the spectral gap in microwave \(n\)-disk systems. Physical Review Letters 110 (2013), 164102
M. Bond, I. Łaba, and J. Zahl. Quantitative visibility estimates for unrectifiable sets in the plane. arXiv:1306.5469 (preprint)
J.-F. Bony and L. Michel. Microlocalization of resonant states and estimates of the residue of the scattering amplitude. Communications in Mathematical Physics 246 (2004), 375–402
D. Borthwick. Spectral Theory of Infinite-Area Hyperbolic Surfaces. Birkhäuser, New York (2007).
D. Borthwick. Distribution of resonances for hyperbolic surfaces. Experimental Mathematics 23 (2014), 25–45
D. Borthwick and T. Weich. Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions. Journal of Spectral Theory 6 (2016), 267–329
J. Bourgain, A. Gamburd, and P. Sarnak. Generalization of Selberg's 3/16 theorem and affine sieve. Acta Mathematica 207 (2011), 255–290
U. Bunke and M. Olbrich. Fuchsian groups of the second kind and representations carried by the limit set. Inventiones mathematicae 127 (1997), 127–154
U. Bunke and M. Olbrich. Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Annals of Mathematics (2)149 (1999), 627–689
M. Coornaert. Mesures de Patterson–Sullivan sur le bord d'un espace hyperbolique au sens de Gromov. Pacific Journal of Mathematics 159 (1993), 241–270
K. Datchev. Resonance free regions for nontrapping manifolds with cusps. Analysis&PDE 9 (2016), 907–953
K. Datchev and S. Dyatlov. Fractal Weyl laws for asymptotically hyperbolic manifolds. Geometric and Functional Analysis 23 (2013), 1145–1206
G. David and S. Semmes. Fractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure. Oxford University Press, Oxford (1997).
J.-M. Delort. FBI Transformation, Second Microlocalization, and Semilinear Caustics. Springer, Berlin (1992).
D. Dolgopyat. On decay of correlations in Anosov flows. Annals of Mathematics (2)147 (1998), 357–390
S. Dyatlov. Resonance projectors and asymptotics for \(r\)-normally hyperbolic trapped sets. Journal of the American Mathematical Society 28 (2015), 311–381
S. Dyatlov, F. Faure, and C. Guillarmou. Power spectrum of the geodesic flow on hyperbolic manifolds. Analysis& PDE 8 (2015), 923–1000
S. Dyatlov and C. Guillarmou. Microlocal limits of plane waves and Eisenstein functions. Annales de l’ENS (4)47 (2014), 371–448
S. Dyatlov and C. Guillarmou. Pollicott–Ruelle resonances for open systems. Annales Henri Poincaré arXiv:1410.5516 (published online)
S. Dyatlov and M. Zworski. Dynamical zeta functions for Anosov flows via microlocal analysis. Annales Scientifiques de l’École Normale Supérieure 49 (2016), 543–577
S. Dyatlov and M. Zworski. Mathematical Theory of Scattering Resonances. http://math.mit.edu/~dyatlov/res/ (book in progress)
G. Freĭman. Foundations of a Structural Theory of Set Addition. Translations of Mathematical Monographs, Vol. 37. American Mathematical Society, Providence (1973).
P. Gaspard and S. Rice. Scattering from a classically chaotic repeller. The Journal of Chemical Physics 90 (1989), 2225–2241
A. Grigis and J. Sjöstrand. Microlocal Analysis for Differential Operators: An Introduction. Cambridge University Press, Cambridge (1994).
C. Guillarmou. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Mathematical Journal 129 (2005), 1–37
V. Guillemin and S. Sternberg. Geometric Asymptotics. Mathematical Surveys, Vol. 14. American Mathematical Society, Providence (1977).
V. Guillemin and S. Sternberg. Semi-Classical Analysis. International Press, Boston (2013).
L. Guillopé and M. Zworski. Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity. Asymptotic Analysis 11 (1995), 1–22
L. Hörmander. The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1990).
L. Hörmander. The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators. Springer, Berlin (1994).
M. Ikawa. Decay of solutions of the wave equation in the exterior of several convex bodies. Annales de lnstitut Fourier 38 (1988), 113–146
D. Jakobson and F. Naud. On the critical line of convex co-compact hyperbolic surfaces. Geometric and Functional Analysis 22 (2012), 352–368
M. Magee, H. Oh, and D. Winter. Expanding maps and continued fractions. arXiv:1412.4284 (preprint)
R. Mazzeo and R. Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. Journal of Functional Analysis 75 (1987), 260–310
C. McMullen. Hausdorff dimension and conformal dynamics III: computation of dimension. American Journal of Mathematics 120 (1998), 691–721
F. Naud. Expanding maps on Cantor sets and analytic continuation of zeta functions. Annales de l’ENS (4)38 (2005), 116–153
S. Nonnenmacher and M. Zworski. Quantum decay rates in chaotic scattering. Acta Mathematica 203 (2009), 149–233
S. Nonnenmacher and M. Zworski. Decay of correlations for normally hyperbolic trapping. Inventiones Mathematicae 200 (2015), 345–438
H. Oh and D. Winter. 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 (2016), 1069–1115
S.J. Patterson. The Laplacian operator on a Riemann surface. I. Compositio Mathematica 31 (1975), 83–107
S.J. Patterson. The Laplacian operator on a Riemann surface. II. Compositio Mathematica 32 (1976), 71–112
S.J. Patterson. The limit set of a Fuchsian group. Acta Mathematica 136 (1976), 241–273
S.J. Patterson and P. Perry. The divisor of Selberg's zeta function for Kleinian groups. Duke Mathematical Journal 106 (2001), 321–390
P. Perry. The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory. Journal of Functional Analysis 75 (1987), 161–187
P. Perry. The Laplace operator on a hyperbolic manifold. II. Eisenstein series and the scattering matrix. Journal für die Reine und Angewandte Mathematik 398 (1989), 67–91
V. Petkov and L. Stoyanov. Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. Analysis PDE 3 (2010), 427–489
G. Petridis. New proofs of Plünnecke-type estimates for product sets in groups. Combinatorica 32 (2012), 721–733
J. Ratcliffe. Foundations of Hyperbolic Manifolds. 2nd edn. Springer, Berlin (2006).
I. Ruzsa. Sums of Finite Sets. In: D.V. Chudnovsky, G.V. Chudnovsky and M.B. Nathanson (eds.) Number Theory: New York Seminar. Springer, pp. 281–293 (1996).
T. Sanders. The structure theory of set addition revisited. Bulletin of the American Mathematical Society 50 (2013), 93–127
J. Sjöstrand and M. Zworski. Fractal upper bounds on the density of semiclassical resonances. Duke Mathematical Journal 137 (2007), 381–459
L. Stoyanov. Spectra of Ruelle transfer operators for axiom A flows. Nonlinearity 24 (2011), 1089–1120
L. Stoyanov. Non-integrability of open billiard flows and Dolgopyat-type estimates. Ergodic Theory and Dynamical Systems 32 (2012), 295–313
D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. Publications Mathématiques de l’IHES 50 (1979), 171–202
T. Tao and V. Vu. Additive Combinatorics. Cambridge Studies in Advanced Mathematics, Vol. 105. Cambridge University Press, Cambridge (2006).
T. Tao and V. Vu. John-type theorems for generalized arithmetic progressions and iterated sumsets. Advances in Mathematics 219 (2008), 428–449
A. Vasy. Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces. Inventiones Mathematicae 194 (2013), 381–513 (with an appendix by Semyon Dyatlov)
A. Vasy. Microlocal Analysis of Asymptotically Hyperbolic Spaces and High Energy Resolvent Estimates. In: G. Uhlmann (ed.) Inverse Problems and Applications. Inside Out II, MSRI publications, Vol. 60. Cambridge University Press, Cambridge (2013).
M. Zworski. Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Inventiones Mathematicae 136 (1999), 353–409
M. Zworski. Semiclassical Analysis. Graduate Studies in Mathematics, Vol. 138. American Mathematical Society, Providence (2012).
M. Zworski. Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited. Journal of Spectral Theory (to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dyatlov, S., Zahl, J. Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26, 1011–1094 (2016). https://doi.org/10.1007/s00039-016-0378-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-016-0378-3