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Sharp polynomial bounds on the number of Pollicott–Ruelle resonances

Published online by Cambridge University Press:  11 March 2013

KIRIL DATCHEV
Affiliation:
Department of Mathematics, 77 Massachusetts Avenue, Massachusetts Institute of Technology, Cambridge, MA 02139, USA email datchev@math.mit.edu
SEMYON DYATLOV
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA email dyatlov@math.berkeley.eduzworski@math.berkeley.edu
MACIEJ ZWORSKI
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA email dyatlov@math.berkeley.eduzworski@math.berkeley.edu

Abstract

We give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.

Type
Research Article
Copyright
Copyright ©2013 Cambridge University Press 

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References

Baladi, V.. Anisotropic Sobolev spaces and dynamical transfer operators: ${C}^{\infty } $ foliations. Algebraic and Topological Dynamics (Contemporary Mathematics, 385). American Mathematical Society, Providence, RI, 2005, pp. 123135.Google Scholar
Baladi, V. and Tsujii, M.. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57 (1) (2007), 127154.CrossRefGoogle Scholar
Bérard, P.. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155 (3) (1977), 249276.Google Scholar
Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15 (6) (2002), 19051973.CrossRefGoogle Scholar
Bony, J.-M. and Chemin, J.-Y.. Espaces fonctionnels associés au calcul de Weyl–Hörmander. Bull. Soc. Math. France 122 (1) (1994), 77118.CrossRefGoogle Scholar
Butterley, O. and Liverani, C.. Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1 (2) (2007), 301322.CrossRefGoogle Scholar
Datchev, K. and Dyatlov, S.. Fractal Weyl laws for asymptotically hyperbolic manifolds. Geom. Funct. Anal., to appear; arXiv:1206.2255v2.Google Scholar
Faure, F., Roy, N. and Sjöstrand, J.. A semiclassical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. J. 1 (2008), 3581.CrossRefGoogle Scholar
Faure, F. and Sjöstrand, J.. Upper bound on the density of Ruelle resonances for Anosov flows. Comm. Math. Phys. 308 (2) (2011), 325364.Google Scholar
Faure, F. and Tsujii, M.. Prequantum transfer operator for Anosov diffeomorphism (preliminary version). Preprint, arXiv:1206.0282.Google Scholar
Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26 (1) (2006), 189217.Google Scholar
Helffer, B. and Sjöstrand, J.. Résonances en limite semi-classique (Resonances in semi-classical limit). Mémoires de la S.M.F. 24/25 (1986).Google Scholar
Hörmander, L.. The Analysis of Linear Partial Differential Operators. Vol. III. Springer, Berlin, 1985.Google Scholar
Leboeuf, P.. Periodic orbit spectrum in terms of Ruelle–Pollicott resonances. Phys. Rev. E (3) 69 (2) (2004), 026204.Google Scholar
Liverani, C.. On contact Anosov flows. Ann. of Math. (2) 159 (3) (2004), 12751312.Google Scholar
Liverani, C.. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete Contin. Dyn. Syst. 13 (5) (2005), 12031215.Google Scholar
Melrose, R. B.. Polynomial bounds on the distribution of poles in scattering by an obstacle, Journées ‘Équations aux Dérivées partielles’, Saint-Jean de Monts, 1984.http://archive.numdam.org/article/JEDP_1984____A3_0.djvu.CrossRefGoogle Scholar
Nonnenmacher, S., Sjöstrand, J. and Zworski, M.. Fractal Weyl law for open quantum chaotic maps. Preprint, arXiv:1105.3128.Google Scholar
Pollicott, M.. On the rate of mixing of Axiom A flows. Invent. Math. 81 (1986), 147164.Google Scholar
Randol, B.. The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator. Trans. Amer. Math. Soc. 236 (1978), 209223.Google Scholar
Ruelle, D.. Resonances of chaotic dynamical systems. Phys. Rev. Lett. 56, 405–407.Google Scholar
Sjöstrand, J.. Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60 (1) (1990), 157.CrossRefGoogle Scholar
Sjöstrand, J. and Zworski, M.. Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137 (3) (2007), 381459.CrossRefGoogle Scholar
Tsujii, M.. Decay of correlations in suspension semi-flows of angle-multiplying maps. Ergod. Th. & Dynam. Sys. 28 (1) (2008), 291317.Google Scholar
Tsujii, M.. Quasi-compactness of transfer operators for Anosov flows. Nonlinearity 23 (7) (2010), 14951545.Google Scholar
Tsujii, M.. Contact Anosov flows and the FBI transform. Preprint, arXiv:1010.0396.Google Scholar
Vodev, G.. Sharp bounds on the number of scattering poles in even-dimensional spaces. Duke Math. J. 74 (1994), 117.CrossRefGoogle Scholar
Zworski, M.. Sharp polynomial bounds on the number of scattering poles. Duke Math. J. 59 (2) (1989), 311323.CrossRefGoogle Scholar
Zworski, M.. Semiclassical Analysis (Graduate Studies in Mathematics, 138). American Mathematical Society, Providence, RI, 2012.Google Scholar