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Upper Bound on the Density of Ruelle Resonances for Anosov Flows

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Abstract

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (−i)V, called Ruelle resonances, close to the real axis and for large real parts.

Résumé

Par une approche semiclassique on montre que le spectre d’un champ de vecteur d’Anosov V sur une variété compacte est discret (dans des espaces de Sobolev anisotropes adaptés). On montre ensuite une majoration de la densité de valeurs propres de l’opérateur (−i)V, appelées résonances de Ruelle, près de l’axe réel et pour les grandes parties réelles.

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References

  1. Aguilar J., Combes J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Arnold, V.I., Avez, A.: Méthodes ergodiques de la mécanique classique. Paris: Gauthier Villars, 1967

  3. Baladi, V.: Anisotropic Sobolev spaces and dynamical transfer operators: C ∞ foliations. In: Kolyada, S. (ed.) et al., Algebraic and topological dynamics. Proceedings of the conference, Bonn, Germany, May 1-July 31, 2004. Providence, RI: Amer. Math. Soc., Contemporary Mathematics, 385, 2005, pp. 123–135

  4. Baladi V., Tsujii M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57, 127–154 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Balslev E., Combes J.M.: Spectral properties of many-body Schrödinger operators with dilatation- analytic interactions. Commun. Math. Phys. 22, 280–294 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Blank M., Keller G., Liverani C.: Ruelle-Perron-Frobenius spectrum for Anosov maps. Nonlinearity 15, 1905–1973 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Bonatti C., Guelman N.: Transitive anosov flows and axiom-a diffeomorphisms. Erg. Th. Dyn. Sys. 29(3), 817–848 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Borthwick, D.: Spectral theory of infinite-area hyperbolic surfaces. Basel-Boston: Birkhauser, 2007

  9. Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge: Cambridge University Press, 2002

  10. Butterley O., Liverani C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2), 301–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators, with application to quantum mechanics and global geometry. (Springer Study ed.). Texts and Monographs in Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1987

  12. Davies, E.B.: Linear operators and their spectra. Cambridge: Cambridge University Press, 2007

  13. Dolgopyat D.: On decay of correlations in Anosov flows. Ann. of Math. (2) 147(2), 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dolgopyat D.: On mixing properties of compact group extensions of hyperbolic systems. Israel J. Math. 130, 157–205 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Faure F.: Semiclassical origin of the spectral gap for transfer operators of a partially expanding map. Nonlinearity 24, 1473–1498 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Faure F., Roy N.: Ruelle-Pollicott resonances for real analytic hyperbolic map. Nonlinearity 19, 1233–1252 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Faure F., Roy N., Sjöstrand J.: A semiclassical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. Journal. 1, 35–81 (2008)

    Article  MATH  Google Scholar 

  18. Field M., Melbourne I., Török A.: Stability of mixing and rapid mixing for hyperbolic flows. Ann. of Math. (2) 166(1), 269–291 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gérard C., Sjöstrand J.: Resonances en limite semiclassique et exposants de Lyapunov. Commun. Math. Phys. 116(2), 193–213 (1988)

    Article  ADS  MATH  Google Scholar 

  20. Ghys E.: Flots d’Anosov dont les feuilletages stables sont différentiables. Ann. Sci. École Norm. Sup. (4) 20(2), 251–270 (1987)

    MathSciNet  MATH  Google Scholar 

  21. Ghys E.: Déformations de flots d’Anosov et de groupes fuchsiens. Ann. Inst. Fourier (Grenoble) 42(1-2), 209–247 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gouzel S., Liverani C.: Banach spaces adapted to Anosov systems. Erg. Th. Dyn. Sys. 26, 189–217 (2005)

    Article  Google Scholar 

  23. Grigis, A., Sjöstrand, J.: Microlocal analysis for differential operators. Volume 196 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1994

  24. Guillope L., Lin K., Zworski M.: The Selberg zeta function for convex co-compact. Schottky groups. Commun. Math. Phys. 245(1), 149–176 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Helffer, B., Sjöstrand, J.: Résonances en limite semi-classique. (resonances in semi-classical limit). Memoires de la S.M.F., 24/25, 1986

  26. Hitrik, M., Sjöstrand, J.: Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2. Ann. Scient. de l’école normale supérieure. http://arxiv.org/abs/math/0703394v1 [math.SP], 2008

  27. Hurder S., Katok A.: Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publ. Math., Inst. Hautes étud. Sci. 72, 5–61 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hörmander L.: Fourier integral operators. I. Acta Math. 127(1), 79–183 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995

  30. Leboeuf P.: Periodic orbit spectrum in terms of Ruelle-Pollicott resonances. Phys. Rev. E (3) 69(2), 026204 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  31. Liverani C.: On contact Anosov flows. Ann. of Math. (2) 159(3), 1275–1312 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liverani C.: Fredholm determinants, Anosov maps and Ruelle resonances. Disc. Cont. Dyn. Sys. 13(5), 1203–1215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Martinez A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. New York, NY, Springer (2002)

    Google Scholar 

  34. McDuff D., Salamon D: Introduction to symplectic topology, 2nd edition. Clarendon press, Oxford (1998)

    Google Scholar 

  35. Nonnenmacher S.: Some open questions in ‘wave chaos’. Nonlinearity 21(8), T113–T121 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Nonnenmacher S., Zworski M.: Distribution of resonances for open quantum maps. Comm. Math. Phys. 269(2), 311–365 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Pesin, Y.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zünch: European Mathematical Society, 2004

  38. Reed, M., Simon, B.: Mathematical methods in physics, Vol. I: Functional Analysis. New York: Academic Press, 1972

  39. Reed, M., Simon, B.: Mathematical methods in physics, Vol. IV: Analysis of operators. New York: Academic Press, 1978

  40. Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium. Statistical mechanics. With a foreword by Giovanni Gallavotti. Reading, MA: Addison-Wesley Publishing Company, 1978

  41. Ruelle D.: Locating resonances for axiom A dynamical systems. J. Stat. Phys. 44, 281–292 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Cannas Da Salva, A.: Lectures on Symplectic Geometry. Berlin-Heidelberg-New York: Springer, 2001

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Sjöstrand, J.: Density of resonances for strictly convex analytic obstacles. Canad. J. Math. 48(2), 397–447, (1996) (with an appendix by M. Zworski)

    Google Scholar 

  45. Sjöstrand, J.: Lecture on resonances. Available on http://www.math.polytechnique.fr/~sjoestrand/NowListe070411.html, 2002

  46. Sjöstrand J.: Resonances associated to a closed hyperbolic trajectory in dimension 2. Asym. Anal. 36(2), 93–113 (2003)

    MATH  Google Scholar 

  47. Sjöstrand J., Zworski M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381–459 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Taylor, M.: Partial differential equations, Vol. I. Berlin-Heidelberg-New York: Springer, 1996

  49. Taylor, M.: Partial differential equations, Vol. II. Berlin-Heidelberg-New York: Springer, 1996

  50. Tsujii M.: Decay of correlations in suspension semi-flows of angle-multiplying maps. Erg. Th. Dyn. Sys. 28, 291–317 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tsujii M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23, 1495 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. Tsujii, M.: Contact Anosov flows and the FBI transform. http://arXiv.org/abs/1010.0396v2 [math.DS], 2010

  53. Wong, M.W.: An introduction to pseudo-differential operators. 2nd ed., River Edge, NJ: World Scientific Publishing Co. Inc., 1999

  54. Wunsch J., Zworski M.: The FBI transform on compact \({\mathcal{C}^\infty}\) manifolds. Trans. Am. Math. Soc. 353(3), 1151–1167 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zelditch, S.: Quantum ergodicity and mixing of eigenfunctions. Elsevier Encyclopedia of Math. Phys., vol. 1, Oxford: Elsevier, 2006, pp. 183–196

  56. Zworski M.: Resonances in physics and geometry. Notices of the A.M.S. 46(3), 319–328 (1999)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut Fourier, UMR 5582, 100 rue des Maths, BP 74, 38402, St Martin d’Hères, France

    Frédéric Faure

  2. IMB, UMR 5584, Université de Bourgogne, 9, Avenue Alain Savary, BP 47870, 21078, Dijon Cedex, France

    Johannes Sjöstrand

Authors
  1. Frédéric Faure
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  2. Johannes Sjöstrand
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Correspondence to Frédéric Faure.

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Communicated by S. Zelditch

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Faure, F., Sjöstrand, J. Upper Bound on the Density of Ruelle Resonances for Anosov Flows. Commun. Math. Phys. 308, 325–364 (2011). https://doi.org/10.1007/s00220-011-1349-z

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  • DOI: https://doi.org/10.1007/s00220-011-1349-z

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