Band structure of the Ruelle spectrum of contact Anosov flows

Frédéric Faure; Masato Tsujii

doi:10.1016/j.crma.2013.04.022

Dynamical Systems/Mathematical Physics

Band structure of the Ruelle spectrum of contact Anosov flows
[Structure en bandes du spectre de Ruelle des flots dʼAnosov de contact]

Présenté par : the Editorial Board

Frédéric Faure ¹ ; Masato Tsujii ²

¹ Institut Fourier, UMR 5582, 100, rue des Maths, BP74, 38402 Saint-Martin-dʼHères, France
² Department of Mathematics, Kyushu University, Moto-oka 744, Nishi-ku, Fukuoka, 819-0395, Japan

Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 385-391.

Résumé
Abstract

Si X est un champ de vecteur dʼAnosov de contact sur une variété compacte lisse M et si $V \in C^{\infty} (M)$ , il est connu que lʼopérateur différentiel $A = - X + V$ a un spectre discret appelé résonances de Ruelle–Pollicott dans des espaces de Sobolev spécifiques. On montre que, pour $| Im z | \to \infty$ , les valeurs propres de A sont incluses dans des bandes verticales et que, dans les gaps entre ces bandes, la résolvante de A est bornée uniformément par rapport à $| Im (z) |$ . Dans chaque bande isolée, la densité des valeurs propres est donnée par une loi de Weyl. Dans la première bande, la plupart des valeurs propres se concentrent sur la ligne verticale $Re (z) = {〈 D 〉}_{M}$ , qui est la moyenne spatiale de la fonction $D (x) = V (x) - \frac{1}{2} div X_{| E_{u} (x)}$ , où $E_{u}$ est la distribution instable. Ce spectre en bande permet dʼexprimer le comportement asymptotique des fonctions de corrélations dynamiques.

If X is a contact Anosov vector field on a smooth compact manifold M and $V \in C^{\infty} (M)$ , it is known that the differential operator $A = - X + V$ has some discrete spectrum called Ruelle–Pollicott resonances in specific Sobolev spaces. We show that for $| Im z | \to \infty$ the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to $| Im (z) |$ . In each isolated band, the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate to the vertical line $Re (z) = {〈 D 〉}_{M}$ , the space average of the function $D (x) = V (x) - \frac{1}{2} div X_{| E_{u} (x)}$ where $E_{u}$ is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.

Reçu le : 2013-01-24
Accepté le : 2013-04-30
Publié le : 2013-05-01

DOI : 10.1016/j.crma.2013.04.022

Affiliations des auteurs :

Frédéric Faure ¹ ; Masato Tsujii ²

¹ Institut Fourier, UMR 5582, 100, rue des Maths, BP74, 38402 Saint-Martin-dʼHères, France
² Department of Mathematics, Kyushu University, Moto-oka 744, Nishi-ku, Fukuoka, 819-0395, Japan

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     author = {Fr\'ed\'eric Faure and Masato Tsujii},
     title = {Band structure of the {Ruelle} spectrum of contact {Anosov} flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {385--391},
     publisher = {Elsevier},
     volume = {351},
     number = {9-10},
     year = {2013},
     doi = {10.1016/j.crma.2013.04.022},
     language = {en},
}

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Frédéric Faure; Masato Tsujii. Band structure of the Ruelle spectrum of contact Anosov flows. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 385-391. doi : 10.1016/j.crma.2013.04.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.022/

Bibliographie
Cité par

[1] O. Butterley; C. Liverani Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., Volume 1 (2007) no. 2, pp. 301-322

[2] K. Datchev; S. Dyatlov; M. Zworski Sharp polynomial bounds on the number of Ruelle resonances, 2012 (preprint) | arXiv

[3] D. Dolgopyat On decay of correlations in Anosov flows, Ann. Math. (2), Volume 147 (1998) no. 2, pp. 357-390

[4] S. Dyatlov Resonance projectors and asymptotics for r-normally hyperbolic trapped sets, 2013 (preprint) | arXiv

[5] F. Faure Prequantum chaos: Resonances of the prequantum cat map, J. Mod. Dyn., Volume 1 (2007) no. 2, pp. 255-285 | arXiv

[6] F. Faure; J. Sjöstrand Upper bound on the density of Ruelle resonances for Anosov flows. A semiclassical approach, Commun. Math. Phys., Volume 308 (2011) no. 2, pp. 325-364 | arXiv

[7] F. Faure; M. Tsujii Prequantum transfer operator for symplectic Anosov diffeomorphism, 2012 (preprint) | arXiv

[8] F. Faure, M. Tsujii, Spectrum and zeta function of contact Anosov flows, paper in preparation.

[9] P. Giulietti; C. Liverani; M. Pollicott Anosov flows and dynamical zeta functions, Ann. Math. (2013) (in press) | arXiv

[10] C. Liverani On contact Anosov flows, Ann. Math. (2), Volume 159 (2004) no. 3, pp. 1275-1312

[11] A. Martinez An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, New York, NY, 2002

[12] S. Nonnenmacher; M. Zworski Decay of correlations for normally hyperbolic trapping, 2013 | arXiv

[13] J. Sjöstrand Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci., Volume 36 (2000) no. 5, pp. 573-611

[14] M. Taylor Partial Differential Equations, vol. I, Springer, 1996

[15] M. Tsujii Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, Volume 23 (2010) no. 7, pp. 1495-1545 | arXiv

[16] M. Tsujii Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform, Ergod. Theory Dyn. Syst., Volume 32 (2012) no. 6, pp. 2083-2118

[17] M. Zworski Semiclassical Analysis, Graduate Studies in Mathematics Series, American Mathematical Society, 2012

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