Dynamical zeta functions  for Anosov flows  via microlocal analysis

Dyatlov, Semyon; Zworski, Maciej

doi:10.24033/asens.2290

Dynamical zeta functions for Anosov flows via microlocal analysis
[Fonctions zêta dynamiques pour les flots d'Anosov en utilisant l'analyse microlocale]

Dyatlov, Semyon ; Zworski, Maciej

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 543-577.

Résumé
Abstract

Cet article donne une courte preuve microlocale du prolongement méromorphe de la fonction zêta de Ruelle pour les flots d'Anosov lisses. Des résultats plus généraux ont été récemment obtenus par Giulietti-Liverani-Pollicott [13] mais notre approche est différente et se base sur l'étude du générateur du flot, que l'on considère comme un opérateur pseudodifférentiel semi-classique.

The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for $C^{\infty}$ Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [13] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator.

Publié le : 2016-11-12

Zbl | 1 citation dans Numdam

DOI : 10.24033/asens.2290

Classification : 37C30, 37D20.
Keywords: Dynamical zeta functions, Anosov flows, Pollicott-Ruelle resonances.
Mot clés : Fonctions zêta dynamiques, flots d'Anosov, résonances de Pollicott-Ruelle.

@article{ASENS_2016__49_3_543_0,
     author = {Dyatlov, Semyon and Zworski, Maciej},
     title = {Dynamical zeta functions  for {Anosov} flows  via microlocal analysis},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {543--577},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {3},
     year = {2016},
     doi = {10.24033/asens.2290},
     zbl = {1369.37028},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2290/}
}

TY  - JOUR
AU  - Dyatlov, Semyon
AU  - Zworski, Maciej
TI  - Dynamical zeta functions  for Anosov flows  via microlocal analysis
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 543
EP  - 577
VL  - 49
IS  - 3
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2290/
DO  - 10.24033/asens.2290
LA  - en
ID  - ASENS_2016__49_3_543_0
ER  -

%0 Journal Article
%A Dyatlov, Semyon
%A Zworski, Maciej
%T Dynamical zeta functions  for Anosov flows  via microlocal analysis
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 543-577
%V 49
%N 3
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2290/
%R 10.24033/asens.2290
%G en
%F ASENS_2016__49_3_543_0

Dyatlov, Semyon; Zworski, Maciej. Dynamical zeta functions  for Anosov flows  via microlocal analysis. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 543-577. doi : 10.24033/asens.2290. http://www.numdam.org/articles/10.24033/asens.2290/

Bibliographie
Cité par

Baladi, V., Algebraic and topological dynamics (Contemp. Math.), Volume 385, Amer. Math. Soc., Providence, RI, 2005, pp. 123-135 | DOI | MR | Zbl

Baladi, V.; Tsujii, M. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), Volume 57 (2007), pp. 127-154 http://aif.cedram.org/item?id=AIF_2007__57_1_127_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Datchev, K.; Dyatlov, S. Fractal Weyl laws for asymptotically hyperbolic manifolds, Geom. Funct. Anal., Volume 23 (2013), pp. 1145-1206 (ISSN: 1016-443X) | DOI | MR | Zbl

Datchev, K.; Dyatlov, S.; Zworski, M. Sharp polynomial bounds on the number of Pollicott-Ruelle resonances, Ergodic Theory Dynam. Systems, Volume 34 (2014), pp. 1168-1183 (ISSN: 0143-3857) | DOI | MR | Zbl

Dyatlov, S.; Faure, F.; Guillarmou, C. Power spectrum of the geodesic flow on hyperbolic manifolds, Anal. PDE, Volume 8 (2015), pp. 923-1000 (ISSN: 2157-5045) | DOI | MR | Zbl

Dyatlov, S.; Guillarmou, C. Pollicott-Ruelle resonances for open systems (preprint arXiv:1410.5516 )

Duistermaat, J. J. On Carleman estimates for pseudo-differential operators, Invent. math., Volume 17 (1972), pp. 31-43 (ISSN: 0020-9910) | DOI | MR | Zbl

Dyatlov, S.; Zworski, M. Stochastic stability of Pollicott-Ruelle resonances, Nonlinearity, Volume 28 (2015), pp. 3511-3533 (ISSN: 0951-7715) | DOI | MR | Zbl

Fried, D. Meromorphic zeta functions for analytic flows, Comm. Math. Phys., Volume 174 (1995), pp. 161-190 http://projecteuclid.org/euclid.cmp/1104275099 (ISSN: 0010-3616) | DOI | MR | Zbl

Faure, F.; Sjöstrand, J. Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., Volume 308 (2011), pp. 325-364 (ISSN: 0010-3616) | DOI | MR | Zbl

Faure, F.; Tsujii, M. The semiclassical zeta function for geodesic flows on negatively curved manifolds (preprint arXiv:1311.4932 )

Faure, F.; Tsujii, M. Band structure of the Ruelle spectrum of contact Anosov flows, C. R. Math. Acad. Sci. Paris, Volume 351 (2013), pp. 385-391 (ISSN: 1631-073X) | DOI | MR | Zbl

Gouëzel, S.; Liverani, C. Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, Volume 26 (2006), pp. 189-217 (ISSN: 0143-3857) | DOI | MR | Zbl

Giulietti, P.; Liverani, C.; Pollicott, M. Anosov flows and dynamical zeta functions, Ann. of Math., Volume 178 (2013), pp. 687-773 (ISSN: 0003-486X) | DOI | MR | Zbl

Guillarmou, C. Invariant distributions and X-ray transform for Anosov flows (preprint arXiv:1408.4732 )

Guillarmou, C. Lens rigidity for manifolds with hyperbolic trapped set (preprint arXiv:1412.1760 )

Guillemin, V. Lectures on spectral theory of elliptic operators, Duke Math. J., Volume 44 (1977), pp. 485-517 (ISSN: 0012-7094) | DOI | MR | Zbl

Guillopé, L.; Zworski, M. Scattering asymptotics for Riemann surfaces, Ann. of Math., Volume 145 (1997), pp. 597-660 (ISSN: 0003-486X) | DOI | MR | Zbl

Guillopé, L.; Zworski, M. The wave trace for Riemann surfaces, Geom. Funct. Anal., Volume 9 (1999), pp. 1156-1168 (ISSN: 1016-443X) | DOI | MR | Zbl

Hörmander, L. On the existence and the regularity of solutions of linear pseudo-differential equations, Enseignement Math., Volume 17 (1971), pp. 99-163 (ISSN: 0013-8584) | MR | Zbl

Hörmander, L., Springer, 1983 | Zbl

Hörmander, L., Springer, 1985 | Zbl

Jin, L.; Zworski, M. A local trace formula for Anosov flows (preprint arXiv:1411.6177, with an appendix by Frédéric Naud)

Lebœuf, P. Periodic orbit spectrum in terms of Ruelle-Pollicott resonances, Phys. Rev. E, Volume 69 (2004), 026204 pages (ISSN: 1539-3755) | DOI | MR

Liverani, C. Fredholm determinants, Anosov maps and Ruelle resonances, Discrete Contin. Dyn. Syst., Volume 13 (2005), pp. 1203-1215 (ISSN: 1078-0947) | DOI | MR | Zbl

Melrose, R. B. Scattering theory and the trace formula of the wave group, J. Funct. Anal., Volume 45 (1982), pp. 429-440 | DOI | Zbl

Melrose, R. B., Spectral and scattering theory (Sanda, 1992) (Lecture Notes in Pure and Appl. Math.), Volume 161, Dekker, New York, 1994, pp. 85-130 | MR | Zbl

Pollicott, M. On the rate of mixing of Axiom A flows, Invent. math., Volume 81 (1985), pp. 413-426 (ISSN: 0020-9910) | DOI | MR | Zbl

Ruelle, D. Zeta-functions for expanding maps and Anosov flows, Invent. math., Volume 34 (1976), pp. 231-242 (ISSN: 0020-9910) | DOI | MR | Zbl

Ruelle, D. Resonances of chaotic dynamical systems, Phys. Rev. Lett., Volume 56 (1986), pp. 405-407 (ISSN: 0031-9007) | DOI | MR

Rugh, H. H. Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems, Ergodic Theory Dynam. Systems, Volume 16 (1996), pp. 805-819 (ISSN: 0143-3857) | DOI | MR | Zbl

Smale, S. Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 (ISSN: 0002-9904) | DOI | MR | Zbl

Sjöstrand, J.; Zworski, M. Lower bounds on the number of scattering poles. II, J. Funct. Anal., Volume 123 (1994), pp. 336-367 (ISSN: 0022-1236) | DOI | MR | Zbl

Unterberger, A. Résolution d'équations aux dérivées partielles dans des espaces de distributions d'ordre de régularité variable, Ann. Inst. Fourier (Grenoble), Volume 21 (1971), pp. 85-128 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Vasy, A. Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov), Invent. math., Volume 194 (2013), pp. 381-513 (ISSN: 0020-9910) | DOI | MR | Zbl

Zworski, M., Graduate Studies in Math., 138, Amer. Math. Soc., Providence, RI, 2012, 431 pages (ISBN: 978-0-8218-8320-4) | MR | Zbl

Zworski, M., Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École polytech., Palaiseau, 1997, 14 pages (exp. n^o 13) | Numdam | MR | Zbl

Cité par Sources :