Ruelle operators and decay of correlations for contact Anosov flows

Luchezar Stoyanov

doi:10.1016/j.crma.2013.09.012

Mathematical analysis/Dynamical systems

Ruelle operators and decay of correlations for contact Anosov flows
[Opérateurs de Ruelle et décroissance des corrélations pour des flots de contact dʼAnosov]

Présenté par : Gilles Lebeau

Luchezar Stoyanov ¹

¹ University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia

Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 669-672.

Abstract (VO)
Résumé

We prove strong spectral estimates for Ruelle transfer operators for arbitrary $C^{2}$ contact Anosov flows. As a consequence of this we obtain: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error; (c) exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure.

On prouve des estimations spectrales fortes pour lʼopérateur de transfert de Ruelle relatif à des flots de contact dʼAnosov arbitraires de classe $C^{2}$ . Comme conséquence, on obtient les trois résultats suivants : (a) lʼexistence dʼun prolongement analytique sans zéros de la fonction zêta de Ruelle dans une bande verticale contenant lʼentropie dans son intérieur et ayant lʼentropie comme ensemble de pôles ; (b) un théorème asymptotique pour le nombre de trajectoires périodiques primitives avec un reste exponentiellement petit ; (c) la décroissance exponentielle des corrélations pour des observables höldériennes par rapport à une mesure de Gibbs quelconque.

Reçu le : 2013-07-30
Accepté le : 2013-09-17
Publié le : 2013-09-01

DOI : 10.1016/j.crma.2013.09.012

Affiliations des auteurs :

Luchezar Stoyanov ¹

¹ University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia

@article{CRMATH_2013__351_17-18_669_0,
     author = {Luchezar Stoyanov},
     title = {Ruelle operators and decay of correlations for contact {Anosov} flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {669--672},
     publisher = {Elsevier},
     volume = {351},
     number = {17-18},
     year = {2013},
     doi = {10.1016/j.crma.2013.09.012},
     language = {en},
}

TY  - JOUR
AU  - Luchezar Stoyanov
TI  - Ruelle operators and decay of correlations for contact Anosov flows
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 669
EP  - 672
VL  - 351
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2013.09.012
LA  - en
ID  - CRMATH_2013__351_17-18_669_0
ER  -

%0 Journal Article
%A Luchezar Stoyanov
%T Ruelle operators and decay of correlations for contact Anosov flows
%J Comptes Rendus. Mathématique
%D 2013
%P 669-672
%V 351
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2013.09.012
%G en
%F CRMATH_2013__351_17-18_669_0

Luchezar Stoyanov. Ruelle operators and decay of correlations for contact Anosov flows. Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 669-672. doi : 10.1016/j.crma.2013.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.09.012/

Bibliographie
Cité par

[1] D. Dolgopyat On decay of correlations in Anosov flows, Ann. Math., Volume 147 (1998), pp. 357-390

[2] P. Giulietti; C. Liverani; M. Pollicott Anosov flows and dynamical zeta functions, Ann. Math., Volume 178 (2013), pp. 687-773

[3] C. Liverani On contact Anosov flows, Ann. Math., Volume 159 (2004), pp. 1275-1312

[4] W. Parry; M. Pollicott Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Volume 187–188 (1990), pp. 1-267

[5] V. Petkov; L. Stoyanov Distribution of periods of closed trajectories in exponentially shrinking intervals, Commun. Math. Phys., Volume 310 (2012), pp. 675-704

[6] M. Pollicott; R. Sharp Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math., Volume 120 (1998), pp. 1019-1042

[7] L. Stoyanov Spectra of Ruelle transfer operators for Axiom A flows on basic sets, Nonlinearity, Volume 24 (2011), pp. 1089-1120

[8] L. Stoyanov Ruelle transfer operators for contact Anosov flows and decay of correlations (Preprint) | arXiv

I. V. Podvigin On convergence rates in the Birkhoff ergodic theorem, Siberian Mathematical Journal, Volume 65 (2024) no. 5, pp. 1170-1186 | DOI:10.1134/s0037446624050161 | Zbl:1557.37001
Masato Tsujii; Zhiyuan Zhang Smooth mixing Anosov flows in dimension three are exponentially mixing, Annals of Mathematics. Second Series, Volume 197 (2023) no. 1, pp. 65-158 | DOI:10.4007/annals.2023.197.1.2 | Zbl:1511.37033
Diego Daltro; Paulo Varandas Exponential decay of correlations for Gibbs measures and semiflows over $C^{1 + α}$ piecewise expanding maps, Annales Henri Poincaré, Volume 22 (2021) no. 7, pp. 2137-2159 | DOI:10.1007/s00023-020-00991-5 | Zbl:1481.37030
Vesselin Petkov; Luchezar Stoyanov Sharp large deviations for hyperbolic flows, Annales Henri Poincaré, Volume 21 (2020) no. 12, pp. 3791-3834 | DOI:10.1007/s00023-020-00956-8 | Zbl:1468.37030
Palle Jorgensen; James Tian Noncommutative boundaries arising in dynamics and representations of the Cuntz relations, Numerical Functional Analysis and Optimization, Volume 41 (2020) no. 5, pp. 571-620 | DOI:10.1080/01630563.2019.1665544 | Zbl:1443.47086
Palle Jorgensen; Feng Tian Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations, Discrete and Continuous Dynamical Systems. Series S, Volume 12 (2019) no. 8, pp. 2307-2348 | DOI:10.3934/dcdss.2019146 | Zbl:1447.81147
Masato Tsujii Geodesic flows on negatively curved manifolds and the semi-classical zeta function, Sugaku Expositions, Volume 31 (2018) no. 1, pp. 69-92 | DOI:10.1090/suga/429 | Zbl:1391.37022
Sergey Bezuglyi; Palle E. T. Jorgensen Introduction and Examples, Transfer Operators, Endomorphisms, and Measurable Partitions, Volume 2217 (2018), p. 1 | DOI:10.1007/978-3-319-92417-5_1
Semyon Dyatlov; Colin Guillarmou Pollicott-Ruelle resonances for open systems, Annales Henri Poincaré, Volume 17 (2016) no. 11, pp. 3089-3146 | DOI:10.1007/s00023-016-0491-8 | Zbl:1367.37038
Anke D. Pohl A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area, Communications in Mathematical Physics, Volume 337 (2015) no. 1, pp. 103-126 | DOI:10.1007/s00220-015-2304-1 | Zbl:1348.37042
Luchezar Stoyanov Ruelle operators and decay of correlations for contact Anosov flows, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 351 (2013) no. 17-18, pp. 669-672 | DOI:10.1016/j.crma.2013.09.012 | Zbl:1306.37025

Cité par 11 documents. Sources : Crossref, zbMATH

Commentaires - Politique