Comptes Rendus
no. 17-18
Mathematical analysis/Dynamical systems
Ruelle operators and decay of correlations for contact Anosov flows
Gilles Lebeau
Luchezar Stoyanov1
1University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia
Comptes Rendus. Mathématique, Volume 351 (2013) no. 17-18, pp. 669-672

We prove strong spectral estimates for Ruelle transfer operators for arbitrary C2 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 C2. 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.09.012

Luchezar Stoyanov  1

1 University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia 
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
@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},
     year = {2013},
     publisher = {Elsevier},
     volume = {351},
     number = {17-18},
     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

[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

Cited by Sources:

Comments - Policy