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

Presented by: 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 (Original language)
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.

Received: 2013-07-30
Accepted: 2013-09-17
Published online: 2013-09-01

DOI: 10.1016/j.crma.2013.09.012

Author's affiliations:

Luchezar Stoyanov ¹

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

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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/

References
Cited by

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

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