We prove strong spectral estimates for Ruelle transfer operators for arbitrary 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 . 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.
Accepted:
Published online:
Luchezar Stoyanov 1
@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}, }
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/
[1] On decay of correlations in Anosov flows, Ann. Math., Volume 147 (1998), pp. 357-390
[2] Anosov flows and dynamical zeta functions, Ann. Math., Volume 178 (2013), pp. 687-773
[3] On contact Anosov flows, Ann. Math., Volume 159 (2004), pp. 1275-1312
[4] Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, Volume 187–188 (1990), pp. 1-267
[5] Distribution of periods of closed trajectories in exponentially shrinking intervals, Commun. Math. Phys., Volume 310 (2012), pp. 675-704
[6] Exponential error terms for growth functions of negatively curved surfaces, Amer. J. Math., Volume 120 (1998), pp. 1019-1042
[7] Spectra of Ruelle transfer operators for Axiom A flows on basic sets, Nonlinearity, Volume 24 (2011), pp. 1089-1120
[8] Ruelle transfer operators for contact Anosov flows and decay of correlations (Preprint) | arXiv
Cited by Sources:
Comments - Policy