Comptes Rendus
Mathematical analysis/Dynamical systems
Ruelle operators with two complex parameters and applications
Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 595-599.

For a C2 weak-mixing Axiom-A flow ϕt:MM on a Riemannian manifold M and a basic set Λ for ϕt, we consider the Ruelle transfer operator Lfsτ+zg, where f and g are real-valued Hölder functions on Λ, τ is the roof function and s,z are complex parameters. Under some assumptions about ϕt for arbitrary Hölder f,g, we establish estimates for the iterations of this Ruelle operator when |Imz|B|Ims|ν for some constants B>0, 0<ν<1 (ν=1 for Lipschitz f,g), in the spirit of the estimates for operators with one complex parameter (see [2,11,12]). Applying these estimates, we obtain a non-zero analytic extension of the zeta function ζ(s,z) for Pfϵ<Re(s)Pf and |z| small enough with a simple pole at s=s(z). Two other applications are considered as well: the first concerns the Hannay–Ozorio de Almeida sum formula, while the second deals with the asymptotic of the counting function πF(T) for weighted primitive periods of the flow ϕt.

Soit ϕt:MM un flot C2, faiblement mélangeant, sur une variété riemannienne M. Soit Λ un ensemble basique pour ϕt. On considère l'opérateur de Ruelle de transfert Lfsτ+zg, où f et g sont des fonctions hölderiennes à valeurs réelles sur Λ, τ est la fonction roof et s,z sont des paramètres complexes. On suppose que ϕt satisfait quelques conditions et, pour des fonctions f,g arbitraires, on prouve des estimations pour les itérations de cet opérateur de Ruelle quand |Imz|B|Ims|ν avec des constantes B>0,0<ν<1 (ν=1 si f,g sont des fonctions lipschitziennes) qui sont analogues aux estimations des opérateurs avec un paramètre complexe (cf. [2,11,12]). En appliquant ces estimations, on obtient un prolongement sans zéros de la fonction zêta ζ(s,z) pour Pfϵ<Re(s)Pf et |z| suffisamment petit avec un pôle simple en s=s(z). Nous proposons aussi deux autres applications : la première concerne la formule de sommation de Hannay–Ozorio de Almeida, tandis que la seconde concerne l'asymptotique de la fonction de comptage πF(T) des périodes primitives du flot ϕt calculées avec des poids.

Published online:
DOI: 10.1016/j.crma.2015.04.005

Vesselin Petkov 1; Luchezar Stoyanov 2

1 IMB, Université de Bordeaux, 33405 Talence cedex, France
2 University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia
     author = {Vesselin Petkov and Luchezar Stoyanov},
     title = {Ruelle operators with two complex parameters and applications},
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     pages = {595--599},
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     year = {2015},
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Vesselin Petkov; Luchezar Stoyanov. Ruelle operators with two complex parameters and applications. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 595-599. doi : 10.1016/j.crma.2015.04.005.

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