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Comptes Rendus Mathématique
Volume 353, n° 7
pages 595-599 (juillet 2015)
Doi : 10.1016/j.crma.2015.04.005
Received : 12 September 2014 ;  accepted : 9 April 2015
Ruelle operators with two complex parameters and applications
Opérateurs de Ruelle avec deux paramètres complexes et applications
 

Vesselin Petkov a, 1 , Luchezar Stoyanov b
a IMB, Université de Bordeaux, 33405 Talence cedex, France 
b University of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia 

Abstract

For a   weak-mixing Axiom-A flow   on a Riemannian manifold M and a basic set Λ for  , we consider the Ruelle transfer operator  , where f and g are real-valued Hölder functions on Λ, τ is the roof function and   are complex parameters. Under some assumptions about   for arbitrary Hölder  , we establish estimates for the iterations of this Ruelle operator when   for some constants  ,   (  for Lipschitz  ), 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   for   and   small enough with a simple pole at  . 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   for weighted primitive periods of the flow  .

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Résumé

Soit   un flot  , faiblement mélangeant, sur une variété riemannienne M . Soit Λ un ensemble basique pour  . On considère l'opérateur de Ruelle de transfert  , où f et g sont des fonctions hölderiennes à valeurs réelles sur Λ, τ est la fonction roof et   sont des paramètres complexes. On suppose que   satisfait quelques conditions et, pour des fonctions   arbitraires, on prouve des estimations pour les itérations de cet opérateur de Ruelle quand   avec des constantes   (  si   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   pour   et   suffisamment petit avec un pôle simple en  . 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   des périodes primitives du flot   calculées avec des poids.

The full text of this article is available in PDF format.
1  V.P. was partially supported by the project ANR 2011 BS01019 01.
2  In fact, one has to define first f and g as functions in   and then extend them as α -Hölder functions on R . In the same way one should proceed with Hölder functions on U .


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