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Ruelle operators with two complex parameters and applications.

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Title: Ruelle operators with two complex parameters and applications.
Authors: Petkov, Vesselin1 petkov@math.u-bordeaux1.fr, Stoyanov, Luchezar2 luchezar.stoyanov@uwa.edu.au
Source: Comptes Rendus. Mathématique. Jul2015, Vol. 353 Issue 7, p595-599. 5p.
Subjects: Ruelle operators, Operator theory, Riemannian manifolds, Parameter estimation, Mathematical functions, Mathematical formulas
Abstract (English): For a C 2 weak-mixing Axiom-A flow ϕ t : M ⟶ M on a Riemannian manifold M and a basic set Λ for ϕ t , we consider the Ruelle transfer operator L f − s τ + z g , 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 ϕ t for arbitrary Hölder f , g , we establish estimates for the iterations of this Ruelle operator when | Im z | ≤ B | Im s | 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 P f − ϵ < Re ( s ) ≤ P f 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 π F ( T ) for weighted primitive periods of the flow ϕ t . [ABSTRACT FROM AUTHOR]
Abstract (French): Résumé Soit ϕ t : M ⟶ M un flot C 2 , faiblement mélangeant, sur une variété riemannienne M . Soit Λ un ensemble basique pour ϕ t . On considère l'opérateur de Ruelle de transfert L f − s τ + z g , 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 ϕ t satisfait quelques conditions et, pour des fonctions f , g arbitraires, on prouve des estimations pour les itérations de cet opérateur de Ruelle quand | Im z | ≤ B | Im s | 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 P f − ϵ < Re ( s ) ≤ P f 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 π F ( T ) des périodes primitives du flot ϕ t calculées avec des poids. [ABSTRACT FROM AUTHOR]
Copyright of Comptes Rendus. Mathématique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Ruelle operators with two complex parameters and applications.
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  Data: For a C 2 weak-mixing Axiom-A flow ϕ t : M ⟶ M on a Riemannian manifold M and a basic set Λ for ϕ t , we consider the Ruelle transfer operator L f − s τ + z g , where f and g are real-valued H&#246;lder functions on Λ, τ is the roof function and s , z are complex parameters. Under some assumptions about ϕ t for arbitrary H&#246;lder f , g , we establish estimates for the iterations of this Ruelle operator when | Im z | ≤ B | Im s | ν for some constants B &gt; 0 , 0 &lt; ν &lt; 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 P f − ϵ &lt; Re ( s ) ≤ P f 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 π F ( T ) for weighted primitive periods of the flow ϕ t . [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label: Abstract (French)
  Group: Ab
  Data: R&#233;sum&#233; Soit ϕ t : M ⟶ M un flot C 2 , faiblement m&#233;langeant, sur une vari&#233;t&#233; riemannienne M . Soit Λ un ensemble basique pour ϕ t . On consid&#232;re l&#39;op&#233;rateur de Ruelle de transfert L f − s τ + z g , o&#249; f et g sont des fonctions h&#246;lderiennes &#224; valeurs r&#233;elles sur Λ, τ est la fonction roof et s , z sont des param&#232;tres complexes. On suppose que ϕ t satisfait quelques conditions et, pour des fonctions f , g arbitraires, on prouve des estimations pour les it&#233;rations de cet op&#233;rateur de Ruelle quand | Im z | ≤ B | Im s | ν avec des constantes B &gt; 0 , 0 &lt; ν &lt; 1 ( ν = 1 si f , g sont des fonctions lipschitziennes) qui sont analogues aux estimations des op&#233;rateurs avec un param&#232;tre complexe (cf. [2,11,12] ). En appliquant ces estimations, on obtient un prolongement sans z&#233;ros de la fonction z&#234;ta ζ ( s , z ) pour P f − ϵ &lt; Re ( s ) ≤ P f et | z | suffisamment petit avec un p&#244;le simple en s = s ( z ) . Nous proposons aussi deux autres applications : la premi&#232;re concerne la formule de sommation de Hannay–Ozorio de Almeida, tandis que la seconde concerne l&#39;asymptotique de la fonction de comptage π F ( T ) des p&#233;riodes primitives du flot ϕ t calcul&#233;es avec des poids. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: &lt;i&gt;Copyright of Comptes Rendus. Math&#233;matique is the property of Academie des Science, Centre Mersenne and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.crma.2015.04.005
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        Text: English
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        StartPage: 595
    Subjects:
      – SubjectFull: Ruelle operators
        Type: general
      – SubjectFull: Operator theory
        Type: general
      – SubjectFull: Riemannian manifolds
        Type: general
      – SubjectFull: Parameter estimation
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      – SubjectFull: Mathematical functions
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      – SubjectFull: Mathematical formulas
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      – TitleFull: Ruelle operators with two complex parameters and applications.
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              Text: Jul2015
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