Ruelle operators with two complex parameters and applications

V. Petkov, L. Stoyanov

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1 Citation (Scopus)


For a C-2 weak-mixing Axiom-A flow phi(t) : M -> M on a Riemannian manifold M and a basic set Lambda for phi(t), we consider the Ruelle transfer operator Lf-s tau+zg, where f and g are real-valued Holder functions on Lambda, tau is the roof function and s, z are complex parameters. Under some assumptions about phi(t) for arbitrary Holder f, g, we establish estimates for the iterations of this Ruelle operator when vertical bar lmz vertical bar 0, 0 <nu <1 (nu = 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 zeta(s, z) for P-f - epsilon <Re (s)

Original languageEnglish
Pages (from-to)595-599
Number of pages5
JournalComptes rendus mathematique
Issue number7
Publication statusPublished - Jul 2015


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