RUELLE TRANSFER OPERATORS WITH TWO COMPLEX PARAMETERS AND APPLICATIONS

摘要

For a C~2 Axiom A flow ø_t : M → M on a Riemannian manifold M and a basic set A for ø_t we consider the Ruelle transfer operator L_(f-Sτ+zg), where f and g are real-valued Holder functions on Λ, τ is the roof function and s,z ∈ C are complex parameters. Under some assumptions about ø_t we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder f, g when |Imz| ≤ B|Ims|~µ for some constants B > 0, 0 < µ < 1 (µ = 1 for Lipschitz f,g), (ii) for Lipschitz f, g when |Ims| ≤ B_1|Imz| for some constant B_1 > 0 . Applying these estimates, we obtain a non zero analytic extension of the zeta function ζ(s, z) for P_f - ∈ < 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 ø_t.