Ruelle transfer operators with two complex parameters and applications

© 2016, Southwest Missouri State University. All rights reserved.For a C2 Axiom A flow ft : M ? M on a Riemannian manifold M and a basic set ? for ft we consider the Ruelle transfer operator Lf-st+zg, where f and g are real-valued Hölder functions on ?, t is the roof function and s, z ? C are complex parameters. Under some assumptions about ft 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 | Im z| = B| Im s|µ for some constants B > 0, 0 <µ <1 (µ = 1 for Lipschitz f, g), (ii) for Lipschitz f, g when | Im s| = B1| Im z| for some constant B1 > 0. Applying these estimates, we obtain a non zero analytic extension of the zeta function ?(s, z) for Pf - e <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 pF (T) for weighted primitive periods of the flow ft.