Spectral estimates for Ruelle transfer operators with two parameters and applications

For $C^2$ weak mixing Axiom A flow ${\varphi }_t:M⟶M$ on a Riemannian manifold $M$ and a basic set $\mathrm{\Lambda }$ for ${\varphi }_t$ we consider the Ruelle transfer operator $L_{f-s\tau +zg}$, where $f$ and $g$ are real-valued H"older functions on $\mathrm{\Lambda }$, $\tau$ is the roof function and $s,z$ are complex parameters. Under some assumptions about ${\varphi }_t$ we establish estimates for the iterations of this Ruelle operator in the spirit of the estimates for operators with one complex parameter (see \cite{D}, \cite{St2}, \cite{St3}). Two cases are covered: (i) for arbitrary H"older $f,g$ when $\mathrm{\mid }\mathrm{\Im }z\mathrm{\mid }\le B\mathrm{\mid }\mathrm{\Im }s{\mathrm{\mid }}^{\mu }$ for some constants $B>0$, $0 < \mu < 1$($\mu = 1$forLipschitz$f,g$),(ii)forLipschitz$f,g$when$|\Im s| \leq B_1 |\Im z|$forsomeconstant$B > 0$ . Applying these estimates, we obtain a non zero analytic extension of the zeta function $\zeta (s,z)$ for $P_f - \epsilon < \Re (s) < P_f$and$|z|$smallenoughwithsimplepoleat$s = s(z)$. 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 ${\pi }_F(T)$ for weighted primitive periods of the flow ${\varphi }_t\mathrm{.}$