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Spectral estimates for Ruelle transfer operators with two parameters and applications

For $C^2$ weak mixing Axiom A flow $ϕ_t: M \longrightarrow M$ 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 $ϕ_t$ 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 Hölder $f,g$ when $|\Im z| \leq B |\Im s|^μ$ for some constants $B > 0$, $0 < μ< 1$ ($μ= 1$ for Lipschitz $f,g$), (ii) for Lipschitz $f,g$ when $|\Im s| \leq B_1 |\Im z|$ for some constant $B > 0$ . 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 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 $ϕ_t.$