Spectral estimates for Ruelle transfer operators with two parameters and applications

TL;DR

This paper develops spectral estimates for Ruelle transfer operators with two complex parameters in hyperbolic dynamics, leading to analytic continuation of zeta functions and applications to periodic orbit counting.

Contribution

It provides new estimates for Ruelle operators with two parameters, enabling analytic extension of zeta functions and applications to dynamical sums and counting functions.

Findings

Established spectral estimates for transfer operators with two parameters.

Obtained analytic continuation of the zeta function with simple poles.

Applied results to orbit counting and sum formulas.

Abstract

For $C^2$ weak mixing Axiom A flow $\phi_t: M \longrightarrow M$ on a Riemannian manifold $M$ and a basic set $\Lambda$ for $\phi_t$ we consider the Ruelle transfer operator $L_{f - s \tau + z g}$, where $f$ and $g$ are real-valued H\"older functions on $\Lambda$, $\tau$ is the roof function and $s, z$ are complex parameters. Under some assumptions about $\phi_t$ 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 $|\Im z| \leq B |\Im s|^\mu$ for some constants $B > 0$, $0 < \mu < 1$ ($\mu = 1$ 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 $\zeta(s, z)$ for $P_f - \epsilon < \Re (s) < P_f$ 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 $\pi_F(T)$ for weighted primitive periods of the flow $\phi_t.$