# American Institute of Mathematical Sciences

November  2016, 36(11): 6413-6451. doi: 10.3934/dcds.2016077

## Ruelle transfer operators with two complex parameters and applications

 1 Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France 2 School of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009

Received  October 2015 Revised  June 2016 Published  August 2016

For a $C^2$ 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 Hölder functions on $\Lambda$, $\tau$ is the roof function and $s, z \in \mathbb{C}$ 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 [4], [21], [22]). Two cases are covered: (i) for arbitrary Hölder $f,g$ 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 $B_1 > 0$ . 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 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 $\pi_F(T)$ for weighted primitive periods of the flow $\phi_t.$
Citation: Vesselin Petkov, Luchezar Stoyanov. Ruelle transfer operators with two complex parameters and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6413-6451. doi: 10.3934/dcds.2016077
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