Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points

Abstract

We consider the Ruelle zeta function R(s) of a genus g hyperbolic Riemann surface with n punctures and v ramification points. R(s) is equal to $Z(s)/Z(s+1)$, where Z(s) is the Selberg zeta function. The main result of this work is the leading behavior of R(s) at $s=0$. If $n_0$ is the order of the determinant of the scattering matrix $\varphi (s)$ at $s=0$, we find that

$$\begin{aligned} \lim _{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2\pi )^{2g-2+n }{\tilde{\varphi }}(0)^{-1} \prod _{j=1}^v m_j, \end{aligned}$$

which says that R(s) has order $2g-2+n-n_0$ at $s=0$, and its leading coefficient can be expressed in terms of $m_1$, $m_2$, $\ldots$, $m_v$, the ramification indices at the ramification points, and ${\tilde{\varphi }}(0)$, the leading coefficient of $\varphi (s)$ at $s=0$. The constant A is an even integer, equal to twice the multiplicity of the eigenvalue $-\,1$ in the scattering matrix $\Phi (s)$ at $s=1/2$, and $(-1)^{\frac{A}{2}}=\varphi \left( \frac{1}{2}\right)$. We also consider the order of the Ruelle zeta function at other integers.