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 $\phi (s)$ at $s=0$, we find that \begin{align*} \lim_{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2π)^{2g-2+n }\tildeφ(0)^{-1} \prod_{j=1}^v m_j, \end{align*}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 $m_1$, $m_2$, $\dots$, $m_v$, the ramification indices at the ramification points, and $\tilde{\phi }(0)$, the leading coefficient of $\phi (s)$ 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 $(-1{)}^{\frac{A}{2}}=\phi \left(\frac{1}{2}\right)$. We also consider the order of the Ruelle zeta function at other integers.

18 pages

References in corpus (2)

• Ruelle zeta function at zero for surfaces
• Local index theorem for orbifold Riemann surfaces

Cited by in corpus (3)

• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• Determinant of Friederichs Dirichlet Laplacians on $2$-dimensional hyperbolic cones
• Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations