Twisted Dirac operators and dynamical zeta functions

Abstract

In this paper, we consider the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd dimensional manifold $X$. These functions are initially defined on one complex variable $s$ in some right half-plane of $\mathbb{C}$. Our goal is the continue meromorphically the dynamical zeta functions to the whole complex plane, using the Selberg trace formula for arbitrary, not necessarily unitary, representations $\chi$ of the fundamental group. First, we prove a trace formula for the integral operator $D^{\sharp}_{\chi}(\sigma)e^{-t(D^{\sharp}_{\chi}(\sigma))^{2}}$, induced by the twisted Dirac operator $D^{\sharp}_{\chi}(\sigma)$ on $X$. Then we use these results to establish the meromorphic continuation of the dynamical zeta functions to $\mathbb{C}$.