Twisted Dirac operators and dynamical zeta functions

TL;DR

This paper extends the meromorphic continuation of dynamical zeta functions associated with geodesic flows on hyperbolic manifolds by employing trace formulas for twisted Dirac operators, linking spectral theory with dynamical systems.

Contribution

It introduces a trace formula for twisted Dirac operators and uses it to achieve meromorphic continuation of dynamical zeta functions on hyperbolic manifolds.

Findings

Established trace formula for twisted Dirac operators.

Proved meromorphic continuation of Ruelle and Selberg zeta functions.

Linked spectral properties of Dirac operators with 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 $\mathbb{C}$.