Functional equations of Selberg and Ruelle zeta functions for non-unitary twists

TL;DR

This paper establishes functional equations and determinant representations for Selberg and Ruelle dynamical zeta functions on hyperbolic manifolds, extending their analytic properties and connecting them to twisted differential operators.

Contribution

It introduces new functional equations for these zeta functions and provides a determinant formula involving regularized determinants of twisted differential operators.

Findings

Proves meromorphic continuation of the zeta functions.

Derives functional equations relating s and -s.

Provides a determinant representation using twisted differential operators.

Abstract

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable $s$ in some right-half plane of $\mathbb{C}$. In [Spi18], it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at $s$ with those at $-s$. We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators.