Dynamics and zeta functions on conformally compact manifolds

TL;DR

This paper investigates the dynamics and zeta functions of conformally compact manifolds with variable negative curvature, providing new meromorphic extensions and counting estimates, and exploring their spectral properties.

Contribution

It introduces optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for closed geodesics on conformally compact manifolds.

Findings

Meromorphic extension of weighted dynamical zeta functions

Asymptotic estimates for weighted closed geodesics

Connections between dynamics and spectral theory

Abstract

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.