Dynamical Zeta Functions in the Nonorientable Case

TL;DR

This paper extends the meromorphicity results of dynamical zeta functions to nonorientable cases and computes the order of vanishing for geodesic flows on nonorientable surfaces, linking it to topological invariants.

Contribution

It introduces a simple argument to extend microlocal proofs to nonorientable cases and calculates the zero order of the zeta function for certain geodesic flows.

Findings

Meromorphicity of dynamical zeta functions extended to nonorientable cases

Order of vanishing at zero equals the first Betti number for nonorientable surfaces

Applicable to geodesic flows on negatively curved nonorientable surfaces

Abstract

We use a simple argument to extend the microlocal proofs of meromorphicity of dynamical zeta functions to the nonorientable case. In the special case of geodesic flow on a connected non-orientable negatively curved closed surface, we compute the order of vanishing of the zeta function at the zero point to be the first Betti number of the surface.