Ruelle zeta function at zero for surfaces

TL;DR

This paper proves that for negatively curved surfaces, the Ruelle zeta function vanishes at zero with an order equal to the absolute Euler characteristic, extending known results from constant curvature cases.

Contribution

It generalizes the understanding of the Ruelle zeta function's behavior at zero from constant curvature surfaces to all negatively curved surfaces.

Findings

Ruelle zeta function vanishes at zero for negatively curved surfaces

Order of vanishing equals the absolute Euler characteristic

Extends previous results from constant to variable curvature

Abstract

We show that the Ruelle zeta function for a negatively curved oriented surface vanishes at zero to the order given by the absolute value of the Euler characteristic. This result was previously known only in constant curvature.