Secondary invariants and the singularity of the Ruelle zeta-function in the central critical point

Abstract

The Ruelle zeta-function of the geodesic flow on the sphere bundle $S(X)$ of an even-dimensional compact locally symmetric space $X$ of rank $1$ is a meromorphic function in the complex plane that satisfies a functional equation relating its values in $s$ and $-s$. The multiplicity of its singularity in the central critical point $s = 0$ only depends on the hyperbolic structure of the flow and can be calculated by integrating a secondary characteristic class canonically associated to the flow- invariant foliations of $S(X)$ for which a representing differential form is given.