The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

Abstract

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Sigma with Betti number b_1, the order of vanishing of the Ruelle zeta function at zero equals 4-b_1, while in the hyperbolic case it is equal to 4-2b_1. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle SSigma with harmonic 1-forms on Sigma .