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

TL;DR

This paper investigates how the order of vanishing of the Ruelle zeta function at zero varies under metric perturbations of hyperbolic 3-manifolds, revealing a topological dependence in 3D contrasting with 2D cases.

Contribution

It establishes a formula for the order of vanishing of the Ruelle zeta function at zero for perturbed hyperbolic 3-manifolds, using microlocal analysis and a new identity involving harmonic forms.

Findings

Order of vanishing equals 4 - b₁ for generic perturbations.

In hyperbolic case, order of vanishing equals 4 - 2b₁.

Introduces a new identity relating resonant forms and harmonic 1-forms.

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 $S\Sigma$ with harmonic 1-forms on $\Sigma$.