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

Abstract

Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold

$$\Sigma $$

Σ

with Betti number

$$b_1$$

b 1

, the order of vanishing of the Ruelle zeta function at zero equals

$$4-b_1$$

4 -

b 1

, while in the hyperbolic case it is equal to

$$4-2b_1$$

4 - 2

b 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

$$S\Sigma $$

S Σ

with harmonic 1-forms on

$$\Sigma $$

Σ

.