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

69 pages; revisions to the exposition following the referee comments. To appear in Inventiones Mathematicae

References in corpus (4)

• Power spectrum of the geodesic flow on hyperbolic manifolds
• Patterson-Sullivan distributions and quantum ergodicity
• Anisotropic Sobolev spaces and dynamical transfer operators: C^infty foliations
• Dynamical Zeta Functions in the Nonorientable Case

Cited by in corpus (4)

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• Twisted Ruelle zeta function at zero for compact hyperbolic surfaces
• Analytic torsion, dynamical zeta function, and the Fried conjecture for admissible twists
• The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion