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

• Main Authors: Cekić, Mihajlo (Author), Delarue, Benjamin (Author), Dyatlov, Semyon (Author), Paternain, Gabriel P. (Author)
• Format: Article
• Language: English
• Published: Springer Berlin Heidelberg, 2022-03-14T14:00:10Z.
• Subjects: Article
• Online Access: Get fulltext

 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 .