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

• Published in: Inventiones mathematicae Vol. 229; no. 1; pp. 303 - 394
• Main Authors: Cekić, Mihajlo, Delarue, Benjamin, Dyatlov, Semyon, Paternain, Gabriel P.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022; Springer Nature B.V
• Subjects: Differential geometry; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Perturbation
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-022-01108-x

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold  Σ 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 - 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 Σ with harmonic 1-forms on  Σ .