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- Title
The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds.
- Authors
Cekić, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P.
- Abstract
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 .
- Subjects
DIFFERENTIAL forms; TOPOLOGICAL property; BETTI numbers; ZETA functions
- Publication
Inventiones Mathematicae, 2022, Vol 229, Issue 1, p303
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-022-01108-x