2020/09/30 by Mihajlo Cekić, Benjamin Delarue, Semyon Dyatlov +1 · 12 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algorithm #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematics #math.AP #math.DG #math.DS #math.SP
paper · pdf · open access · doi:10.1007/s00222-022-01108-x
published in Inventiones mathematicae 229(1), 303-394 (Springer Science+Business Media) · 69 pages; revisions to the exposition following the referee comments. To appear in Inventiones Mathematicae
arxiv created 2022/02/09 · openalex publication_date 2022/03/11 · arxiv updated 2022/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold Σ with Betti number b1, the order of vanishing of the Ruelle zeta function at zero equals 4-b1, while in the hyperbolic case it is equal to 4-2b1. 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 SΣ with harmonic 1-forms on Σ.