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

• 成果类型: Article
• 署名作者: Cekic, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P.
• 署名单位: University of Zurich; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; University of Paderborn; Massachusetts Institute of Technology (MIT); University of Cambridge
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01108-x
• 发表日期: 2022
• 页码: 303-394
• 关键词: anosov-flows; spectrum; torsion; SPACES

摘要

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. Let (Sigma, g) be a compact connected oriented 3-dimensional Riemannian manifold of negative sectional curvature. The Ruelle zeta function zeta(R)(lambda) = Pi(gamma) (1 - e(i lambda T gamma)), Im lambda >> 1 (1.1) is a converging product for Im lambda large enough and continues meromorphically to lambda epsilon C as proved by Giulietti-Liverani-Pollicott [34] and Dyatlov-Zworski [20]. Here the product is taken over all primitive closed geodesics gamma on (Sigma, g) and T-gamma is the length of gamma. In this paper we study the order of vanishing of zeta(R) at lambda = 0, defined as the unique integer m(R)(0) such that lambda(-mR(0)) zeta(R)(lambda) is holomorphic and nonzero at 0. Our main result is