{
  "id": "2009.08558",
  "version": "v1",
  "published": "2020-09-17T23:30:08.000Z",
  "updated": "2020-09-17T23:30:08.000Z",
  "title": "The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds",
  "authors": [
    "Mihajlo Cekić",
    "Semyon Dyatlov",
    "Benjamin Küster",
    "Gabriel P. Paternain"
  ],
  "comment": "69 pages",
  "categories": [
    "math.DS",
    "math.AP",
    "math.DG",
    "math.SP"
  ],
  "abstract": "We show that for a generic conformal metric perturbation of a hyperbolic 3-manifold $\\Sigma$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1(\\Sigma)$, contrary to the hyperbolic case where it is equal to $4-2b_1(\\Sigma)$. The result is proved by developing a suitable perturbation theory that exploits the natural pairing between resonant and co-resonant differential forms. To obtain a metric conformal perturbation we need to establish the non-vanishing of the pushforward of a certain product of resonant and co-resonant states and we achieve this by a suitable regularisation argument. Along the way we describe geometrically all resonant differential forms (at zero) for a closed hyperbolic 3-manifold and study the semisimplicity of the Lie derivative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-17T23:30:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ruelle zeta function",
      "generic conformal metric perturbation",
      "metric conformal perturbation",
      "co-resonant differential forms",
      "hyperbolic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}