{
  "id": "2211.06255",
  "version": "v1",
  "published": "2022-11-11T14:51:32.000Z",
  "updated": "2022-11-11T14:51:32.000Z",
  "title": "Resonant forms at zero for dissipative Anosov flows",
  "authors": [
    "Mihajlo Cekić",
    "Gabriel P. Paternain"
  ],
  "comment": "67 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.AP",
    "math.DG",
    "math.GT",
    "math.SP"
  ],
  "abstract": "We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $\\Omega_{\\text{SRB}}^{\\pm}$, and the cohomology classes $[\\iota_{X}\\Omega_{\\text{SRB}}^{\\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant $1$-forms. When both classes vanish we associate to the flow a $\\textit{helicity}$ that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$-forms and SRB-measures as well as the local geometry of the maps $X\\mapsto [\\iota_{X}\\Omega_{\\text{SRB}}^{\\pm}]$ near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant $1$-forms at zero, we show that $[\\iota_{X}\\Omega_{\\text{SRB}}^{\\pm}]=0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $-\\chi(M)$, the same as in the geodesic flow case. In contrast, we show that if $(M,g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$-form has a Ruelle zeta function whose order of vanishing at zero is $-\\chi(M)-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-11T14:51:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C30",
      "37D40",
      "58C40",
      "53C65",
      "35P05"
    ],
    "keywords": [
      "dissipative anosov flows",
      "resonant forms",
      "null-homologous volume preserving flow",
      "ruelle zeta function",
      "study resonant differential forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}