{
  "id": "1009.4655",
  "version": "v2",
  "published": "2010-09-23T16:40:11.000Z",
  "updated": "2011-03-24T16:00:05.000Z",
  "title": "A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle",
  "authors": [
    "Giovanni Forni"
  ],
  "comment": "46 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove a geometric criterion on a $\\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $\\SL$-orbits of all algebraically primitive Veech surfaces (see also \\cite{Bouw:Moeller}) and of all Prym eigenforms discovered in \\cite{McMullen2}, as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also \\cite{Ftwo}, \\cite{Avila:Viana}). The argument simplifies and generalizes our proof for the case of canonical measures \\cite{Ftwo}. In an Appendix Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-24T16:00:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30F60"
    ],
    "keywords": [
      "non-uniform hyperbolicity",
      "kontsevich-zorich cocycle",
      "geometric criterion",
      "appendix carlos matheus discusses",
      "moduli space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.4655F"
    }
  }
}