{
  "id": "1305.2033",
  "version": "v4",
  "published": "2013-05-09T08:49:24.000Z",
  "updated": "2014-11-10T10:20:44.000Z",
  "title": "A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces",
  "authors": [
    "Carlos Matheus",
    "Martin Moeller",
    "Jean-Christophe Yoccoz"
  ],
  "comment": "69 pages, Appendix C by Samuel Leli\\`evre. Final version based on the referees' reports. To appear in Invent. Math",
  "categories": [
    "math.DS"
  ],
  "abstract": "We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Leli\\`evre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus $>0$) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-08T18:45:43.000Z",
      "comment": "68 pages, Appendix C by Samuel Leli\\`evre. V3: a mistake (kindly pointed out to us by Gabriela Schmithuesen) in the previous version of our Lemma 6.10 is corrected in this version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-11-10T10:20:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "square-tiled surface",
      "simplicity",
      "simple lyapunov spectrum",
      "teichmueller flow",
      "moduli spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.2033M"
    }
  }
}