{
  "id": "1011.3217",
  "version": "v1",
  "published": "2010-11-14T11:34:01.000Z",
  "updated": "2010-11-14T11:34:01.000Z",
  "title": "Looking for a Billiard Table which is not a Lattice Polygon but Satisfies Veech's Dichotomy",
  "authors": [
    "Meital Cohen"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Over the course of studying billiard dynamics, several questions were raised. One of the questions was, which surfaces satisfy the following property (which is called Veech's dichotomy): Any direction is either completely periodic or uniquely ergodic. In an important paper Veech gave a sufficient condition for this dichotomy. He showed that if the stabilizer of a translation surface is a lattice in $SL_2(\\R)$, then the surface satisfies Veech's dichotomy. Later, Smillie and Weiss proved that this condition is not necessary. They constructed a translation surface which satisfies Veech's dichotomy but is not a lattice surface. Their construction was based on previous work of Hubert and Schmidt, by taking a branched cover over a lattice surface, where the branch locus is a single non-periodic connection point. In this work we tried to answer the following question: Is there a flat structure obtained from a billiard table that satisfies Veech's dichotomy, but its Veech group is not a lattice? We prove that in the entire list of possible candidates for such a construction, an example does not exist.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-14T11:34:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lattice polygon",
      "billiard table",
      "lattice surface",
      "single non-periodic connection point",
      "translation surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.3217C"
    }
  }
}