{
  "id": "2411.02106",
  "version": "v1",
  "published": "2024-11-04T14:17:16.000Z",
  "updated": "2024-11-04T14:17:16.000Z",
  "title": "Length averages for codimension one foliations",
  "authors": [
    "Masayuki Asaoka",
    "Yushi Nakano",
    "Paulo Varandas",
    "Tomoo Yokoyama"
  ],
  "comment": "34 pages, 13 figures",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group structure itself, to obtain irregular behavior of ball averages for certain non-amenable group actions. Several geometric realization results show that any such groups can appear connected with the topology of leaves which are connected sums of plugs with a special geometry, namely nearly equidistant boundary components. This is used to produce the first examples of codimension one $\\mathcal C^\\infty$ regular foliations on a compact Riemannian manifold $M$ for which the length average of some continuous function does not exist on a non-empty open subset of $M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-04T14:17:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "length average",
      "codimension",
      "ball averages",
      "geometric realization results",
      "equidistant boundary components"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}