{
  "id": "2202.09178",
  "version": "v1",
  "published": "2022-02-18T13:12:31.000Z",
  "updated": "2022-02-18T13:12:31.000Z",
  "title": "Refined horoball counting and conformal measure for Kleinian group actions",
  "authors": [
    "Jonathan M. Fraser",
    "Liam Stuart"
  ],
  "comment": "21 pages, 4 figures",
  "categories": [
    "math.DS",
    "math.CA",
    "math.MG"
  ],
  "abstract": "Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the `inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small $r>0$ there are $r^{-\\delta}$ many horoballs of size approximately $r$, where $\\delta$ is the Poincar\\'e exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately $r$ inside a given ball $B(z,R)$. Roughly speaking, if $r \\lesssim R^2$, then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of $B(z,R)$). However, for larger values of $r$, the count depends in a subtle way on $z$. Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several `fractal dimensions' of certain $s$-conformal measures for $s>\\delta$ and use this to examine continuity properties of $s$-conformal measures at $s=\\delta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-18T13:12:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30F40",
      "28A80",
      "11J83"
    ],
    "keywords": [
      "conformal measure",
      "kleinian group actions",
      "refined horoball counting",
      "limit set",
      "parabolic fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}