{
  "id": "1909.08913",
  "version": "v1",
  "published": "2019-09-19T10:41:59.000Z",
  "updated": "2019-09-19T10:41:59.000Z",
  "title": "Quantitative recurrence properties for self-conformal sets",
  "authors": [
    "Simon Baker",
    "Michael Farmer"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we study the quantitative recurrence properties of self-conformal sets $X$ equipped with the map $T:X\\to X$ induced by the left shift. In particular, given a function $\\varphi:\\mathbb{N}\\to(0,\\infty),$ we study the metric properties of the set $$R(T,\\varphi)=\\left\\{x\\in X:|T^nx-x|<\\varphi(n)\\textrm{ for infinitely many }n\\in \\mathbb{N}\\right\\}.$$ Our main result is that under the open set condition, for the natural measure $\\mu$ supported on $X$, $\\mu$-almost every $x\\in X$ is contained in $R(T,\\varphi)$ when an appropriate volume sum diverges. We also prove a complementary result which states that for self-similar sets satisfying the open set condition, when the volume sum converges, then $\\mu$-almost every $x\\in X$ is not contained in $R(T,\\varphi).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-19T10:41:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantitative recurrence properties",
      "self-conformal sets",
      "open set condition",
      "appropriate volume sum diverges",
      "volume sum converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}