{
  "id": "1509.03589",
  "version": "v1",
  "published": "2015-09-11T17:42:09.000Z",
  "updated": "2015-09-11T17:42:09.000Z",
  "title": "Inhomogeneous self-similar sets with overlaps",
  "authors": [
    "Simon Baker",
    "Jonathan M. Fraser",
    "András Máthé"
  ],
  "comment": "15 pages, 2 figures",
  "categories": [
    "math.CA",
    "math.DS"
  ],
  "abstract": "It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of a spectral gap property that holds for certain subgroups of $SO(d)$ for $d\\geq 3$. We also obtain new upper bounds for the upper box dimension of an inhomogeneous self-similar set which hold in general. Moreover, our counterexamples demonstrate that these bounds are optimal. In the final section we show that if the \\emph{weak separation property} is satisfied, ie. the overlaps are controllable, then the `expected formula' does hold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-11T17:42:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "37C45"
    ],
    "keywords": [
      "upper box dimension",
      "open set condition",
      "iterated function system satisfies",
      "spectral gap property",
      "expected formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}