{
  "id": "1901.01691",
  "version": "v1",
  "published": "2019-01-07T07:53:14.000Z",
  "updated": "2019-01-07T07:53:14.000Z",
  "title": "Dimension of invariant measures for affine iterated function systems",
  "authors": [
    "De-Jun Feng"
  ],
  "comment": "58 pages",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $\\{S_i\\}_{i\\in \\Lambda}$ be a finite contracting affine iterated function system (IFS) on ${\\Bbb R}^d$. Let $(\\Sigma,\\sigma)$ denote the two-sided full shift over the alphabet $\\Lambda$, and $\\pi:\\Sigma\\to {\\Bbb R}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $\\sigma$-invariant measure on $\\Sigma$ under $\\pi$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-07T07:53:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant measure",
      "contracting affine iterated function system",
      "finite contracting affine iterated function",
      "hausdorff dimension satisfies",
      "ledrappier-young type formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}