{
  "id": "1511.06893",
  "version": "v1",
  "published": "2015-11-21T15:52:12.000Z",
  "updated": "2015-11-21T15:52:12.000Z",
  "title": "On self-affine measures with equal Hausdorff and Lyapunov dimensions",
  "authors": [
    "Ariel Rapaport"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\mu$ be a self-affine measure on $\\mathbb{R}^{d}$ associated to a self-affine IFS $\\{\\varphi_{\\lambda}(x) = A_{\\lambda}x + v_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and a probability vector $p=(p_{\\lambda})_{\\lambda}>0$. Assume the strong separation condition holds. Let $\\gamma_{1}\\ge...\\ge\\gamma_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and $p^{\\mathbb{N}}$, and let $\\mathbf{G}$ be the group generated by $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$. We show that if $\\gamma_{m+1}>\\gamma_{m}=...=\\gamma_{d}$, if $\\mathbf{G}$ acts irreducibly on the vector space of alternating $m$-forms, and if the Furstenberg measure $\\mu_{F}$ satisfies $\\dim_{H}\\mu_{F}+D>(m+1)(d-m)$, then $\\mu$ is exact dimensional with $\\dim\\mu=D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-21T15:52:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C45",
      "28A80"
    ],
    "keywords": [
      "self-affine measure",
      "equal hausdorff",
      "lyapunov dimensions",
      "strong separation condition holds",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151106893R"
    }
  }
}