{
  "id": "1307.2841",
  "version": "v3",
  "published": "2013-07-10T16:14:47.000Z",
  "updated": "2014-12-15T15:10:35.000Z",
  "title": "Projections of self-similar sets with no separation condition",
  "authors": [
    "Ábel Farkas"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate how the Hausdorff dimension and measure of a self-similar set $K\\subseteq\\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\\mathcal{T}$ generated by the orthogonal parts of the defining maps of $K$. We show that if $\\mathcal{T}$ is finite then every linear image of $K$ is a graph directed attractor and there exists at least one projection of $K$ such that the dimension drops under the image of the projection. In general, with no restrictions on $\\mathcal{T}$ we establish that $\\mathcal{H}^{t}(L\\circ O(K))=\\mathcal{H}^{t}(L(K))$ for every element $O$ of the closure of $\\mathcal{T}$, where $L$ is a linear map and $t=\\dim_{H}K$. We also prove that for disjoint subsets $A$ and \\textbf{$B$} of $K$ we have that $\\mathcal{H}^{t}(L(A)\\cap L(B))=0$. Hochman and Shmerkin showed that if $\\mathcal{T}$ is dense in $SO(d,\\mathbb{R})$ and the strong separation condition is satisfied then $\\dim_{H}(g(K))=\\min\\{\\dim_{H}K,l\\} $ where $g$ is a continuously differentiable map of rank $l$. We deduce the same result without any separation condition and we generalize a result of Ero$\\breve{\\mathrm{g}}$lu by obtaining that $\\mathcal{H}^{t}(g(K))=0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-09T13:47:39.000Z",
      "title": "Projections and other images of self-similar sets with no separation condition",
      "abstract": "We investigate how the Hausdorff dimension and measure of a self-similar set $K\\subseteq\\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\\mathcal{T}$ generated by the orthogonal parts of the defining maps of $K$. We show that if $\\mathcal{T}$ is finite then every linear image of $K$ is a graph directed attractor and there exists at least one projection of $K$ such that the dimension drops under the image of the projection. In general, with no restrictions on $\\mathcal{T}$ we establish that $\\mathcal{H}^{t}\\left(L\\circ O(K)\\right)=\\mathcal{H}^{t}\\left(L(K)\\right)$ for every element $O$ of the closure of $\\mathcal{T}$, where $L$ is a linear map and $t=\\dim_{H}K$. We also prove that for disjoint subsets $A$ and \\textbf{$B$} of $K$ we have that $\\mathcal{H}^{t}\\left(L(A)\\cap L(B)\\right)=0$. Hochman and Shmerkin showed that if $\\mathcal{T}$ is dense in $SO(d,\\mathbb{R})$ and the strong separation condition is satisfied then $\\dim_{H}\\left(g(K)\\right)=\\min\\left\\{ \\dim_{H}K,l\\right\\} $ where $g$ is a continuously differentiable map of rank $l$. We deduce the same result without any separation condition and we generalize a result of Ero$\\breve{\\mathrm{g}}$lu by obtaining that $\\mathcal{H}^{t}(g(K))=0$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-15T15:10:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80",
      "28A78",
      "37C45"
    ],
    "keywords": [
      "self-similar set",
      "projection",
      "linear image",
      "strong separation condition",
      "dimension drops"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.2841F"
    }
  }
}