{
  "id": "1406.5318",
  "version": "v1",
  "published": "2014-06-20T08:54:23.000Z",
  "updated": "2014-06-20T08:54:23.000Z",
  "title": "Affine embeddings and intersections of Cantor sets",
  "authors": [
    "De-Jun Feng",
    "Wen Huang",
    "Hui Rao"
  ],
  "comment": "The paper will appear in J. Math. Pure. Appl",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $E, F\\subset \\R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\\dim_HE\\cap f(F)<\\min\\{\\dim_HE, \\dim_HF\\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\\geq 2$ are two integers with $\\log p/\\log q\\not \\in \\Q$. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-20T08:54:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37C45",
      "28A75"
    ],
    "keywords": [
      "cantor sets",
      "affine embeddings",
      "intersection",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5318F"
    }
  }
}