{
  "id": "1709.03906",
  "version": "v1",
  "published": "2017-09-12T15:18:43.000Z",
  "updated": "2017-09-12T15:18:43.000Z",
  "title": "Affine embeddings of Cantor sets in the plane",
  "authors": [
    "Amir Algom"
  ],
  "comment": "51 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $F,E\\subseteq \\mathbb{R}^2$ be two self similar sets. First, assuming $F$ is generated by an IFS $\\Phi$ with strong separation, we characterize the affine maps $g:\\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ such that $g(F)\\subseteq F$. Our analysis depends on the cardinality of the group $G_\\Phi$ generated by the orthogonal parts of the similarities in $\\Phi$. When $|G_\\Phi|=\\infty$ we show that any such self embedding must be a similarity, and so (by the results of Elekes, Keleti and M\\'ath\\'{e}) some power of its orthogonal part lies in $G_\\Phi$. When $|G_\\Phi| < \\infty$ and $\\Phi$ has a uniform contraction $\\lambda$, we show that the linear part of any such embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of $\\lambda$. We also study the existence and properties of affine maps $g$ such that $g(F)\\subseteq E$, where $E$ is generated by an IFS $\\Psi$. In this direction, we provide more evidence for a Conjecture of Feng, Huang and Rao, that such an embedding exists only if the contraction ratios of the maps in $\\Phi$ are algebraically dependent on the contraction ratios of the maps in $\\Psi$. Furthermore, we show that, under some conditions, if $|G_\\Phi|=\\infty$ then $|G_\\Psi|=\\infty$ and if $|G_\\Phi|<\\infty$ then $|G_\\Psi|<\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-12T15:18:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cantor sets",
      "affine embeddings",
      "contraction ratios",
      "affine maps",
      "self similar sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}