{
  "id": "2002.09791",
  "version": "v1",
  "published": "2020-02-22T23:27:53.000Z",
  "updated": "2020-02-22T23:27:53.000Z",
  "title": "Projective spectrum and spectral dynamics",
  "authors": [
    "Bryan Goldberg",
    "Rongwei Yang"
  ],
  "categories": [
    "math.FA",
    "math.DS"
  ],
  "abstract": "For a tuple $A= (A_0, A_1, \\ldots , A_n)$ of elements in a unital Banach algebra $\\mathcal{B}$, its \\textit{projective (joint) spectrum} $p(A)$ is the collection of $z\\in\\mathbb{P}^{n}$ such that $A(z)=z_0A_0+z_1 A_1 + \\ldots z_n A_n$ is not invertible. If the tuple $A$ is associated with the generators of a finitely generated group, then $p(A)$ is simply called the projective spectrum of the group. This paper investigates a connection between self-similar group representations and an induced polynomial map on the projective space that preserves the projective spectrum of the group. The focus is on two groups: the infinite dihedral group $D_\\infty$ and the Grigorchuk group ${\\mathcal G}$ of intermediate growth. The main theorem shows that for $D_\\infty$ the Julia set of the induced rational map $F$ is equal to the union of the projective spectrum with the extended indeterminacy set. Moreover, the limit function of the iteration sequence $\\{F^{\\circ n}\\}$ on the Fatou set is determined explicitly. The result has an application to the group ${\\mathcal G}$ and gives rise to a conjecture about its associated Julia set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-22T23:27:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "projective spectrum",
      "spectral dynamics",
      "julia set",
      "self-similar group representations",
      "unital banach algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}