{
  "id": "2101.02981",
  "version": "v1",
  "published": "2021-01-08T12:23:49.000Z",
  "updated": "2021-01-08T12:23:49.000Z",
  "title": "Contraction groups and the big cell for endomorphisms of Lie groups over local fields",
  "authors": [
    "Helge Glockner"
  ],
  "comment": "34 pages, LaTeX; former title: Contraction groups of analytic endomorphisms and dynamics on the big cell",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a Lie group over a totally disconnected local field and $\\alpha$ be an analytic endomorphism of $G$. The contraction group of $\\alpha$ ist the set of all $x\\in G$ such that $\\alpha^n(x)\\to e$ as $n\\to\\infty$. Call sequence $(x_{-n})_{n\\geq 0}$ in $G$ an $\\alpha$-regressive trajectory for $x\\in G$ if $\\alpha(x_{-n})=x_{-n+1}$ for all $n\\geq 1$ and $x_0=x$. The anti-contraction group of $\\alpha$ is the set of all $x\\in G$ admitting an $\\alpha$-regressive trajectory $(x_{-n})_{n\\geq 0}$ such that $x_{-n}\\to e$ as $n\\to\\infty$. The Levi subgroup is the set of all $x\\in G$ whose $\\alpha$-orbit is relatively compact, and such that $x$ admits an $\\alpha$-regressive trajectory $(x_{-n})_{n\\geq 0}$ such that $\\{x_{-n}\\colon n\\geq 0\\}$ is relatively compact. The big cell associated to $\\alpha$ is the set $\\Omega$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $\\pi$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $\\Omega$ which maps $(x,y,z)$ to $xyz$. We show: $\\Omega$ is open in $G$ and $\\pi$ is \\'{e}tale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-08T12:23:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E20",
      "22D05",
      "22E25",
      "22E35",
      "32P05",
      "37B05",
      "37C05",
      "37C86",
      "37D10"
    ],
    "keywords": [
      "big cell",
      "lie group",
      "local field",
      "anti-contraction group",
      "immersed lie subgroup structures"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}