{
  "id": "2304.06892",
  "version": "v1",
  "published": "2023-04-14T01:59:45.000Z",
  "updated": "2023-04-14T01:59:45.000Z",
  "title": "Transitivity and bifurcation sets for the $β$-transformation with a hole at $0$",
  "authors": [
    "Pieter Allaart",
    "Derong Kong"
  ],
  "comment": "60 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Given $\\beta\\in(1,2]$, let $T_\\beta$ be the $\\beta$-transformation on the unit circle $[0,1)$ such that $T_\\beta(x)=\\beta x\\pmod 1$. For each $t\\in[0,1)$ let $K_\\beta(t)$ be the survivor set consisting of all $x\\in[0,1)$ whose orbit $\\{T^n_\\beta(x): n\\ge 0\\}$ never enters the interval $[0,t)$. Letting $\\mathscr{E}_\\beta$ denote the bifurcation set of the set-valued map $t\\mapsto K_\\beta(t)$, Kalle et al. [{\\em Ergodic Theory Dynam. Systems}, 40 (9): 2482--2514, 2020] conjectured that \\[ \\dim_H\\big(\\mathscr{E}_\\beta\\cap[t,1]\\big)=\\dim_H K_\\beta(t) \\qquad \\forall\\,t\\in(0,1). \\] The main purpose of this article is to prove this conjecture. We do so by investigating dynamical properties of the symbolic equivalent of the survivor set $K_\\beta(t)$, in particular its entropy and topological transitivity. In addition, we compare $\\mathscr{E}_\\beta$ with the bifurcation set $\\mathscr{B}_\\beta$ of the map $t\\mapsto \\dim_H K_\\beta(t)$ (which is a decreasing devil's staircase by a theorem of Kalle et al.), and show that, for Lebesgue-almost every $\\beta\\in(1,2]$, the difference $\\mathscr{E}_\\beta\\backslash\\mathscr{B}_\\beta$ has positive Hausdorff dimension, but for every $k\\in\\{0,1,2,\\dots\\}\\cup\\{\\aleph_0\\}$, there are infinitely many values of $\\beta$ such that the cardinality of $\\mathscr{E}_\\beta\\backslash\\mathscr{B}_\\beta$ is exactly $k$. Some connections with other topics in dynamics, such as kneading invariants of Lorenz maps and the doubling map with an arbitrary hole, are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-14T01:59:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "28A78",
      "68R15",
      "26A30",
      "37E05",
      "37B40"
    ],
    "keywords": [
      "bifurcation set",
      "transitivity",
      "transformation",
      "survivor set",
      "ergodic theory dynam"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}