{
  "id": "1901.00327",
  "version": "v1",
  "published": "2019-01-02T11:31:45.000Z",
  "updated": "2019-01-02T11:31:45.000Z",
  "title": "Asymptotic pairs in positive-entropy systems",
  "authors": [
    "François Blanchard",
    "Bernard Host",
    "Sylvie Ruette"
  ],
  "comment": "Published in 2002",
  "journal": "Ergod. Th. & Dynam. Syst., 22, 671-686, 2002",
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that in a topological dynamical system $(X,T)$ of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs $(x,y)$ such that $x\\not= y$ and $\\lim_{n\\to +\\infty} d(T^n x,T^n y)=0$. More precisely we consider a $T$-ergodic measure $\\mu$ of positive entropy and prove that the set of points that belong to a proper asymptotic pair is of measure $1$. When $T$ is invertible, the stable classes (i.e., the equivalence classes for the asymptotic equivalence) are not stable under $T^{-1}$: for $\\mu$-almost every $x$ there are uncountably many $y$ that are asymptotic with $x$ and such that $(x,y)$ is a Li-Yorke pair with respect to $T^{-1}$. We also show that asymptotic pairs are dense in the set of topological entropy pairs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-02T11:31:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B40",
      "37A35"
    ],
    "keywords": [
      "positive-entropy systems",
      "positive entropy",
      "proper asymptotic pair",
      "asymptotic equivalence",
      "ergodic measure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}