{
  "id": "1912.09213",
  "version": "v1",
  "published": "2019-12-19T14:22:46.000Z",
  "updated": "2019-12-19T14:22:46.000Z",
  "title": "Revisiting the asymptotics of the flow for some dynamical systems on the torus",
  "authors": [
    "Marc Briane",
    "Loïc Hervé"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper we study the large time asymptotics of the flow of a dynamical system $X'=b(X)$ posed in the $d$-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if $b$ vanishes at different points, we only assume that the set of the averages of $b$ with respect to the invariant probability measures for the flow is reduced to a singleton. We also rewrite the Liouville theorem which holds for any invariant probability measure $\\mu$, namely $\\mu\\,b$ is divergence free, as a divergence-curl formula satisfied by any regular periodic function. The combination of these two tools turns out to be a new approach to get the asymptotics for some flows. This allows us to obtain the desired asymptotics in any dimension when $b = a\\,\\xi$ with $a$ a possibly vanishing periodic nonnegative function and $\\xi$ a nonzero vector in $R^d$, or when $b = A\\nabla v$ with $A$ a periodic nonnegative symmetric matrix-valued function and $v$ a periodic function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-19T14:22:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dynamical system",
      "asymptotics",
      "nonnegative symmetric matrix-valued function",
      "vanishing periodic nonnegative function",
      "invariant probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}