{
  "id": "1310.4942",
  "version": "v1",
  "published": "2013-10-18T08:18:08.000Z",
  "updated": "2013-10-18T08:18:08.000Z",
  "title": "On a non-linear $p$-adic dynamical system",
  "authors": [
    "U. A. Rozikov",
    "I. A. Sattarov"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate the behavior of trajectories of a $(3,2)$-rational $p$-adic dynamical system in the complex $p$-adic filed ${\\mathbb C}_p$, when there exists a unique fixed point $x_0$. We study this $p$-adic dynamical system by dynamics of real radiuses of balls (with the center at the fixed point $x_0$). We show that there exists a radius $r$ depending on parameters of the rational function such that: when $x_0$ is an attracting point then the trajectory of an inner point from the ball $U_r(x_0)$ goes to $x_0$ and each sphere with a radius $>r$ (with the center at $x_0$) is invariant; When $x_0$ is a repeller point then the trajectory of an inner point from a ball $U_r(x_0)$ goes forward to the sphere $S_r(x_0)$. Once the trajectory reaches the sphere, in the next step it either goes back to the interior of $U_r(x_0)$ or stays in $S_r(x_0)$ for some time and then goes back to the interior of the ball. As soon as the trajectory goes outside of $U_r(x_0)$ it will stay (for all the rest of time) in the sphere (outside of $U_r(x_0)$) that it reached first.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-18T08:18:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46S10",
      "12J12",
      "11S99",
      "30D05",
      "54H20"
    ],
    "keywords": [
      "adic dynamical system",
      "non-linear",
      "inner point",
      "trajectory reaches",
      "repeller point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.4942R"
    }
  }
}