{
  "id": "1309.6009",
  "version": "v1",
  "published": "2013-09-23T23:35:47.000Z",
  "updated": "2013-09-23T23:35:47.000Z",
  "title": "Selections and their Absolutely Continuous Invariant Measures",
  "authors": [
    "A. Boyarsky",
    "P. Góra",
    "Zh. Li"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $% I\\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects onto $I_{i.}$ We define the lower and upper maps $\\tau_{1},$ $\\tau_{2}$ by the lower and upper boundaries of $G_{i},i=1,....,n,$ respectively. We assume $\\tau_{1}$, $\\tau_{2}$ to be piecewise monotonic and preserving continuous invariant measures $\\mu_{1}$ and $\\mu_{2}$, respectively. Let $% F^{(1)}$ and $F^{(2)}$ be the distribution functions of $\\mu_{1}$ and $\\mu_{2}.$ The main results shows that for any convex combination $F$ of $% F^{(1)} $ and $F^{(2)}$ we can find a map $\\eta $ with values between the graphs of $\\tau_{1}$ and $\\tau_{2}$ (that is, a selection) such that $F$ is the $\\eta $-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of multi-valued maps to random maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-23T23:35:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05"
    ],
    "keywords": [
      "absolutely continuous invariant measures",
      "invariant distribution function",
      "disjoint closed regions",
      "random maps",
      "upper maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.6009B"
    }
  }
}