{
  "id": "1404.7801",
  "version": "v2",
  "published": "2014-04-30T17:04:42.000Z",
  "updated": "2015-07-09T21:27:11.000Z",
  "title": "Duality results for Iterated Function Systems with a general family of branches",
  "authors": [
    "Jairo K. Mengue",
    "Elismar R. Oliveira"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.OC"
  ],
  "abstract": "For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\\{\\tau_x: Z\\to Z,\\, x\\in x\\}$ and $\\{\\tau_y:W\\to W,\\, y\\in Y\\}$. For a fixed $\\alpha \\in \\mathcal{P}(X)$ with $supp(\\alpha)=X$ we define the entropy of a holonomic measure $\\pi \\in \\mathcal{P}(X\\times Z)$ relative to $\\alpha$, the pressure of a continuous cost function $c(x,z)$ and show that for $c$ Lipschitz this pressure coincides with the spectral radius of the associated transfer operator. The same approach can be applied to the pair $Y,W$. For fixed probabilities $\\alpha \\in \\mathcal{P}(X)$ and $\\beta \\in \\mathcal{P}(Y)$ with $supp(\\alpha)=X,\\,supp(\\beta)=Y$ we denote by $H_\\alpha(\\pi), \\pi \\in \\Pi(\\cdot,\\cdot,\\tau)$, the entropy of the $(X,Z)-$marginal of $\\pi$ relative to $\\alpha$ and denote by $H_\\beta(\\pi)$, the entropy of the $(Y,W)-$marginal of $\\pi$ relative to $\\beta$. The marginal pressure of a continuous cost function $c \\in C(X\\times Y \\times Z \\times W)$ relative to $(\\alpha,\\beta)$ will be defined by $P^{m}(c) = \\sup_{\\pi\\in\\Pi(\\cdot,\\cdot,\\tau)} \\int c\\, d\\pi + H_{\\alpha}(\\pi) +H_{\\beta}(\\pi)$ and we will show the following duality result: \\[\\inf_{P^{m}(c -\\varphi(x) -\\psi(y))=0} \\int \\varphi(x)\\,d\\mu +\\int \\psi(y)\\,d\\nu = \\sup_{\\pi\\in\\Pi(\\mu,\\nu,\\tau)} \\int c\\, d\\pi + H_{\\alpha}(\\pi) +H_{\\beta}(\\pi).\\] When $Z$ and $W$ have only one point and the entropy is unconsidered this equality can be rewritten as the Kantorovich Duality for compact spaces $X,Y$ and continuous cost $-c$: \\[\\inf_{c -\\varphi(x) -\\psi(y)\\leq 0} \\int \\varphi(x)\\,d\\mu +\\int \\psi(y)\\,d\\nu = \\sup_{\\pi\\in\\Pi(\\mu,\\nu)} \\int c\\, d\\pi .\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-30T17:04:42.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-07-09T21:27:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "iterated function systems",
      "duality result",
      "general family",
      "continuous cost function",
      "compact metric spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7801M"
    }
  }
}