{
  "id": "1303.2705",
  "version": "v1",
  "published": "2013-03-11T22:20:40.000Z",
  "updated": "2013-03-11T22:20:40.000Z",
  "title": "Random Iteration of Rational Functions",
  "authors": [
    "David Simmons"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "It is a theorem of Denker and Urba\\'nski ('91) that if $T:\\mathbb C\\to\\mathbb C$ is a rational map of degree at least two and if $\\phi:\\mathbb C\\to\\mathbb R$ is H\\\"older continuous and satisfies the \"thermodynamic expanding\" condition $P(T,\\phi) > \\sup(\\phi)$, then there exists exactly one equilibrium state $\\mu$ for $T$ and $\\phi$, and furthermore $(\\mathbb C,T,\\mu)$ is metrically exact. We extend these results to the case of a holomorphic random dynamical system on $\\mathbb C$, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogensch\\\"utz ('92/'93). Specifically, if $(T,\\Omega,\\textbf P,\\theta)$ is a holomorphic random dynamical system on $\\mathbb C$ and $\\phi:\\Omega\\to H_\\alpha(\\mathbb C)$ is a H\\\"older continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of $(\\mathbb X,\\mathbb T,\\phi)$ over $(\\Omega,\\textbf P,\\theta)$. Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on $\\mathbb C$, generalizing several basic results from the deterministic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-11T22:20:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational functions",
      "random iteration",
      "holomorphic random dynamical system",
      "random potential function satisfying",
      "continuous random potential function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2705S"
    }
  }
}