{
  "id": "1510.04681",
  "version": "v1",
  "published": "2015-10-15T19:46:51.000Z",
  "updated": "2015-10-15T19:46:51.000Z",
  "title": "Almost sure convergence of maxima for chaotic dynamical systems",
  "authors": [
    "M. P. Holland",
    "M. Nicol",
    "A. Török"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Suppose $(f,\\mathcal{X},\\nu)$ is a measure preserving dynamical system and $\\phi:\\mathcal{X}\\to\\mathbb{R}$ is an observable with some degree of regularity. We investigate the maximum process $M_n:=\\max\\{X_1,\\ldots,X_n\\}$, where $X_i=\\phi\\circ f^i$ is a time series of observations on the system. When $M_n\\to\\infty$ almost surely, we establish results on the almost sure growth rate, namely the existence (or otherwise) of a sequence $u_n\\to\\infty$ such that $M_n/u_n\\to 1$ almost surely. The observables we consider will be functions of the distance to a distinguished point $\\tilde{x}\\in \\mathcal{X}$. Our results are based on the interplay between shrinking target problem estimates at $\\tilde{x}$ and the form of the observable (in particular polynomial or logarithmic) near $\\tilde{x}$. We determine where such an almost sure limit exists and give examples where it does not. Our results apply to a wide class of non-uniformly hyperbolic dynamical systems, under mild assumptions on the rate of mixing, and on regularity of the invariant measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-15T19:46:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chaotic dynamical systems",
      "sure convergence",
      "shrinking target problem estimates",
      "sure growth rate",
      "wide class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151004681H"
    }
  }
}