{
  "id": "1902.03441",
  "version": "v1",
  "published": "2019-02-09T16:38:24.000Z",
  "updated": "2019-02-09T16:38:24.000Z",
  "title": "The complete $L^q$-spectrum and large deviations for return times for equilibrium states with summable potentials",
  "authors": [
    "M. Abadi",
    "J. -R. Chazottes",
    "S. Gallo"
  ],
  "comment": "37 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\{X_k\\}_{k\\geq 1}$ be a stationary and ergodic process with joint distribution $\\mu$ where the random variables $X_k$ take values in a finite set $\\mathcal{A}$. Let $R_n$ be the first time this process repeats its first $n$ symbols of output. It is well-known that $n^{-1} \\log R_n$ converges almost surely to the entropy of the process. Refined properties of $R_n$ (large deviations, multifractality, etc) are encoded in the return-time $L^q$-spectrum defined as \\[ \\mathcal{R}(q)=\\lim_n\\frac{1}{n}\\log\\int R_n^q\\, d\\mu \\] provided the limit exists. When $\\{X_k\\}_{k\\geq 1}$ is distributed according to the equilibrium state of a potential $\\varphi$ with summable variation, we are able to compute $\\mathcal{R}(q)$ for all $q\\in\\mathbb{R}$, namely \\[ \\mathcal{R}(q) = \\begin{cases} P((1-q)\\varphi) & \\text{for}\\;\\; q\\geq q^*\\\\ \\sup_\\eta \\int \\varphi \\, d\\eta & \\text{for}\\;\\; q\\le q^* \\end{cases} \\] where the supremum is taken over all shift-invariant probability measures, $P((1-q)\\varphi)$ is the topological pressure of $(1-q)\\varphi$ ($q\\in\\mathbb{R})$, and $q^*=q^*_\\varphi\\in [-1,0)$ is the unique solution of $P((1-q)\\varphi) =\\sup_\\eta \\int \\varphi \\, d\\eta$, as $q$ runs through $\\mathbb{R}$. Surprisingly, this spectrum does not coincide with the hitting-time $L^q$-spectrum for all $q<q^*$. The two spectra coincide if and only if the equilibrium state of $\\varphi$ is the measure of maximal entropy. As a by-product, we prove large deviation asymptotics for $n^{-1}\\log R_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-09T16:38:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equilibrium state",
      "return times",
      "summable potentials",
      "shift-invariant probability measures",
      "large deviation asymptotics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}