{
  "id": "1811.07748",
  "version": "v2",
  "published": "2018-11-19T15:29:27.000Z",
  "updated": "2019-01-22T14:54:57.000Z",
  "title": "The infinite dimensional manifold of Hölder equilibrium probabilities has non-negative curvature",
  "authors": [
    "Artur O. Lopes",
    "Rafael O. Ruggiero"
  ],
  "categories": [
    "math.DS",
    "cond-mat.stat-mech",
    "math-ph",
    "math.DG",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Here we consider the discrete time dynamics described by a transformation $T:M \\to M$, where $T$ is either the action of shift $T=\\sigma$ on the symbolic space $M=\\{1,2,...,d\\}^\\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \\to S^1$ of class $C^{1+\\alpha}$ (\\,for example $x \\to T(x) =d\\, x $ (mod $1) $\\,), where $M=S^1$ is the unitary circle. It is known that the infinite dimensional manifold $\\mathcal{N}$ of H\\\"older equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a certain normalized H\\\"older potential $A$ denote by $\\mu_A \\in \\mathcal{N}$ the associated equilibrium probability. The set of tangent vectors $X$ (a function $X: M \\to \\mathbb{R}$) to the manifold $\\mathcal{N}$ at the point $\\mu_A$ coincides with the kernel of the Ruelle operator for the normalized potential $A$. The Riemannian norm $|X|=|X|_A$ of the vector $X$, which is tangent to $\\mathcal{N}$ at the point $\\mu_A$, is described via the asymptotic variance, that is, satisfies \\smallskip $\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,|X|^2\\,\\,= \\,\\,<X,X>\\,\\,=\\,\\,\\lim_{n \\to \\infty} \\frac{1}{n} \\int (\\sum_{i=0}^{n-1} X\\circ T^i )^2 \\,d \\mu_A$. Given two unitary tangent vectors to the manifold $\\mathcal{N}$ at $\\mu_A$, denoted by $X$ and $Y$, we will show that the sectional curvature $K(X,Y)$ equals to $\\int X^{2}Y^{2}d\\mu_{A}$, so it is always non-negative. The curvature vanishes, if and only if, the supports of the functions $X$ and $Y$ are disjoint. In our proof for the above expression for the curvature it is necessary in some moment to show the existence of geodesics for such Riemannian metric.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-01-22T14:54:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37A60"
    ],
    "keywords": [
      "infinite dimensional manifold",
      "equilibrium probability",
      "hölder equilibrium probabilities",
      "non-negative curvature",
      "unitary tangent vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}