{
  "id": "2407.04658",
  "version": "v1",
  "published": "2024-07-05T17:14:34.000Z",
  "updated": "2024-07-05T17:14:34.000Z",
  "title": "Thermodynamic Formalism for a family of cellular automata and duality with the shift",
  "authors": [
    "Artur O. Lopes",
    "Elismar R. Oliveira",
    "Marcelo Sobottka"
  ],
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.PR",
    "nlin.CG"
  ],
  "abstract": "We will consider a family of cellular automata $\\Phi: \\{1,2,...,r\\}^\\mathbb{N}\\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $\\sigma$-invariant and $\\Phi$-invariant, where $\\sigma$ is the full shift. Via the use of versions of the Ruelle operator $\\mathcal{L}_{A,\\sigma}$ and $\\mathcal{L}_{B,\\Phi}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $\\sigma$ and $\\Phi$. Via the use of a version of the involution kernel $W$ for a $(\\sigma,\\Phi)$-mixed skew product $\\hat{\\Phi}: \\{1,2,...,r\\}^\\mathbb{Z}\\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $\\rho_A$ for $(\\mathcal{L}_{A,\\sigma})^*$ and eigenfunctions $\\psi_B$ for $\\mathcal{L}_{B,\\Phi}$. In another direction, considering the non-mixed extension $\\hat{\\Phi}_n : \\{1,2,...,r\\}^\\mathbb{Z}\\circlearrowright$ of $\\Phi$, given a Lispchitz potential $\\hat{A} : \\{1,2,...,r\\}^\\mathbb{Z}\\to \\mathbb{R}$, we can identify a Lipschitz potential $A:\\{1,2,...,r\\}^\\mathbb{N} \\to \\mathbb{R} $, in such away that relates the variational problem of $\\hat{\\Phi}_n$-Topological Pressure for $\\hat{A}$ with the $\\Phi$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $\\Phi$ (or $\\hat{\\Phi})$ can eventually be conjugated with another shift of finite type is irrelevant in our context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-05T17:14:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37B15",
      "68Q80"
    ],
    "keywords": [
      "cellular automata",
      "thermodynamic formalism",
      "lispchitz potential",
      "topological pressure",
      "full shift"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}