{
  "id": "1803.04060",
  "version": "v1",
  "published": "2018-03-11T22:48:20.000Z",
  "updated": "2018-03-11T22:48:20.000Z",
  "title": "Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation",
  "authors": [
    "Scott Schmieding"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X_{A},\\sigma_{A})$ be a shift of finite type and $\\text{Aut}(\\sigma_{A})$ its corresponding automorphism group. Associated to $\\phi \\in \\text{Aut}(\\sigma_{A})$ are certain Lyapunov exponents $\\alpha^{-}(\\phi), \\alpha^{+}(\\phi)$ which describe asymptotic behavior of the sequence of coding ranges of $\\phi^{n}$. We give lower bounds on $\\alpha^{-}(\\phi), \\alpha^{+}(\\phi)$ in terms of the spectral radius of the corresponding action of $\\phi$ on the dimension group associated to $(X_{A},\\sigma_{A})$. We also give lower bounds on the topological entropy $h_{top}(\\phi)$ in terms of a distinguished part of the spectrum of the action of $\\phi$ on the dimension group, but show that in general $h_{top}(\\phi)$ is not bounded below by the logarithm of the spectral radius of the action of $\\phi$ on the dimension group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-11T22:48:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lyapunov exponents",
      "dimension representation",
      "dimension group",
      "lower bounds",
      "spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}