{
  "id": "2301.04909",
  "version": "v1",
  "published": "2023-01-12T10:04:35.000Z",
  "updated": "2023-01-12T10:04:35.000Z",
  "title": "Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters",
  "authors": [
    "Dinis Amaro",
    "Mario Bessa",
    "Helder Vilarinho"
  ],
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $\\Phi:\\mathbb{R}\\times M\\to \\text{GL}(2,\\mathbb{R})$ induced by second order linear homogeneous differential equations $\\ddot x+\\alpha(\\varphi^t(\\omega))\\dot x+\\beta(\\varphi^t(\\omega))x=0$ are almost everywhere distinct. The coefficients $\\alpha,\\beta$ evolve along the $\\varphi^t$-orbit for $\\omega\\in M$ and $\\varphi^t: M\\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\\ddot x+\\beta(\\varphi^t(\\omega))x=0$ and for a Schr\\\"odinger equation $\\ddot x+(E-Q(\\varphi^t(\\omega)))x=0$, inducing a cocycle $\\Phi:\\mathbb{R}\\times M\\to \\text{SL}(2,\\mathbb{R})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-12T10:04:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lyapunov spectrum",
      "lp parameters",
      "order linear homogeneous differential equations",
      "second order linear homogeneous differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}