{
  "id": "math/0608033",
  "version": "v1",
  "published": "2006-08-01T18:42:53.000Z",
  "updated": "2006-08-01T18:42:53.000Z",
  "title": "Regularity of weak foliations for thermostats",
  "authors": [
    "Gabriel P. Paternain"
  ],
  "doi": "10.1088/0951-7715/20/1/006",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $\\phi$ determined by the motion of a particle under the influence of a magnetic field $\\Omega$ and a thermostat with external field ${\\bf e}$. We show that if $\\phi$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and only if the external field ${\\bf e}$ has a global potential $U$, $g_{1}:=e^{-2U}g$ has constant curvature and $e^{-U}\\Omega$ is a constant multiple of the area form of $g_1$. We also give necessary and sufficient conditions for just one of the weak foliations to be of class $C^{1,1}$ and we show that the {\\it combined} effect of a thermostat and a magnetic field can produce an Anosov flow with a weak stable foliation of class $C^{\\infty}$ and a weak unstable foliation which is {\\it not} $C^{1,1}$. Finally we study Anosov thermostats depending quadratically on the velocity and we characterize those with smooth weak foliations. In particular, we show that quasi-fuchsian flows as defined by Ghys in \\cite{Ghy1} can arise in this fashion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-01T18:42:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "anosov thermostats depending",
      "magnetic field",
      "regularity",
      "external field",
      "smooth weak foliations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}