{
  "id": "1710.06712",
  "version": "v1",
  "published": "2017-10-18T13:03:50.000Z",
  "updated": "2017-10-18T13:03:50.000Z",
  "title": "Linear response for Dirac observables of Anosov diffeomorphisms",
  "authors": [
    "Matthieu Porte"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a $\\mathcal{C}^3$ family $t\\mapsto f_t$ of $\\mathcal{C}^4$ Anosov diffeomorphisms on a compact Riemannian manifold $M$. Denoting by $\\rho_t$ the SRB measure of $f_t$, we prove that the map $t\\mapsto\\int \\theta d\\rho_t$ is differentiable if $\\theta$ is of the form $\\theta(x)=h(x)\\delta(g(x)-a)$, with $\\delta$ the Dirac distribution, $g:M\\rightarrow \\mathbb{R}$ a $\\mathcal{C}^4$ function, $h:M\\rightarrow\\mathbb{R}$ a $\\mathcal{C}^3$ function and $a$ a regular value of $g$. We also require a transversality condition, namely that the intersection of the support of $h$ with the level set $\\{g(x)=a\\} $ is foliated by 'admissible stable leaves'.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-18T13:03:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "anosov diffeomorphisms",
      "dirac observables",
      "linear response",
      "compact riemannian manifold",
      "srb measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}