{
  "id": "1905.08344",
  "version": "v1",
  "published": "2019-05-20T20:51:21.000Z",
  "updated": "2019-05-20T20:51:21.000Z",
  "title": "Regularity of the Density of SRB Measures for Solenoidal Attractors",
  "authors": [
    "Carlos Bocker",
    "Ricardo Bortolotti"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\\mathbb{T}$, $C$ is a linear contracting map of $\\mathbb{R}^d$, $f$ is in $C^r(\\mathbb{T}^u,\\mathbb{R}^d)$ and $r \\geq 2$. We prove that if $|(\\det C)(\\det E)| \\|C^{-1}\\|^{-2s}>1$ for some $s<r-(\\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $H^s(\\mathbb{T}^u\\times \\mathbb{R}^d)$, in particular, if $s>\\frac{u+d}{2}$ then the density is $C^k$ for every $k<s-\\frac{u+d}{2}$. We also exhibit a condition involving $E$ and $C$ under which this tranversality condition is valid for almost every $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-20T20:51:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "srb measure",
      "solenoidal attractors",
      "absolutely continuous invariant probabilities",
      "regularity",
      "higher-dimensional hyperbolic endomorphisms admit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}