{
  "id": "2208.04576",
  "version": "v1",
  "published": "2022-08-09T07:35:02.000Z",
  "updated": "2022-08-09T07:35:02.000Z",
  "title": "A Dichotomy for the dimension of SRB measure",
  "authors": [
    "Haojie Ren"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We study dynamical systems generated by skew products: $$T: [0,1)\\times\\mathbb{R}\\to [0,1)\\times\\mathbb{R} \\quad\\quad T(x,y)=(bx\\mod1,\\gamma y+\\phi(x))$$ where integer $b\\ge2$, $0<\\gamma<1$ and $\\phi$ is a real analytic $\\mathbb{Z}$-periodic function. We prove the following dichotomy for the SRB measure $\\omega$ for $T$: Either the support of $\\omega$ is a graph of real analytic function, or the dimension of $\\omega$ is equal to $\\min\\{2,1+\\frac{\\log b}{\\log1/\\gamma}\\}$. Furthermore, given $b$ and $\\phi$, the former alternative only happens for finitely many $\\gamma$ unless $\\phi$ is constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-09T07:35:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "srb measure",
      "real analytic function",
      "periodic function",
      "skew products",
      "study dynamical systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}