{
  "id": "math/0402333",
  "version": "v1",
  "published": "2004-02-20T13:03:48.000Z",
  "updated": "2004-02-20T13:03:48.000Z",
  "title": "Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on ${\\bf T}times SL(2,{\\bf R})$",
  "authors": [
    "Raphaël Krikorian"
  ],
  "comment": "80 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Given $\\alpha$ in some set $\\Sigma$ of total (Haar) measure in ${\\bf T}={\\bf R}/{\\bf Z}$, and $A\\in C^{\\infty}({\\bf T},SL(2,{\\bf R}))$ which is homotopic to the identity, we prove that if the fibered rotation number of the skew-product system $(\\alpha,A):{\\bf T}\\times SL(2,{\\bf R})\\to {\\bf T}\\times SL(2,{\\bf R})$, $(\\alpha,A)(\\theta,y)=(\\theta+\\alpha,A(\\theta)y)$ is diophantine with respect to $\\alpha$ and if the fibered products are uniformly bounded in the $C^0$-topology then the cocycle $(\\alpha,A)$ is $C^\\infty$-reducible --that is $A(\\cdot)=B(\\cdot+\\alpha)A_0 B(\\cdot)^{-1}$, for some $A_0\\in SL(2,{\\bf R})$, $B\\in C^{\\infty}({\\bf T},SL(2,{\\bf R}))$. This result which can be seen as a non-pertubative version of a theorem by L.H. Eliasson has two interesting corollaries: the first one is a result of differentiable rigidity: if $\\alpha\\in\\Sigma$ and the cocycle $(\\alpha,A)$ is $C^0$-conjugated to a constant cocycle $(\\alpha,A_0)$ with $A_0$ in a set of total measure in $SL(2,{\\bf R})$ then the conjugacy is $C^\\infty$; the second consequence is: if $\\alpha\\in \\Sigma$ is fixed then the set of $A\\in C^\\infty({\\bf T},SL(2,{\\bf R}))$ for which $(\\alpha,A)$ has positive Lyapunov exponent is $C^\\infty$-dense. A similar result is true for the Schr\\\"odinger cocycle and for 2-frequencies conservative differential equations in the plane.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-02-20T13:03:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "differentiable rigidity",
      "quasi-periodic cocycles",
      "reducibility",
      "total measure",
      "fibered rotation number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 80,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......2333K"
    }
  }
}