{
  "id": "1806.08873",
  "version": "v1",
  "published": "2018-06-22T22:39:17.000Z",
  "updated": "2018-06-22T22:39:17.000Z",
  "title": "Collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles",
  "authors": [
    "Cecilia González-Tokman",
    "Anthony Quas"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "In this paper, we study random Blaschke products, acting on the unit circle and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. We then consider a perturbation in which the Perron-Frobenius operators are composed with a convolution with a normal distribution with small variance (this corresponds to replacing a map by an annealed version where the image point is moved by a random normal amount). Combining the above, we obtain a very natural cocycle, where the unperturbed cocycle has infinitely many distinct Lyapunov exponents, but arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum except for the exponent 0 associated with the absolutely continuous invariant measure. The phenomenon is superficially similar to the finite-dimensional phenomenon, discovered by Bochi, where a generic area-preserving diffeomorphism is either hyperbolic, or has all zero Lyapunov exponents. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-22T22:39:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37H15"
    ],
    "keywords": [
      "lyapunov spectrum",
      "perron-frobenius operator cocycles",
      "study random blaschke products",
      "arbitrarily small natural perturbations",
      "distinct lyapunov exponents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}