{
  "id": "1907.12950",
  "version": "v1",
  "published": "2019-07-30T13:59:18.000Z",
  "updated": "2019-07-30T13:59:18.000Z",
  "title": "Periodic points and measures for a class of skew products",
  "authors": [
    "Maria Carvalho",
    "Sebastián A. Pérez"
  ],
  "comment": "1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider the open set constructed by M. Shub in [42] of partially hyperbolic skew products on the space $\\mathbb{T}^2\\times \\mathbb{T}^2$ whose non-wandering set is not stable. We show that there exists an open set $\\mathcal{U}$ of such diffeomorphisms such that if $F_S\\in \\mathcal{U}$ then its measure of maximal entropy is unique, hyperbolic and, generically, describes the distribution of periodic points. Moreover, the non-wandering set of such an $F_S\\in \\mathcal{U}$ contains closed invariant subsets carrying entropy arbitrarily close to the topological entropy of $F_S$ and within which the dynamics is conjugate to a subshift of finite type. Under an additional assumption on the base dynamics, we verify that $F_S$ preserves a unique SRB measure, which is physical, whose basin has full Lebesgue measure and coincides with the measure of maximal entropy. We also prove that there exists a residual subset $\\mathcal{R}$ of $\\mathcal{U}$ such that if $F_S\\in \\mathcal{R}$ then the topological and periodic entropies of $F_S$ are equal, $F_S$ is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T13:59:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37A35",
      "37D30",
      "37A05",
      "37A30"
    ],
    "keywords": [
      "periodic points",
      "skew products",
      "invariant subsets carrying entropy",
      "strongly faithful symbolic extension",
      "carrying entropy arbitrarily close"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}