{
  "id": "2012.15528",
  "version": "v1",
  "published": "2020-12-31T10:29:20.000Z",
  "updated": "2020-12-31T10:29:20.000Z",
  "title": "Almost blenders and parablenders",
  "authors": [
    "Sébastien Biebler"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and D{\\'i}az in the 90s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper, we introduce analogous notions in a measurable point of view. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of R2 leaving invariant the continuation of a hyperbolic basic set. When some inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak and Urba{\\'n}ski, we study families of fiberwise unipotent skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-31T10:29:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic basic set",
      "parablender",
      "study families",
      "local unstable manifolds contains",
      "positive measure inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}