{
  "id": "1204.3071",
  "version": "v3",
  "published": "2012-04-13T18:35:08.000Z",
  "updated": "2019-09-14T19:09:37.000Z",
  "title": "Asymptotic expansion of smooth interval maps",
  "authors": [
    "Juan Rivera-Letelier"
  ],
  "comment": "To appear in Asterisque volume dedicated to the memory of J.C. Yoccoz",
  "categories": [
    "math.DS"
  ],
  "abstract": "We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various different ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unicritical maps. This also solves a conjecture of Luzzatto in dimension 1. Combined with a result of Nowicki and Przytycki, these considerations imply that several natural nonuniform hyperbolicity conditions are invariant under topological conjugacy. Another consequence is for the thermodynamic formalism of non-degenerate smooth interval maps: A non-degenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-13T13:02:50.000Z",
      "abstract": "We show that several different ways to quantify the asymptotic expansion of a non-degenerate smooth interval map coincide. A consequence is an extension to multimodal maps of the remarkable result of Nowicki and Sands giving several characterizations of the Collet-Eckmann condition for unimodal maps. Combined with a result of Nowicki and Przytycki, this implies that several natural non-uniform hyperbolicity conditions are invariant under topological conjugacy. Another consequence is for the thermodynamic formalism of non-degenerate smooth interval maps: A high-temperature phase transition occurs precisely when the Topological Collet-Eckmann condition fails.",
      "comment": "Added corollary on regularity and statistical properties of arbitrary exponentially mixing acip's. A throughout discussion on the optimality of the hypothesis of the Main Theorem'",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2019-09-14T19:09:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic expansion",
      "phase transition occurs",
      "non-degenerate smooth interval map coincide",
      "collet-eckmann condition",
      "natural non-uniform hyperbolicity conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.3071R"
    }
  }
}