{
  "id": "1707.04311",
  "version": "v1",
  "published": "2017-07-13T20:35:55.000Z",
  "updated": "2017-07-13T20:35:55.000Z",
  "title": "Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps",
  "authors": [
    "Vitor Araujo",
    "Felipe Santos"
  ],
  "comment": "49 pages, 5 figures, results from PhD thesis",
  "categories": [
    "math.DS"
  ],
  "abstract": "We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: for weak-expanding maps such that $\\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic SRB-like measures satisfy Pesin's Entropy Formula. It follows that the same holds for $C^1$ non-uniformly expanding maps. Furthermore, all the necessarily existing SRB-like measures are equilibrium states for the potential $\\psi=-\\log|\\det Df|$. In particular, this holds for any C$^1$-expanding map of finite dimensional compact Riemannian manifolds and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak$^*$-closed convex hull of the ergodic SRB-like measures. In this work no Markov structure is assumed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-13T20:35:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D25",
      "37D35",
      "37D20",
      "37C40"
    ],
    "keywords": [
      "non-uniformly expanding maps",
      "satisfy pesins entropy formula",
      "dimensional compact riemannian manifolds",
      "measures satisfy pesins entropy",
      "ergodic srb-like measures satisfy"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}