{
  "id": "1806.05245",
  "version": "v1",
  "published": "2018-06-13T19:55:51.000Z",
  "updated": "2018-06-13T19:55:51.000Z",
  "title": "Lyapunov spectrum with constant sign",
  "authors": [
    "Vitor Araujo"
  ],
  "comment": "34 pages; 5 figures",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "Let $f:M\\to M$ be a $C^1$ map of a compact manifold $M$, with dimension at least $2$, admitting some point whose future trajectory has only negative Lyapunov exponents. Then this trajectory converges to a periodic sink. We need only assume that $Df$ is never the null map at any point (in particular, we need no extra smoothness assumption on $Df$), encompassing a wide class of possible critical behavior. Similarly, a trajectory having only positive Lyapunov exponents for a $C^1$ diffeomorphism is itself a periodic repeller (source). Analogously for a $C^1$ open and dense subset of vector field on finite dimensional manifolds: for a flow $\\phi_t$ generated by such a vector field, if a trajectory admits weak asymptotic sectional contraction (the extreme rates of expansion of the Linear Poincar\\'e Flow are all negative), then this trajectory belongs either to the basin of attraction of a periodic hyperbolic attracting orbit (a periodic sink or an attracting equilibrium); or the trajectory accumulates a codimension one saddle singularity. Similar results hold for weak sectional expanding trajectories. Both results extend part of the non-uniform hyperbolic theory (Pesin's Theory) from the $C^{1+}$ diffeomorphism setting to $C^1$ endomorphisms and $C^1$ flows. Some ergodic theoretical consequences are discussed. The proofs use versions of Pliss' Lemma for maps and flows translated as (reverse) hyperbolic times, and a condition ensuring that certain subadditive cocycles over vector fields are in fact additive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-13T19:55:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D25",
      "37D30",
      "37D20"
    ],
    "keywords": [
      "lyapunov spectrum",
      "constant sign",
      "vector field",
      "trajectory admits weak asymptotic sectional",
      "admits weak asymptotic sectional contraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}