{
  "id": "2408.11314",
  "version": "v1",
  "published": "2024-08-21T03:40:10.000Z",
  "updated": "2024-08-21T03:40:10.000Z",
  "title": "Homoclinic tangencies in $\\mathbb{R}^n$",
  "authors": [
    "Victoria Rayskin"
  ],
  "journal": "Discrete and Continuous Dynamical Systems, 2005, 12(3): 465-480",
  "doi": "10.3934/dcds.2005.12.465",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $f: M \\to M$ denote a diffeomorphism of a smooth manifold $M$. Let $p$ in $M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$, respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact. It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T03:40:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "37C05",
      "37C15",
      "37D10"
    ],
    "keywords": [
      "homoclinic tangencies",
      "horseshoe structure arbitrarily close",
      "similar planar result",
      "smooth manifold",
      "finite order"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}