{
  "id": "math/0507073",
  "version": "v2",
  "published": "2005-07-04T15:23:15.000Z",
  "updated": "2005-07-06T15:38:54.000Z",
  "title": "Complex Horseshoes and the Dynamics of Mappings of Two Complex Variables",
  "authors": [
    "Ralph W. Oberste-Vorth"
  ],
  "comment": "26 pages, PhD thesis with addendum",
  "journal": "Cornell University 1987",
  "categories": [
    "math.DS"
  ],
  "abstract": "In this study, a theory analogous to both the theories of polynomial-like mappings and Smale's real horseshoes is developed for the study of the dynamics of mappings of two complex variables. In partial analogy with polynomials in a single variable there are the H\\'enon mappings in two variables as well as higher dimensional analogues. From polynomial-like mappings, H\\'enon-like and quasi-H\\'enon-like mappings are defined following this analogy. A special form of the latter is the complex horseshoe. The major results about the real horseshoes of Smale remain true in the complex setting. In particular: (1) Trapping fields of cones(which are sectors in the real case) in the tangent spaces can be defined and used to find horseshoes. (2) The dynamics of a horseshoe is that of the two-sided shift on the symbol space on some number of symbols which depends on the type of the horseshoe. (3) Transverse intersections of the stable and unstable manifolds of a hyperbolic periodic point guarantee the existence of horseshoes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-07-06T15:38:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex variables",
      "complex horseshoe",
      "hyperbolic periodic point guarantee",
      "smale remain true",
      "polynomial-like mappings"
    ],
    "tags": [
      "dissertation",
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}