{
  "id": "math/9902159",
  "version": "v1",
  "published": "1999-02-26T22:53:33.000Z",
  "updated": "1999-02-26T22:53:33.000Z",
  "title": "Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits",
  "authors": [
    "Vadim Kaloshin"
  ],
  "comment": "12 pages, 7 postscript figures",
  "doi": "10.1007/s002200050811",
  "categories": [
    "math.DS"
  ],
  "abstract": "Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhouse domain ${\\cal N} \\subset Diff^r(M)$, an arbitrary quick growth of the number of periodic points holds on a residual set. This result follows from a theorem of Gonchenko-Shilnikov-Turaev, a detailed proof of which is also presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-02-26T22:53:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic orbits",
      "generic diffeomorphisms",
      "superexponential growth",
      "a-m property",
      "hyperbolic periodic points"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Commun. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}