{
  "id": "1508.03625",
  "version": "v1",
  "published": "2015-08-14T19:57:31.000Z",
  "updated": "2015-08-14T19:57:31.000Z",
  "title": "Semi-parabolic tools for hyperbolic Hénon maps and continuity of Julia sets in $\\mathbb{C}^{2}$",
  "authors": [
    "Remus Radu",
    "Raluca Tanase"
  ],
  "comment": "45 pages, incl. references; 6 figures; continues arXiv:1411.3824",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of strongly dissipative H\\'enon maps which have a fixed point with one eigenvalue $(1+t)\\lambda$, where $\\lambda$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the H\\'enon map is hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and $J^{+}$ depend continuously on the parameters as $t\\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of H\\'enon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-14T19:57:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F45",
      "37D99",
      "32A99",
      "47H10"
    ],
    "keywords": [
      "julia set",
      "hyperbolic hénon maps",
      "semi-parabolic tools",
      "complex henon map",
      "fixed point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803625R"
    }
  }
}