{
  "id": "1609.02119",
  "version": "v1",
  "published": "2016-09-07T19:16:00.000Z",
  "updated": "2016-09-07T19:16:00.000Z",
  "title": "Degeneration of Dynamical Degrees in Families of Maps",
  "authors": [
    "Gregory Call",
    "Joseph H. Silverman"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "The dynamical degree of a dominant rational map $f:\\mathbb{P}^N-rightarrow\\mathbb{P}^N$ is the quantity $\\delta(f):=\\lim(\\text{deg} f^n)^{1/n}$. We study the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make three conjectures concerning, respectively, the set of $t$ such that: (1) $\\delta(f_t)\\le\\delta(f_T)-\\epsilon$; (2) $\\delta(f_t)<\\delta(f_T)$; (3) $\\delta(f_t)<\\delta(f_T)$ and $\\delta(g_t)<\\delta(g_T)$ for \"independent\" families of maps. We prove our first conjecture for monomial maps and give evidence for our second and third conjectures by proving them for certain non-trivial families.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-07T19:16:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P05",
      "37P30",
      "37P55"
    ],
    "keywords": [
      "dynamical degree",
      "degeneration",
      "dominant rational map",
      "monomial maps",
      "third conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}