{
  "id": "2005.13368",
  "version": "v1",
  "published": "2020-05-27T13:57:55.000Z",
  "updated": "2020-05-27T13:57:55.000Z",
  "title": "Invariant subvarieties with small dynamical degree",
  "authors": [
    "Yohsuke Matsuzawa",
    "Sheng Meng",
    "Takahiro Shibata",
    "De-Qi Zhang",
    "Guolei Zhong"
  ],
  "comment": "30 pages; comments are welcome!",
  "categories": [
    "math.AG",
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $f:X\\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $\\Sigma_{f^{\\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\\infty}}$ of maximal elements in $\\Sigma_{f^{\\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\\sharp P_{f^n}\\le d_1(f)^n(1+o(1))$$ as $n\\to \\infty$, where $d_1(f)$ is the first dynamic degree of $f$. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety (with $f$ stabilizing the big torus), we give an optimal upper bound $$\\sharp S_{f^n}\\le d_1(f)^{n\\cdot\\dim(X)}(1+o(1))$$ as $n \\to \\infty$, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-27T13:57:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J50",
      "08A35",
      "32H50",
      "37B40",
      "11G10",
      "14M25"
    ],
    "keywords": [
      "small dynamical degree",
      "invariant subvarieties",
      "optimal upper bound",
      "invariant prime divisors",
      "first dynamic degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}