{
  "id": "2409.07826",
  "version": "v1",
  "published": "2024-09-12T08:16:45.000Z",
  "updated": "2024-09-12T08:16:45.000Z",
  "title": "Intersection of orbits of loxodromic automorphisms of affine surfaces",
  "authors": [
    "Marc Abboud"
  ],
  "categories": [
    "math.AG",
    "math.CV",
    "math.DS",
    "math.NT"
  ],
  "abstract": "We show the following result: If $X_0$ is an affine surface over a field $K$ and $f, g$ are two loxodromic automorphisms with an orbit meeting infinitely many times, then $f$ and $g$ must share a common iterate. The proof uses the preliminary work of the author in [Abb23] on the dynamics of endomorphisms of affine surfaces and arguments from arithmetic dynamics. We then show a dynamical Mordell-Lang type result for surfaces in $X_0 \\times X_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-12T08:16:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "37F80",
      "37P05",
      "37P30",
      "32H50",
      "37P55",
      "11G50"
    ],
    "keywords": [
      "affine surface",
      "loxodromic automorphisms",
      "intersection",
      "dynamical mordell-lang type result",
      "common iterate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}