{
  "id": "2307.13875",
  "version": "v1",
  "published": "2023-07-26T00:44:50.000Z",
  "updated": "2023-07-26T00:44:50.000Z",
  "title": "On the dynamics of endomorphisms of the direct product of two free groups",
  "authors": [
    "André Carvalho"
  ],
  "comment": "21 pages. Comments are welcome!",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that Brinkmann's problems are decidable for endomorphisms of $F_n\\times F_m$: given $(x,y),(z,w)\\in F_n\\times F_m$ and $\\Phi\\in \\text{End}(F_n\\times F_m)$, it is decidable whether there is some $k\\in \\mathbb N$ such that $(x,y)\\Phi^k=(z,w)$ (or $(x,y)\\Phi^k\\sim(z,w)$). We also prove decidability of a two-sided version of Brinkmann's conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\\times F_m$ at the infinity, proving that every point in a continuous extension of an automorphism to the completion is either periodic or wandering, implying that the dynamics is asymptotically periodic, as occurs in the free and free-abelian times free cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-26T00:44:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free groups",
      "direct product",
      "endomorphisms",
      "free-abelian times free cases",
      "brinkmanns conjugacy problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}