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  "body": "\n## Statement\n\nLet $X$ be a [[quasi-projective variety|EQSHJPE7H8]] over $\\C$ and let $\\Phi \\colon X \\to X$ be a [[morphism|CDS1VWB7A3]].\nWrite $\\Phi^n$ for the $n$-th [[iterate|AYVTR2ES5B]] of $\\Phi$, with $\\Phi^0 = \\id$.\nLet $V \\subseteq X$ be a closed [[subvariety|4YSCK1DF26]] and let $x \\in X(\\C)$.\nThen the set\n$$\\set{n \\in \\N : \\Phi^n(x) \\in V}$$\nis a finite union of sets of the form $\\set{a k + b : k \\in \\N}$ with $a, b \\in \\N$.\n\n## Remarks\n\nA set $\\set{ak + b : k \\in \\N}$ with $a = 0$ is the single point $\\set{b}$, so finite sets are allowed.\nAn equivalent form concerns the orbit $\\set{\\Phi^n(x) : n \\in \\N}$ of $x$.\nIf the orbit meets $V$ in infinitely many points, then $V$ contains a positive-dimensional subvariety $W$ with $\\Phi^a(W) \\subseteq W$ for some $a \\ge 1$ and $\\Phi^b(x) \\in W$ for some $b$.\nThe conjecture is open in general.\nBell, Ghioca and Tucker proved it when $\\Phi$ is étale.\nWhen $X = \\C^N$ and $\\Phi$ is linear, it is the Skolem–Mahler–Lech theorem on the zeros of a [[linear recurrence|0NDVHBVBXB]].\nWhen $\\Phi$ is a translation on a semiabelian variety, it follows from the Mordell–Lang theorem of Faltings and Vojta, which gives the name.\nThe main tool is $p$-adic.\nOn suitable residue classes of $n$, the map $n \\mapsto \\Phi^n(x)$ is interpolated by a $p$-adic analytic function.\nA nonzero analytic function on $\\Z_p$ has finitely many zeros.\nOver fields of positive characteristic the statement fails.\nFor example, over $\\F_p(t)$ let $\\Phi(x, y) = (tx, (1 + t)y)$ on $\\mathbb{G}_m^2$, let $V$ be the line $y = x + 1$ and start at $(1, 1)$.\nThen $\\Phi^n(1, 1) = (t^n, (1 + t)^n)$ lies in $V$ exactly when $(1 + t)^n = t^n + 1$, that is, when $n$ is a power of $p$.\n\n## Explanations\n\n### L1 human\n\nStart at a point and apply the same polynomial map again and again.\nAsk at which steps the point lands on a fixed curve or surface.\nThe conjecture predicts that these steps are never irregular: they form finitely many arithmetic progressions, plus finitely many extra steps.\nSo either the orbit hits the surface only a few times, or it hits it in a regular repeating pattern.\n",
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    "history": {
      "first_stated": "2009",
      "posed_by": "Dragos Ghioca and Thomas J. Tucker"
    },
    "kind": "conjecture",
    "msc": [
      "37P55",
      "37P15"
    ],
    "name": "Dynamical Mordell–Lang conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Arithmetic_dynamics"
      }
    ],
    "tag": "6TV7FR553H",
    "uses": [
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    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/6TV7FR553H/",
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  "markdown": "https://tamnd.github.io/math2/node/6TV7FR553H.md",
  "msc": [
    "37P55",
    "37P15"
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  "name": "Dynamical Mordell–Lang conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/6TV/6TV7FR553H.md",
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  "tag": "6TV7FR553H",
  "uri": "m2://6TV7FR553H",
  "url": "https://tamnd.github.io/math2/wiki/dynamical-mordell-lang-conjecture/",
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    "level": "W0",
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