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Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture. (English) Zbl 1191.14027
The main result of this paper is that if \(f\) and \(g\) are nonlinear complex polynomials of the same degree and if there are \(x_0,y_0\in\mathbb C\) such that the orbits \(\mathcal O_f(x_0)=\{f^n:n\in\mathbb N\}\) and \(\mathcal O_g(y_0)\) have infinite intersection, then \(f\) and \(g\) have a common iterate (i.e., \(f^n=g^n\) for some \(n>0\), where in this review \(f^n\) denotes the \(n^{\text{th}}\) iterate).
This result gives a special case of a conjecture of the first two authors: Let \(f_1,\dots,f_k\) be polynomials in \(\mathbb C[X]\), and let \(V\) be a subvariety of \(\mathbb A^k\). Assume that no positive-dimensional subvariety of \(V\) is periodic under the action of \((f_1,\dots,f_k)\) on \(\mathbb A^k\). Then \(V(\mathbb C)\) has finite intersection with each orbit of \((f_1,\dots,f_k)\) on \(\mathbb A^k\). This conjecture, in turn, fits into far-reaching conjectures in arithmetic dynamics due to S. Zhang [Yau, Shing Tung (ed.), Essays in geometry in memory of S. S. Chern, Surveys in Differential Geometry 10, 381–430 (2006; Zbl 1207.37057)].
The above result is obtained as a corollary of the following theorem. Let \(K\) be a field of characteristic zero, let \(\alpha,\beta,x_0,y_0\in K\) with \(\alpha\neq0\), and let \(f,g\in K[X]\) with \(\deg f=\deg g>1\). If infinitely many points of \(\mathcal O_f(x_0)\times\mathcal O_g(y_0)\) lie on the line \(Y=\alpha X+\beta\), then \(g^k(\alpha X+\beta)=\alpha f^k(X)+\beta\) for some positive integer \(k\).
The latter theorem is proved by first reducing to the case in which \(K\) is a number field, and then applying a result of Y. F. Bilu and R. F. Tichy [Acta Arith. 95, 261–288 (2000; Zbl 0958.11049)]. This latter result describes all \(F,G\in K[X]\) for which \(\deg f=\deg g>1\) and \(F(X)=G(Y)\) has infinitely many solutions over the ring of \(S\)-integers of \(K\), where \(K\) is a number field and \(S\) is a finite set of places of \(K\).
The paper also gives a proof of the following theorem. Let \(K\) be a field of characteristic zero, let \(f,g\in K[X]\), and let \(x_0,y_0\in K\). If the set \(\{(f^n(x_0),g^n(y_0)):n\in\mathbb N\}\) has infinite intersection with a line \(L\) in \(\mathbb A^2\) defined over \(K\), then \(L\) is periodic under the action of \((f,g)\) on \(\mathbb A^2\). This theorem is proved by again reducing to the number field case, and then using Siegel’s theorem to reduce to the earlier theorem.

MSC:
14G25 Global ground fields in algebraic geometry
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15 Dynamical systems over global ground fields
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