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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.

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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