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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
##### Keywords:
Mordell-Lang conjecture; arithmetic dynamics
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##### References:
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