Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve:
Intersections of polynomial orbits, and a dynamical Mordell–Lang conjecture,
Inventiones Math. 171 (2008), 463–483. MR 2008k:37099

(Both the published version and the arXiv version are available online.)

We study the orbits of a polynomial g(X) in C[X], namely the sets {eg(e), g(g(e)), ...} with e in C. We prove that if nonlinear complex polynomials g,h of the same degree have orbits with infinite intersection, then g and h have a common iterate. We also present a dynamical analogue of the Mordell–Lang conjecture, and deduce a special case of this conjecture from our result.

Our proof involves a dynamical analogue of Silverman's specialization theorem, which we prove by means of the Tate/Call–Silverman theory of canonical heights of morphisms of varieties. This specialization result allows us to reduce to the case that both orbits are contained in a number field K, and hence in a ring R of S-integers of K. It follows that, for every n, the equation gn(x)=hn(y) has infinitely many solutions with x,y in R, where gn denotes the n-th iterate of g. According to a result of Bilu and Tichy, this gives information about the functional decompositions of gn and hn. We obtain our result by combining the information deduced for all n with some additional arguments involving polynomial decomposition and the structure of the class group and unit group of the ring of integers of K.

Additional comment added July 2008:  See our subsequent paper for a generalization of the result in this paper, in which we need not assume that g and h have the same degree.

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