Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve:
Linear relations between polynomial orbits,
Duke Math. J. 161 (2012), 1379–1410.

(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 and h have orbits with infinite intersection, then g and h have a common iterate. More generally, we describe the intersection of any line in Cd with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.

In our previous paper, we proved the first result in case g and h have the same degree. The proof in the present paper combines this previous result with Siegel's theorem on integral points on curves, the Bilu–Tichy classification of Diophantine equations G(x)=H(y) having infinitely many S-integer solutions, and several new results on polynomial decomposition, including a recent result of Müller and Zieve. We also give an alternate method for deducing our result from the result of our previous paper, which works in the non-isotrivial case (meaning that there is no linear change of variables after which both orbits are contained in a number field); this alternate approach replaces the use of Bilu–Tichy and polynomial decomposition with arguments involving canonical heights.

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