(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
{*e*, *g*(*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
**C**^{d} 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.

*Michael Zieve*:
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