(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*,*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 *g ^{n}*(

*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.

*Michael Zieve*:
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