  We prove a special case of a dynamical analogue of the classical Mordell-Lang
conjecture. In particular, let $\phi$ be a rational function with no
superattracting periodic points other than exceptional points. If the
coefficients of $\phi$ are algebraic, we show that the orbit of a point outside
the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite
intersection with any curve contained in $(\bP^1)^g$. We also show that our
result holds for indecomposable polynomials $\phi$ with coefficients in $\bC$.
Our proof uses results from $p$-adic dynamics together with an integrality
argument. The extension to polynomials defined over $\bC$ uses the method of
specializations coupled with some new results of Medvedev and Scanlon for
describing the periodic plane curves under the action of $(\phi,\phi)$ on
$\bA^2$.
