The Dynamical Mordell-Lang Conjecture

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