A Case of the Dynamical Mordell-lang Conjecture

We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (P). If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (P) has only finite intersection with any curve contained in (P). We also show that our result holds for indecomposable polynomials φ with coefficients in C. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over C uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ,φ) on A.