A case of the dynamical Mordell–Lang conjecture

Abstract

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 $${(\mathbb P^1)^g}$$ . If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of $${(\mathbb P^1)^g}$$ has only finite intersection with any curve contained in $${(\mathbb P^1)^g}$$ . We also show that our result holds for indecomposable polynomials φ with coefficients in $${\mathbb C}$$ . Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over $${\mathbb C}$$ 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 $${\mathbb A^2}$$ .