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 (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 (P 1 ) g has only finite intersection with any curve contained in (P 1 ) g . 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 2 .

The dynamical Mordell-Lang problem for Noetherian spaces

“…(see [4,5,13,23]). In almost all known ramified cases, Φ is given by the coordinatewise action through one-variable rational maps on (P 1 ) m , i.e.…”

Periods of rational maps modulo primes

“…This project originated in the summer of 2009, when four of the authors (R.B., D.G., P.K., and T.T.) were working to extend their results from [BGKT11] to other cases of the Dynamical Mordell-Lang Conjecture. At that time, T.S.…”

On some extensions of the Ailon–Rudnick theorem

“…for an arbitrary variety V ⊆ C 2 and a sufficiently large integer N ≥ 1. For a fixed t and the diagonal case n = m, this is of the same flavour as the uniform dynamical Mordell-Lang conjecture, which, for a fixed (t 1 , t 2 ) ∈ C 2 asserts that the iterates n, m ≥ 1 such that f (n) (t 1 ), g (m) (t 2 ) ∈ V , see [3,16,17] and references therein, lie in finitely many arithmetic progressions (which number does not depend on t 1 , t 2 ).…”