Linear relations between polynomial orbits

Abstract

We study the orbits of a polynomial f∈C[X], namely, the sets {α,f(α),f(f(α)),…} with α∈C. We prove that if two nonlinear complex polynomials f,g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in Cd with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.