Linear relations between polynomial orbits

Abstract

We study the orbits of a polynomial $f\in \mathbb{C}\left[X\right]$, namely, the sets $\{\alpha ,f(\alpha ),f(f\left(\alpha \right)),\dots \}$ with $\alpha \in \mathbb{C}$. 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 ${\mathbb{C}}^d$ with a $d$-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.