Preperiodic points and unlikely intersections

Abstract

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of $1$-dimensional complex dynamical systems. We show that for any fixed $a,b\in \mathbb{C}$ and any integer $d\ge 2$, the set of $c\in \mathbb{C}$ for which both $a$ and $b$ are preperiodic for $z^d+c$ is infinite if and only if $a^d=b^d$. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions $\varphi ,\psi \in \mathbb{C}(z)$ have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that $\varphi$ and $\psi$ are defined over $\overline{\mathbb{Q}}$. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.