Bifurcations, intersections, and heights

Abstract

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_t: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$, parameterized by $t$ in a quasi-projective complex variety. We use this to prove one implication in the if-and-only-if statement of Conjecture 1.10 in [Baker-DeMarco, 2013] on unlikely intersections in the moduli space of rational maps; we present the conjecture here in a more general form.