Bifurcations, intersections, and heights

Abstract

We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps $f_t:{\mathbb{P}}^1\left(ℂ\right)\to {\mathbb{P}}^1\left(ℂ\right)$, parameterized by $t$ in a quasiprojective complex variety. We use this to prove one implication in the if-and-only-if statement of a certain conjecture on unlikely intersections in the moduli space of rational maps (see “Special curves and postcritically finite polynomials”, Forum Math. Pi 1 (2013), e3). We present the conjecture here in a more general form.