We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps , parameterized by 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.
"Bifurcations, intersections, and heights." Algebra Number Theory 10 (5) 1031 - 1056, 2016. https://doi.org/10.2140/ant.2016.10.1031