Abstract
In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ft : P 1 (C)! P 1 (C), parame- terized by t in a quasi-projective complex variety. We use this to prove one implication in the if-and-only-if statement of (BD2, Conjecture 1.10) on unlikely intersections in the moduli space of rational maps; we present the conjecture here in a more general form.
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