We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps [math] , parameterized by [math] 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.
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Laura DeMarco (2016) studied this question.
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