#### Vol. 10, No. 5, 2016

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Bifurcations, intersections, and heights

### Laura DeMarco

Vol. 10 (2016), No. 5, 1031–1056
##### Abstract

We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ${f}_{t}:{ℙ}^{1}\left(ℂ\right)\to {ℙ}^{1}\left(ℂ\right)$, 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.

##### Keywords
dynamics of rational maps, canonical height, stability
##### Mathematical Subject Classification 2010
Primary: 37P30
Secondary: 37F45, 11G05