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In [DKY], it was conjectured that there is a uniform bound $B$, dependingonly on the degree $d$, so that any pair of holomorphic maps $f, g:\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of theirpreperiodic points or have at most $B$ in common. Here we show that thisuniform bound holds for a Zariski open and dense set in the space of all pairs,$\mathrm{Rat}_d \times \mathrm{Rat}_d$, for each degree $d\geq 2$. The proofinvolves a combination of arithmetic intersection theory and complex-dynamicalresults, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, andMavraki-Schmidt. In addition, we present alternate proofs of recent results ofDeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of aconjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systemsand elliptic curves.

Dynamics on $\mathbb{P}^1$: preperiodic points and pairwise stability