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

In [DKY], it was conjectured that there is a uniform bound $B$, depending only 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 their preperiodic points or have at most $B$ in common. Here we show that this uniform 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 proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, and Mavraki-Schmidt. In addition, we present alternate proofs of recent results of DeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of a conjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systems and elliptic curves.