Let fc(z) = z2 + c for c 2 C. We show there exists a uniform bound on the number of points in P1(C) that can be preperiodic for both fc1 and fc2 with c1 6= c2 in C. The proof combines arithmetic ingredients with complex-analytic; we estimate an adelic energy pairing when the parameters lie in Q, building on the quantitative arithmetic equidistribution theorem in [FRL], and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proof is effective, and we provide explicit constants for each of the results.
|Original language||English (US)|
|State||Published - Nov 6 2019|
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