COMMON PREPERIODIC POINTS FOR QUADRATIC POLYNOMIALS

Let fc(z) = z2 + c for c is an element of C. We show there exists a uniform upper bound on the number of points in P1(C) that can be preperiodic for both fc1 and fc2, for any pair c1 =6 c2 in C. The proof combines arithmetic ingredi-ents with complex-analytic: we estimate an adelic energy pairing when the parameters lie in Q, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in com-plex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.