H\"older estimates and uniformity in arithmetic dynamics
Abstract
In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees d1,d2≥2, outside a Zariski closed subset of the space of pairs of rational maps (f,g) of degree d1 and d2 respectively, the maps f and g share at most a uniformly bounded number of common preperiodic points. This generalizes a result of DeMarco and Mavraki to maps of possibly different degrees. Our main contribution is the use of H\"older properties of the Green function of a rational map to obtain height estimates.
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