In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of -dimensional complex dynamical systems. We show that for any fixed and any integer , the set of for which both and are preperiodic for is infinite if and only if . This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that and are defined over . The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.
Matthew Baker. Laura Demarco. "Preperiodic points and unlikely intersections." Duke Math. J. 159 (1) 1 - 29, 15 July 2011. https://doi.org/10.1215/00127094-1384773