  In this article, we combine complex-analytic and arithmetic tools to study
the preperiodic points of one-dimensional complex dynamical systems. We show
that for any fixed complex numbers a and b, and any integer d at least 2, the
set of complex numbers c for which both a and b are preperiodic for z^d+c is
infinite if and only if a^d = b^d. 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 two complex rational functions f and g have
infinitely many preperiodic points in common, then they must have the same
Julia set. This generalizes a theorem of Mimar, who established the same result
assuming that f and g are defined over an algebraic extension of the rationals.
The main arithmetic ingredient in the proofs is an adelic equidistribution
theorem for preperiodic points over number fields and function fields, with
non-archimedean Berkovich spaces playing an essential role.
