Given two rational maps @math and @math on @math of degree at least two, we study a symmetric, nonnegative-real-valued pairing @math which is closely related to the canonical height functions @math and @math associated to these maps. Our main results show a strong connection between the value of @math and the canonical heights of points which are small with respect to at least one of the two maps @math and @math . Several necessary and sufficient conditions are given for the vanishing of @math . We give an explicit upper bound on the difference between the canonical height @math and the standard height @math in terms of @math , where @math denotes the squaring map. The pairing @math is computed or approximated for several families of rational maps @math .
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.