The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for endomorphisms of (P^1)^n

We prove Zhang’s Dynamical Manin-Mumford Conjecture and Dynamical Bogomolov Conjecture for dominant endomorphisms of (P^1)^n. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with an analysis of the symmetries of the Julia set for a rational function.