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 Phi 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.