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We prove Zhang's Dynamical Manin-Mumford Conjecture and Dynamical BogomolovConjecture for dominant endomorphisms of (P^1)^n. We use the equidistributiontheorem for points of small height with respect to an algebraic dynamicalsystem, combined with an analysis of the symmetries of the Julia set for arational function.

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