  We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times
P^1 defined over a number field. We use the equidistribution theorem for points
of small height with respect to an algebraic dynamical system, combined with a
theorem of Levin regarding symmetries of the Julia set. Using a specialization
theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for
endomorhisms of P^1\times P^1 defined over the complex numbers.
