  We study dynamical properties of asymptotically extremal polynomials
associated with a non-polar planar compact set E. In particular, we prove that
if the zeros of such polynomials are uniformly bounded then their Brolin
measures converge weakly to the equilibrium measure of E. In addition, if E is
regular and the zeros of such polynomials are sufficiently close to E then we
prove that the filled Julia sets converge to polynomial convex hull of E in the
Klimek topology.
