  We describe the topological behavior of typical orbits of complex quadratic
polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type.
Here we prove that for such Brjuno values of alpha the closure of the critical
orbit, which is the measure theoretic attractor of the map, has zero area. Then
combining with Part I of this work, we show that the limit set of the orbit of
a typical point in the Julia set is equal to the closure of the critical orbit.
