The solar Julia sets of basic quadratic Cremer polynomials
  In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression. 
A. Blokh, X. Buff, A.Ch\'eritat, L. Oversteegen