Let P be a polynomial of degree d with a Cremer point p and norepelling or parabolic periodic bi-accessible points. We show that there aretwo types of such Julia sets JP. The red dwarf JP are nowhere connected imkleinen and such that the intersection of all impressions of external angles is acontinuum containing p and the orbits of all critical images. The solar JP aresuch that every angle with dense orbit has a degenerate impression disjointfrom other impressions and JP is connected im kleinen at its landing point.We study bi-accessible points and locally connected models of JP and showthat such sets JP appear through polynomial-like maps for generic polynomialswith Cremer points. Since known tools break down for d 2 (if d 2, it isnot known if there are small cycles near p, while if d = 2, this result is dueto Yoccoz), we introduce wandering ray continua in JP and provide a newapplication of Thurston laminations. © 2009 American Mathematical Society.