Cubic rational maps with escaping critical points, Part Ⅱ: Classification of Julia sets in the case of a parabolic fixed point

Julia sets of bi-critical rational maps are either connected or a Cantor set. This Julia set dichotomy has been extended to any cubic rational map with all critical points escaping to an attracting fixed point. In this paper, we classified the Julia set of an arbitrary cubic rational map with all critical points escaping to a parabolic fixed point. We proved a Julia set trichotomy (connected, Cantor set, semi-Cantor set) for such cubic rational maps when the multiplier at the parabolic fixed point is 1.