Decomposition of rational maps by stable multicurves

A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are disjoint if and only if they are separated by a coiling curve. Furthermore, we prove that a post-critically finite rational map with a coiling curve is renormalizable. Using a similar argument, we give a sufficient condition for a Fatou domain to qualify as a Jordan domain. By tuning polynomails in such a Fatou domain, we provide examples of post-critically finite rational maps with coiling curves.