Julia and John revisited
We show that the Fatou components of a semi-hyperbolic rational map are John domains. The converse does not hold. This compares to a famous result of Carleson, Jones and Yoccoz for polynomials, in which case the two conditions are equivalent.
We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet–Eckmann maps, topological Collet–Eckmann maps and maps satisfying a summability condition (as considered by Graczyk and Smirnov).