  We show that Fatou components of a semi-hyperbolic rational map are John
domains and that the converse does not hold. This generalizes a famous result
of Carleson, Jones and Yoccoz.
  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 and Topological Collet-Eckmann
maps and maps verifying a summability condition (as considered by Graczyk and
Smirnov).
