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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-1-4,
author = {Nicolae Mihalache},
title = {Julia and John revisited},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {67-86},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-1-4}
}
Nicolae Mihalache. Julia and John revisited. Fundamenta Mathematicae, Tome 215 (2011) pp. 67-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm215-1-4/