Julia and John revisited

Nicolae Mihalache (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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).

Domaines: Systèmes dynamiques [math.DS]

Nicolae Mihalache. Julia and John revisited. Fundamenta Mathematicae, 2011, 215 (1), pp.67-86. ⟨10.4064/fm215-1-4⟩. ⟨hal-00796882⟩