Julia and John revisited

Tom 215 / 2011

Nicolae Mihalache Fundamenta Mathematicae 215 (2011), 67-86 MSC: Primary 37F10. DOI: 10.4064/fm215-1-4


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).


  • Nicolae MihalacheUniversité Paris Est – Créteil
    61 avenue du Général de Gaulle
    94010 Créteil Cedex France

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