A family of rational maps with buried Julia components

Sébastien Godillon (1)

1 Departament de Matemàtica Aplicada i Anàlisi [Barcelona]

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least 5. This paper introduce a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of critically separating Julia components is encoded by a weighted Hubbard tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover this family contains some rational maps of degree 3 with explicit formula that answers a question McMullen raised.

Mots clés: rational maps; Julia sets; exchanging dynamics of Julia components; topology of Julia components

Domaines: Systèmes dynamiques [math.DS]

Sébastien Godillon. A family of rational maps with buried Julia components. 2013. ⟨hal-00812775⟩