On Fatou-Julia decompositions

Asuke, Taro

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 22 (2013) no. 1, pp. 155-195.

Nous proposons une décomposition de Fatou-Julia pour les pseudosemigroupes holomorphes. On montre que les ensembles limites des groupes kleiniens de type fini, les ensembles de Julia de fonctions holomorphes et ceux des feuilletages réguliers transversalement holomorphes de codimension un sont des cas particuliers de cette décomposition. La décomposition est utilisée pour introduire une décomposition de Fatou-Julia pour les feuilletages holomorphes singuliers. Dans les cas étudiés, le comportement de la décomposition est comme attendu.

We propose a Fatou-Julia decomposition for holomorphic pseudosemigroups. It will be shown that the limit sets of finitely generated Kleinian groups, the Julia sets of mapping iterations and Julia sets of complex codimension-one regular foliations can be seen as particular cases of the decomposition. The decomposition is applied in order to introduce a Fatou-Julia decomposition for singular holomorphic foliations. In the well-studied cases, the decomposition behaves as expected.

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