A Fatou-Julia decomposition of transversally holomorphic foliations

Asuke, Taro

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 1057-1104.

Une décomposition de Fatou-Julia d’un feuilletage transversalement holomorphe de codimension complexe un a été obtenue par Ghys, Gomez-Mont et Saludes. Dans cet article, nous proposons une autre décomposition en utilisant des familles normales. Ces deux décompositions ont des propriétés communes, ainsi que certaines différences. Il est montré que l’ensemble de Fatou pour notre décomposition contient toujours celui pour la décomposition de Ghys, Gomez-Mont et Saludes, et aussi que l’inclusion est stricte pour certains exemples. Cette propriété est importante pour une version du théorème de Duminy en relation avec les classes caractéristiques secondaires. Quelques similitudes et différences entre les ensembles de Julia de feuilletages et ceux d’itérations d’applications sont présentées. Une application aux études de la métrique transversale de Kobayashi est aussi donnée.

A Fatou-Julia decomposition of transversally holomorphic foliations of complex codimension one was given by Ghys, Gomez-Mont and Saludes. In this paper, we propose another decomposition in terms of normal families. Two decompositions have common properties as well as certain differences. It will be shown that the Fatou sets in our sense always contain the Fatou sets in the sense of Ghys, Gomez-Mont and Saludes and the inclusion is strict in some examples. This property is important when discussing a version of Duminy’s theorem in relation to secondary characteristic classes. The structure of Fatou sets is studied in detail, and some properties of Julia sets are discussed. Some similarities and differences between the Julia sets of foliations and those of mapping iterations will be shown. An application to the study of the transversal Kobayashi metrics is also given.

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