On Fatou and Julia sets of foliations

Abstract

The Fatou–Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition has been considered for foliations in a unified manner by Ghys–Gomez-Mont–Saludes, Haefliger, the author, et al. Although the decomposition will be fundamental in the study, it is not easy to determine the decomposition. In this article, we give a sufficient condition for open sets to be contained in Fatou sets. We also discuss relations between Fatou–Julia decompositions and minimal sets.