On Fatou and Julia sets of foliations

TL;DR

This paper explores the Fatou-Julia decomposition for foliations, providing conditions for identifying Fatou sets and examining their relation to minimal sets, advancing understanding in complex dynamics.

Contribution

It introduces a sufficient condition for open sets to be part of Fatou sets and discusses their relation to minimal sets in the context of foliations.

Findings

Provided a sufficient condition for open sets to be in Fatou sets

Analyzed the relationship between Fatou--Julia decompositions and minimal sets

Enhanced understanding of the dynamics of foliations

Abstract

The Fatou-Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition can be also considered for foliations in a unified manner. Although the decomposition will be fundamental in the study, it is not easy to decide 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.