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ON FATOU-JULIA DECOMPOSITIONS ABSTRACT. We propose a Fatou-Julia decomposition for holomorphic pseudo-

ABSTRACT. We propose a Fatou-Julia decomposition for holomorphic pseudo-
semigroups. 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 be-
haves as expected.
Iterations of rational mappings and actions of finitely generated Kleinian groups
are typical dynamical systems on CP1. The notion of the Julia sets [15], [16] and
the limit sets [14] are significant in their study. Sullivan's dictionary [18] says that
they are in a close correspondence (see also [12, pp. 9899]). More generally, the
Julia sets are defined also for actions of semigroups generated by rational maps on
CP1 (cf. [9], [19]). These complex dynamical systems are one-dimensional and
on closed manifolds. Transversely holomorphic foliations of complex codimension
one yield dynamical systems of a similar kind. Indeed, the holonomy pseudogroups
of such foliations act on one-dimensional complex manifolds. If foliations are given
on closed manifolds, then the holonomy pseudogroups have certain compactness


Source: Asuke, Taro - Graduate School of Mathematical Sciences, University of Tokyo


Collections: Mathematics