 
Summary: ON FATOUJULIA DECOMPOSITIONS
TARO ASUKE
ABSTRACT. We propose a FatouJulia 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 FatouJulia decomposition for
singular holomorphic foliations. In the wellstudied cases, the decomposition be
haves as expected.
INTRODUCTION
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 onedimensional 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 onedimensional complex manifolds. If foliations are given
on closed manifolds, then the holonomy pseudogroups have certain compactness
