 
Summary: ON FATOUJULIA DECOMPOSITIONS OF
PSEUDOSEMIGROUPS
TARO ASUKE
Abstract. According to Sullivan's dictionary, the Julia sets for iterations
of rational mappings and the limit sets of Kleinian groups are in a close
relationship. In this article, we will give a rough idea which relates these
two notions more concretely. This is an announcement of [2], and based on
a talk given at `2009 Complex Dynamics conference Integrated Research
on Complex Dynamics and its Related Fields ' held at Kyoto University.
Introduction
According to Sullivan's dictionary, the Julia sets for iterations of rational map
pings and the limit sets of Kleinian groups are in a close relationship [13]. Re
cently, the notion of Julia sets is also introduced for complex codimensionone
transversely holomorphic foliations of closed manifolds [4], [1], and it is shown
that they have some common properties to the Julia sets and the limit sets
as above. It is quite natural to expect there is a concept which unifies these
notions. In order to find such a concept, we will need to deal with semigroups,
groups and pseudogroups. Hence one way will be to consider pseudosemigroups
and define their Julia sets (another approach can be found in [3]). We propose
in [2] a definition of such Julia sets. In this article, we will sketch a rough
