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ON FATOUJULIA DECOMPOSITIONS OF PSEUDOSEMIGROUPS II
 

Summary: ON FATOU­JULIA DECOMPOSITIONS OF
PSEUDOSEMIGROUPS II
TARO ASUKE
Dedicated to the 60th birthday of professor Ushiki
Abstract. According to Sullivan's dictionary [9], the Julia sets for iter-
ations of rational mappings and the limit sets of Kleinian groups are in
a close relationship. In [2], we proposed a framework which unifies these
dynamical systems and explains the dictionary. More concretely, notions of
pseudosemigroups and their Fatou-Julia decompositions are introduced. In
this article, we will introduce a result that the action of a pseudosemigroup
on the Fatou set is non-expanding with respect to a Hermitian metric or a
volume form on the Fatou set, and give a rough sketch of the proof. This
article is an announcement of [2] and is a sequel to [3]. This is also based on
a talk given at `2010 Complex Dynamics conference ­ Integrated Research
on Complex Dynamics and its Related Fields ­' held at Kyoto University.
Introduction
According to Sullivan's dictionary [9], the Julia sets for iterations of rational
mappings and the limit sets of Kleinian groups are in a close relationship. The
Julia sets can be also defined for transversely holomorphic foliations of complex
codimension one [4], [6], [1]. Properties of the Julia sets of foliations are not yet

  

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

 

Collections: Mathematics