The Dynamics of Semigroups of Rational Functions I
The Dynamics of Semigroups of Rational Functions I
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有理函数半群的动力学 I
DOI：
10.1112/plms/s3-73.2.358
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发表时间：
1996
影响因子：
1.8
通讯作者：
G. Martin
中科院分区：
文献类型：
--
作者：
A. Hinkkanen;G. Martin
本文关注的是推广的经典理论的动力学迭代的有理映射的黎曼球，更一般的设置的动力学相关的任意半群的有理映射。我们的部分动机的结果Gehring和马丁表明，某些参数空间的KJeinian群基本上是稳定的盆无穷的某些多项式半群。本文讨论了Fatou集和Julia集的结构及其基本性质。我们调查在何种程度上沙利文的'无游荡域'定理仍然有效。我们得到了一个完整的概括的经典结果分类盆地及其相关的动力学下的代数假设类似的群论概念的“几乎阿贝尔”。证明了多项式半群一般可以有游荡域。我们提出了一些关于我们认为可能是真实的假设。我们还证明了一个定理的存在性填充Julia集某些多项式半群的具体应用的理论的Kleinian群铭记。
This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for KJeinian groups are essentially the stable basins of infinity for certain polynomial semigroups. Here we discuss the structure of the Fatou and Julia sets and their basic properties. We investigate to what extent Sullivan's 'no wandering domains' theorem remains valid. We obtain a complete generalisation of the classical results concerning classification of basins and their associated dynamics under an algebraic hypothesis analogous to the group-theoretical notion of 'virtually abelian'. We show that, in general, polynomial semigroups can have wandering domains. We put forward some conjectures regarding what we believe might be true. We also prove a theorem about the existence of filled in Julia sets for certain polynomial semigroups with specific applications to the theory of Kleinian groups in mind.