Laminations in holomorphic dynamics
Laminations in holomorphic dynamics
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DOI：
10.4310/jdg/1214460037
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发表时间：
1994-12
影响因子：
2.5
通讯作者：
M. Lyubich;Y. Minsky
中科院分区：
文献类型：
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作者：
M. Lyubich;Y. Minsky
我们提出了一种方法来关联到一个合理的映射的黎曼球面的三维物体称为双曲orbifold 3-层压。这个对象与映射的关系类似于双曲三维流形与克莱因群的关系。为了构造3-层，我们分析了有理映射的自然扩张及其所包含的规范二维叶空间上的复仿射结构，并对后临界有限映射进行了全面的构造。我们表明，相应的叠层有一个紧凑的凸芯。作为第一个应用，我们通过对双曲三维流形的Mostow和马尔登刚性及同构定理证明的“叠层推广”，给出了Thurston刚性在后临界有限映射上的三维证明.沿着应用了一个Ahlfors型的Julia集的零测度参数。该方法也为Lattes变形实例提供了一个新的视角。
We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a Kleinian group. In order to construct the 3-lamination we analyze the natural extension of a rational map and the complex affine structure on the canonical 2-dimensional leaf space contained in it. In this paper the construction is carried out in full for post-critically finite maps. We show that the corresponding laminations have a compact convex core. As a first application we give a three-dimensional proof of Thurston's rigidity for post-critically finite mappings, via the "lamination extension" of the proofs of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic 3-manifolds. An Ahlfors-type argument for zero measure of the Julia set is applied along the way. This approach also provides a new point of view on the Lattes deformable examples.