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Study of Dynamics of Branched Coverings on the Sphere and Dynamical Zeta Function

球体上分支覆盖物的动力学及动态Zeta函数研究

基本信息

批准号：
15540204
负责人：
KAMEYAMA Atsushi
金额：
$ 1.09万
依托单位：
Gifu University (2004-2005)Osaka University (2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

Among the ends of this research is to classify branched coverings on the 2-dimensional sphere up to "isotopy." In the 1-dimensional case, we have a good invariant, called a kneading sequence, which divides maps on the interval into "isotopy" classes. However, we face the difficulty that a kneading sequence has no standard extension in 2-dimension. Thus we consider all possible geometric semiconjugacy from a symbolic dynamics to the Julia set.We have the following results. Let f be a subhyperbolic rational map. Denote by J the Julia set of f, and by J^* the lift of J by the universal covering. Consider the set Cod(f) of codings of J obtained by geometric coding trees.Then1. If the attractor K of an IFS constructed by lifts of a collection of inverses of f has a positive measure, then K tiles J^*.2. A coding map is an n-to-one except on a null set, where n is an integer.3. The collapsing of a coding map is described by a finite directed graph.4. Cod(f) is isomorphic to the quotient of the set of trees by some action of a subgroup of the fundamental group. Moreover, the monoid of rational maps commuting with f naturally acts on Cod (f).Another direction of our study is to investigate nontrivial symmetries of fractal sets. The figure obtained by gluing two copies of Sierpinski's gasket at their "boundaries" has infinitely many automorphisms, while Sierpinski's gasket itself has the symmetry of the regular triangle. We show when a glued fractal has nontrivial automorphisms and how to construct such a fractal. Furthermore we describe the structure of the automorphism group, and proved that under some assumption,. the group can be realized by Moebius.transforms.

本研究的目的之一是将二维球面上的分支覆盖分类到“合痕”。在1维的情况下，我们有一个很好的不变量，称为揉序列，它将区间上的映射划分为“合痕”类。然而，我们面临的困难，揉捏序列没有标准的扩展在二维。因此，我们考虑了从符号动力系统到Julia集的所有可能的几何双偶性，得到了如下结果。设f是一个次双曲有理映射。用J表示f的Julia集，用J^* 表示J通过泛覆盖的提升。考虑由几何编码树得到的J的编码的集合Cod（f）。如果由f的逆集合的提升构造的IFS的吸引子K具有正测度，则K个瓦片J^*。一个编码映射是一个n对一的映射，除了在一个空集合上，其中n是一个整数。编码映射的折叠由有限有向图描述。Cod（f）同构于由基本群的子群的某个作用所构成的树的集合的商。此外，与f交换的有理映射的幺半群自然地作用在Cod（f）上。我们研究的另一个方向是研究分形集的非平凡对称性。将两个谢尔宾斯基垫片的副本粘在它们的“边界”上得到的图形有无限多个自同构，而谢尔宾斯基垫片本身具有正三角形的对称性。我们展示了当一个胶合分形有非平凡的自同构，以及如何构建这样的分形。进一步描述了自同构群的结构，并证明了在一定的假设下，该群可以通过Moebius变换来实现。