Iterated Monodromy Groups

迭代单调群于2001年被引入，作为与拓扑空间的(部分)自覆盖自然相关的群，例如，由复有理函数在穿孔黎曼球面上的作用而产生的。这些群被用作关于动力系统的组合信息的代数编码。如果覆盖是扩张的，那么所有的基本信息(例如Julia集)都可以从迭代的单元群中恢复出来。该项目致力于研究迭代单向群及其在拓扑学、动力学和群论中的应用。与部分自覆盖相比，迭代单群被自然地定义为更一般的结构。这个广义定义可以用来构造多维动力系统Julia集的单纯逼近。对多元有理函数的Julia集的拓扑结构有一个令人满意的了解的例子很少。该项目将提供一种构造此类Julia集的近似的一般方法，并将导致动力系统的新的同调不变量。该项目的目的是研究代数和几何群论与动力系统及其相关的分形Julia集之间的新联系。迭代单元群在这两个数学分支之间架起了一座桥梁。几何群论研究对称群的大规模性质。动力系统研究映射迭代的混沌动力学，这是《科学》中混沌系统的模型。迭代单向群以一种计算有效的方式对关于动力系统的组合信息进行编码。特别地，他们给出了一种用多面体构造与动力系统相关的复杂分形图的近似的方法。这种方法可以用于多维动态系统的研究，在这些系统中，通常的计算机可视化是不能使用的。迭代正则群也是群的非常有趣和奇异的例子。这样，动力系统可以用来更好地理解群论。这个项目是由拓扑学计划和分析计划共同资助的。

Iterated monodromy groups were introduced in 2001 as groups naturally associated with (partial) self-coverings of topological spaces, for instance arising from the action of complex rational functions on a punctured Riemann sphere. These groups are used as algebraic encoding of combinatorial information about the dynamical system. If the covering is expanding, then all the essential information (for instance the Julia set) can be recovered from the iterated monodromy groups. The project is devoted to the study of iterated monodromy groups and their application to topology, dynamics and group theory. Iterated monodromy groups are naturally defined for more general structures than partial self-covering. This generalized definition can be used to construct simplicial approximations of Julia sets of multi-dimensional dynamical systems. There are very few examples of rational functions of several variables for which there is a satisfactory understanding of the topology of their Julia set. The project will provide a general method of constructing approximations of such Julia sets and will lead to new homological invariants of dynamical systems.The aim of the project is to study new connections between algebra and geometric group theory on one side and dynamical systems and the associated fractal Julia sets on the other. The iterated monodromy groups provide a bridge between these two branches of mathematics. Geometric group theory studies large-scale properties of groups of symmetries. Dynamical systems study chaotic dynamics of iterations of maps, which are models of chaotic systems in Science. Iterated monodromy groups encode in a computationally effective way combinatorial information about the dynamical systems. In particular, they give a method to construct approximations by polyhedra of complicated fractals associated with the dynamical systems. This approach can be used in the study of multi-dimensional dynamical systems, where usual computer visualization can not be applied. Iterated monodromy groups are also very interesting and exotic examples of groups. This way dynamical systems can be used to reach better understanding of group theory.This project is jointly funded by the Topology Program and the Analysis Program.