Spaces of Rational Maps: Dynamics, combinatorcs and topology

复杂有理映射的动力学既非常丰富，也很容易研究。由于这个原因，在过去的100年里，特别是最近，由于计算机实验，它们得到了深入的研究。本文的目的是通过结合复动力学和群论的工具来考察有理映射空间，并展示其丰富的代数和组合结构。我们期望“迭代单元群”的语言是建立有理映射空间中各种动态有效轨迹的精确组合模型的基础。除了这一哲学之外，有理映射空间应该通过“后临界有限映射”来理解，即使得临界点的前向迭代形成有限集的映射。这应该被认为是类似于实数中的有理数，虽然稠密，但易于计算。特别是，我们将在有理映射空间中定义和考虑“切片”：由后临界有限映射的有限集合控制的一维子簇。然后，有理映射的空间结构将通过这些切片之间的排列而显现出来。

The dynamics of complex rational maps is both very rich, and accessible to study. For this reason, they have been thoroughly investigated since the last 100 years, and particularly so recently, thanks to computer experimentation.The purpose of this proposal is to examine the space of rational maps, and exhibit its rich algebraic and combinatorial structure by combining tools from complex dynamics and group theory. We expect the language of “iterated monodromy groups” to be fundamental in building accurate combinatorial models of various dynamically-significant loci in the space of rational maps.An aside of this philosophy is that the space of rational maps should be understood via “post-critically finite maps”, namely maps such that the forward iterates of critical points form a finite set. Theseshould be thought of as akin to rational numbers among the reals dense, yet amenable to calculation.In particular, we will define and consider “slices” within the space of rational maps: One-dimensional subvarieties controlled by a finite collection of post-critically finite maps. The structure of the space of rational maps will then appear through the arrangement of these slices among themselves.