The Structure of Expanding Rational Maps as Holomorphic Dynamical Systems

我们将研究黎曼球面上作为全纯动力系统的扩张或双曲有理映射的结构。这种方法是由麦克马伦和沙利文发展的有理映射理论和克莱因群之间的类比而产生的。对于一个扩张的、(等价的，几何上有限的，没有尖点的)Kleinan群，我们可以把一个拓扑对象--通过对双曲空间进行商化而得到的商紧致三维流形--与无穷远处Riemann球面上的不连续区域在群的作用下联系起来。对于扩张的有理映射，人们还可以将一个拓扑对象--被视为球面到其自身的分支覆盖的映射--联系在一起，直到同伦论形式的等价。这些分支覆盖等价于扩张有理映射的特征是已知的。然而，与Kleian群的情况相反，关于这些分支覆盖的组合性质如何反映在相应有理映射的几何和动力学以及它们的参数空间中，一般知之甚少。受不可压缩曲面理论在研究双曲三维流形中的作用的启发，我们将研究分支覆盖的组合学。这一分析将被用来发展有理映射的一般组合和分解理论，以及有理映射参数空间中双曲分支的邻接和紧性的一般刻画。动力系统是目前在许多学科中广泛使用的一种数学模型，用于描述生物、物理和社会现象。本主题的主题是，即使是一个非常简单的模型也可以产生极其复杂的行为。因此，人们通常首先简化模型，以使问题易于处理。接下来，人们需要工具来解决这些问题。最后，人们寻求将问题归结为一个组合问题，即原则上计算机可以解决的问题。这个项目涉及将一维全纯函数(这是应用了强大的复分析工具的最简单的动力系统)的动力学归结为一个组合问题。这个问题，如果表述正确，看起来与另一个领域的问题非常相似，后者的解决方案已经知道。这个项目的主要目标是利用这些已知的结果，通过类比推理，形成和证明全纯动力学的新结果。

The structure of expanding, or hyperbolic, rational maps as holomorphic dynamical systems on the Riemann sphere will be investigated. The approach is motivated by the analogy between the theories of rational maps and Kleinian groups developed by McMullen and Sullivan. To an expanding, (equivalently, geometrically finite without cusps) torsion-free Kleinian group one may associate a topological object-the quotient compact three-manifold obtained by quotienting hyperbolic space together with the domain of discontinuity on the Riemann sphere at infinity by the action of the group. To an expanding rational map one may also associate a topological object-the map viewed as a branched covering of the sphere to itself, up to a homotopy-theoretic form of equivalence. A characterization of those branched coverings which are equivalent to an expanding rational map is known. However, in contrast to the case of Kleinian groups, little is known in general about how combinatorial properties of these branched coverings are reflected in the geometry and dynamics of the corresponding rational maps and in their parameter spaces. Motivated by the role played by the theory of incompressible surfaces in the study of hyperbolic 3-manifolds, the combinatorics of branched coverings will studied. This analysis will be used to develop general combination and decomposition theories for ratimnal maps and general characterizations of adjacencies and compactness properties of hyperbolic components in parameter spaces of rational maps. A dynamical system is a mathematical model now widely used in many disciplines to describe biological, physical, and social phenomena. The main theme in the subject is that even a very simple model can yield extremely complicated behavior. Therefore one often simplifies the model first to make the problem tractable. Next, one needs tools to attack these problems. Finally, one looks for a reduction of the problem to a combinatorial one, i.e., one which, in principle, a computer can sol ve. This project concerns reducing the dynamics of one-dimensional holomorphic functions (which are the simplest dynamical systems to which the powerful tools of complex analysis apply) to a combinatorial problem. This problem, when formulated correctly, looks remarkably similar to one in a different field whose solution is already known. The main goal of this project is to take these known results and, reasoning by analogy, formulate and prove new results in holomorphic dynamics.