Research in Kleinian Groups and Conformal Dynamics

抽象的敏锐 本文主要研究三个不同的领域：第一，特殊亚纯函数族中的双曲密度;第二，Kleinian群的拟共形变形空间的模的定义;第三，单参数亚纯函数族中的离散性。 2-生成克莱因群 共形动力系统中的一个突出问题是证明对于给定的族，双曲系统是稠密的。 我们 本文将我们的结果推广到其他有限维亚纯函数族{f（z，c）= ctan（z）}中，特别是推广到没有临界点且有多个渐近值的亚纯函数族. ii -我们将把褶坐标的定义从Kleinian群的一维和二维空间推广到Kleinian群的任意空间。 褶坐标的优点是，它们给出了由群确定的双曲3-流形的几何与群的复解析迹模之间的直接关系。 iii -我们考虑PSL（2，C）的两个生成子群的族，其中一个是固定的抛物生成元，另一个是依赖于单个复参数c的抛物生成元. 我们建议扩展我们在工作中使用的那种分析褶坐标找到必要和充分的条件C组离散。 建议在三个不同领域进行研究。 第一个涉及扩展以前的工作在该地区的共形动力系统。这些系统是作为各种物理、经济和社会现象的模型出现的。 一个动态系统可以用一个函数族f（z，c）来描述，它依赖于两个变量。对于c的每个固定值，我们 对给定点z的轨道感兴趣;也就是说，我们考虑点z，f（c，z），f（f（c，z））等的集合。如果输入数据z的微小变化仅轻微改变轨道，我们说系统是稳定的，或可预测的。 另一方面，如果z的微小变化引起轨道的剧烈变化，我们说系统是混沌的。 如果轨道满足 在一定的技术条件下，该系统称为双曲系统。 双曲线行为似乎是常态，但这往往很难证明。 我们已经证明，这是一个特殊的功能类，并建议将这项工作扩展到一个更大的类。 第二个研究领域涉及研究一类称为双曲三维流形的三维物体。 它们之所以被称为双曲几何，是因为它们被赋予了一种称为双曲几何的非欧几何。在这样的几何图形中，三角形的角之和小于直角。 双曲3-流形可以改变，使得几何形状改变，但保持双曲。 在以前的工作中，基于对复杂的计算机程序生成的例子的仔细分析，我们发现了一种很好的方法来描述双曲三维流形的一个和两个参数族。 我们建议将这项工作扩展到依赖于任意数量的参数的家庭。 在我们的工作过程中的3流形，我们遇到了一类代数对象称为团体。 这些群体出现在许多不同的数学领域，并已被许多研究人员研究。然而，仍然有一些悬而未决的问题，关于他们和我们的第三个领域的研究，我们建议扩展我们以前的分析，这些群体来回答这些问题。

Abstract Keen The proposed research is in three different areas: first, hyperbolic density in special families of meromorphic functions; second, the definition of moduli for quasi-conformal deformation spaces of Kleinian groups; and third, discreteness in a one parameter family of 2-generator Kleinian groups. i - One of the outstanding problems in conformal dynamical systems is to prove that, for a given family, the hyperbolic systems are dense. We propose to extend our result that the functions in the family {f(z,c) = c tan(z)} are dense to other finite dimensional families of meromorphic functions; in particular, to families with no critical points and finitely many asymptotic values. ii - We shall extend our definition of Pleating Coordinates from one and two dimensional spaces of Kleinian groups to arbitrary spaces of Kleinian groups. The advantage of pleating coordinates is that they give a direct relation between the geometry of the hyperbolic 3-manifold determined by the group and complex analytic trace moduli for the group. iii - We consider a family of two generator subgroups of PSL(2,C) with one fixed parabolic generator and one parabolic generator depending on a single complex parameter c. We propose to extend the kind of analysis we have used in our work on pleating coordinates to find necessary and sufficient conditions on c for the groups to be discrete. Research is proposed in three different areas. The first involves extending previous work in the area of conformal dynamical systems. Such systems arise as models of various physical, economic and social phenomena. A dynamical system may be described by a family of functions, f(z,c), depending on two variables. For each fixed value of c we are interested in the orbits of a given point z; that is, we consider the set of points z, f(c,z), f(f(c,z)), and so on. If a small change in the input data, z, changes the orbit only slightly, we say the system is stable, or predictable. If a small change in z, on the other hand, causes a drastic change in the orbit, we say the system is chaotic. If the orbits satisfy a certain technical condition, the system is called hyperbolic. It would seem that hyperbolic behavior is the norm, but this is often difficult to prove. We have proved that it is for a special class of functions and propose to extend this work to a larger class. The second area of research involves studying a class of 3 dimensional objects called hyperbolic 3-manifolds. They are so-called because they are endowed with a non-Euclidean geometry called a hyperbolic geometry. In such a geometry the sum of the angles in a triangle is less than a straight angle. The hyperbolic 3-manifold may be varied so that the geometry changes, but remains hyperbolic. In previous work, based on careful analysis of examples generated by sophisticated computer programs, we found a good way to describe one and two parameter families of hyperbolic 3-manifolds. We propose to extend this work to families depending on a arbitrary number of parameters. In the course of our work on 3-manifolds we encountered a class of algebraic objects called groups. These groups arise in a number of different areas of mathematics and have been studied by many researchers. There are, however, still a number of open questions concerning them and for our third area of research we propose to extend our previous analysis of these groups to answer some of these questions.