Mathematical Sciences: Topics in Dynamical Systems

9401538 Boyle将研究涉及有限类型移位的自同构、马尔可夫过程之间的良好映射、扩展Z^n作用、动力系统的有限表示、拓扑轨道等价和有序上同调、非负矩阵的逆问题以及一般符号动力学的问题。作为后三类的具体例子，他希望与合作者和学生-表征作为零维动力系统的同胚的有序第一切赫上同调群出现的有序群-通过解决具有积分系数的形式幂级数环中的相关因数分解问题来表征积分非负矩阵的非零谱-将剩余熵的定义扩展到光滑动力系统并确定在这种情况下它是否为a非平凡不变量（就像它在零维情况下一样）。Rudolph提出的研究可测动力学的课题包括：受限轨道等价和广义遍历定理的特征因子。限制轨道等价的概念是一种通过考虑轨道结构的微扰来研究动力系统性质的方法。对所允许的扰动类型的限制选择控制着所观察到的结构类型。在研究各种广义遍历定理的收敛性时，人们寻找自然分裂成收敛于零的部分，以及具有非平凡收敛但非常有限的部分，通常是代数结构。后一种平均方法的“特征因子”不仅控制了收敛性，而且还提供了对底层系统动态的洞察。这是在动力系统和矩阵理论的一般领域的纯数学研究。动力学部分主要涉及符号动力学的理论问题；这个抽象的课题涉及到磁性介质中数据编码的实际问题，它有一个有限的方面，这使它本身具有适用性。（抽象分类方案产生了一种算法，该算法现在是IBM产品的一部分。）矩阵研究是符号动力学技术和观点在另一学科非负矩阵理论中一些古老的、基本的和困难的问题上的成功的、新颖的应用的延续。可测量动力学的基本原理是使用对系统随时间的行为的观察来深入了解系统的结构。在这种情况下，人们可以放松“时间”的概念，以包括晶体的空间维度或DNA分子的线性维度。这种简单的视角变化可以导致对系统结构如何在其时间行为中编码的深刻而富有成效的见解。这项工作的中心主题是使用这种简单的现实世界模型来更深入地了解动力系统。＊＊＊

9401538 Boyle Boyle will work on problems involving automorphisms of a shift of finite type, good maps betwen Markov processes, expansive Z^n actions, finite presentations of dynamical systems, topological orbit equivalence and ordered cohomology, inverse problems for nonnegative matrices, and general symbolic dynamics. As specific examples in the last three categories, he hopes with collaborators and students to --characterize the ordered groups which arise as the ordered first Cech cohomology group of a homeomorphism of a zero dimensional dynamical system --characterize the nonzero spectra of integral nonnonegative matrices by solving an associated factorization problem in the ring of formal power series with integral coefficients --extend the definition of residual entropy to smooth dynamical systems and determine whether in that case it is a nontrivial invariant (as it can be in the zero dimensional case). Among the topics proposed by Rudolph for study in measurable dynamics are restricted orbit equivalence and characteristic factors for generalized ergodic theorems. The notion of restricted orbit equivalence is intended as a method to investigate the nature of a dynamical system by considering the perturbations of its orbit structure. The choice of restriction one places on the kind of perturbation allowed controls the type of structure observed. -- In investigating the convergence of various kinds of generalized ergodic theorems, one searches for natural splittings into parts converging to zero, and parts with nontrivial convergence, but very limiting, usually algebraic structure. This latter "characteristic factor" of the averaging method not only controls the convergence, but also gives insight into the dynamics of the underlying system. This is pure mathematical research in the general areas of dynamical systems and matrix theory. The dynamical parts largely involve theoretical problems in symbolic dynamics; this abstract s ubject is involved in practical problems of encoding data in magnetic media, and it has a finite aspect which has lent itself to applicability. (Abstract classficiation schemes gave rise to an algorithm which is now part of an IBM product.) The matrix research is a continuation of a successful, novel application of symbolic dynamics techniques and viewpoint to some old, basic and difficult problems in another subject, the theory of nonnegative matrices. The underlying philosophy of measurable dynamics is to use observations of the behavior of a system over time to gain insight into the structure of the system. In this one can loosen the notion of "time" to include the spatial dimensions of a crystal or the linear dimension of a DNA molecule. Such simple changes of perspective can lead to deep and fruitful insights into how the structure of the system is encoded in its temporal behavior. A central theme of the work is to use such simple real-world models to obtain deeper understanding of dynamical systems. ***