Mathematical Sciences: Topics in Dynamical Systems

9401538波义耳·波义耳将致力于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、扩张的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、非负矩阵的反问题以及一般符号动力学。作为后三类中的具体例子，他希望与合作者和学生一起刻画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，通过解决具有整系数的形式幂级数环中的相关因式分解问题来刻画整数非负矩阵的非零谱--将剩余熵的定义推广到光滑动力系统，并确定在这种情况下它是否是一个非平凡不变量(就像在零维情况下一样)。在鲁道夫提出的研究可测动力学的主题中，包括广义遍历定理的受限轨道等价和特征因子。限制轨道等价的概念旨在通过考虑轨道结构的扰动来研究动力系统的性质。对允许的扰动的种类施加的限制的选择控制观察到的结构的类型。--在研究各种广义遍历定理的收敛时，人们寻找自然分裂为收敛到零的部分，以及具有非平凡收敛但非常有限的、通常是代数结构的部分。平均法的后一种“特征因子”不仅控制着收敛，而且还能洞察底层系统的动态。这是对动力系统和矩阵理论一般领域的纯数学研究。动力学部分主要涉及符号动力学中的理论问题；这个抽象的S主题涉及到在磁性介质中编码数据的实际问题，并且它具有有限的方面，这使得它具有适用性。(抽象的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对另一门学科--非负矩阵理论中一些古老的、基本的和困难的问题的成功的、新颖的应用的继续。可测量动力学的基本哲学是使用对系统随时间的行为的观察来洞察系统的结构。其中可以放松“时间”的概念，将晶体的空间维度或DNA分子的线性维度包括在内。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进行编码的深入而富有成效的洞察。这项工作的一个中心主题是使用这种简单的现实世界模型来加深对动力系统的理解。***

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. ***