Symbolic Dynamics and Related Topics

符号动力学是一门在动力学中充当工具的学科，也是开发模型和例子的舞台。这门学科与代数、编码理论和矩阵理论有着密切的联系，它在概率论和遍历理论中有很强的根基，因为它的基本对象之一(拓扑马尔可夫链或有限类型移位)是马尔可夫链的拓扑支撑和热力学形式应用的主要工具。在这一主题的核心以及新的领域中，仍然存在基本的未决问题。该项目涉及两种类型的问题，将根据学科的性质要求使用代数、分析和遍历理论工具来解决。这门学科的核心是有限类型移位的分类这个公开问题；拒绝威廉姆斯长期存在的移位等价猜想所涉及的思想使这个问题可能不像以前那么棘手。与此相关的是各种符号动力系统的一组分类问题。这项提议的核心是在“正K理论”内的工作，这是一个既能解决这些问题又能重建学科基础的环境。分类被重塑为不同环上矩阵乘以满足正性条件的初等矩阵链的等价性。符号动力学是对给定的动力系统进行符号扩张的一种动力学工具。现在有了符号扩展的一般熵理论，所提出的研究将提供对系统可能的符号扩展的更深层次的理解。该建议还旨在解决涉及ZD符号作用的某些问题；有限状态马氏链的分类直到良好的有限同构；正递归可数状态马氏链的分类直到熵可忽略的共轭；以及正递归可数状态马氏链的分类直到可测同构。拟议的研究将在教育的背景下进行，主要针对研究生，但也针对本科生。这方面的一个工具是P.I.的“研究互动小组”，自2002年9月以来，他们每学年都会召开一到两次会议。这一群体包括本科生和研究生，以及其他可用的客座教师或博士后。

ABSTRACTBoyleSymbolic dynamics is a subject that serves as a tool within dynamics and as an arena for the development of models and examples. The subject has significant connections with algebra, coding theory and matrix theory and it has strong roots in probability and ergodic theory, as one of the basic objects (topological Markov chain or shift of finite type) is the topological support of a Markov chain and a major tool for application of the thermodynamic formalism. There remain basic open problems at the heart of the subject as well as new frontiers. The project is concerned with problems of both sorts, to be addressed with algebraic, analytic and ergodic theoretic tools, as is demanded by the nature of the subject. At the core of the subject is the open problem of classifying shifts of finite type; the ideas involved in the rejection of Williams' longstanding shift equivalence conjecture leave this problem perhaps less intractable than before. Related to this is a set of classification problems for various symbolic dynamical systems. At the heart of this proposal is work within "Positive K-theory", a setting for both solving these problems and restructuring the foundations of the subject. Classification is recast as equivalence of matrices over various rings under multiplication by chains of elementary matrices subject to positivity conditions. Symbolic dynamics is a dynamical tool by way of symbolic extensions of given dynamical systems. There is now a general entropy theory of symbolic extensions, and the proposed research would provide a still deeper understanding of the possible symbolic extensions of a system. The proposal also aims to progress on certain problems involving Zd symbolic actions; the classification of finite state Markov chains up to good finitary isomorphism; the classification of positively recurrent countable state Markov chains up to entropy-negligible conjugacy; and the classification of toral endomorphisms up to measurable isomorphism. The proposed research is to be done in the context of education, primarily of graduate students but also undergraduates. A vehicle for this has been the "research interaction team" of the P.I., meeting in the academic year once or twice weekly since September 2002. This group includes undergraduate and graduate students, and others such as visiting faculty or postdocs as available.