Symbolic Dynamics and Related Topics

波义耳符号动力学是动力学中的一个工具，也是一个发展模型和实例的竞技场。 该主题与代数，编码理论和矩阵理论有着重要的联系，并且它在概率和遍历理论中有着很强的根源，因为基本对象之一（拓扑马尔可夫链或有限型移位）是马尔可夫链的拓扑支持，也是热力学形式主义应用的主要工具。在这一主题的核心仍然存在一些基本的未决问题，也存在一些新的领域。该项目涉及这两种类型的问题，要解决的代数，分析和遍历的理论工具，这是所要求的性质的主题。 在核心的主题是开放的问题分类转移的有限类型;的想法涉及拒绝威廉姆斯的长期转移等价猜想离开这个问题也许不那么棘手比以前。与此相关的是一组各种符号动力系统的分类问题。 这项建议的核心是在“正K理论”中的工作，这是一个解决这些问题和重建学科基础的环境。分类被改写为等价的矩阵在各种环下乘法链的初等矩阵的正性条件。 符号动力学是一种动力学工具，通过符号扩展给定的动力系统。现在有一个一般的熵理论的符号扩展，和拟议的研究将提供一个更深入的了解可能的符号扩展的系统。该提案还旨在取得进展的某些问题，涉及Zd符号行动;分类有限状态马尔可夫链良好有限同构;分类积极经常可数状态马尔可夫链熵可忽略共轭;和分类的toral自同态可测同构。 拟议的研究是在教育的背景下进行的，主要是研究生，但也有本科生。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，每学年每周举行一次或两次会议。这个群体包括本科生和研究生，以及其他如访问教师或博士后。

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.