Symbolic 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.
