9322498 Wagoner This project will investigate the dynamics of shift spaces and flows, their symmetry groups, and connections with algebraic K-theory. There are generally many ways of describing or constructing shift systems which have essentially the same dynamical behaviour. Even in the one-dimensional case, effectively recognizing when two concrete models are equivalent is a fundamental open problem. Another natural and related problem is to understand the different equivalences or symmetries between two systems. One approach to the classification and symmetry group problems for one-dimensional shift systems which has led to progress on some of the main problems in the field in recent years involves analogies from algebraic K-theory. In turn, the geometry of both discrete-time and continuous-time dynamical systems and their symmetry groups has very naturally given rise to a new type of positive algebraic K-theory for ordered rings, using the usual row and column operations on matrices, but with certain inequalities imposed. The main mathematical structures under investigation here, shift dynamical systems and Markov chains, are useful models for phenomena in areas ranging from differential equations, to statistical mechanics, to information and coding theory. However, their usefulness is impaired by the surprising difficulty of deciding when two shift systems which arose in different ways enjoy essentially the same dynamical behaviour. This is related to understanding the different equivalences or symmetries between two systems. (Symmetries of a given system are often called reversible cellular automata.) Now the investigator has shown that a very abstract algebraic theory known as algebraic K-theory can be invaluable in addressing some of these problems. Analogies from algebraic K-theory have led unexpectedly to exciting progress on some of the main problems in symbolic dynamics. Moreover, looking at matters the other way round has led to a new type of K-theory, whose problems are of interest both intrinsically and for the insight they will bring back to dynamical systems and their symmetries. ***
