Further Topics

Abstract

In this chapter we introduce some further topics which are closely tied to subshifts of finite type. In the first section we look at sofic systems. The continuous symbolic image of a subshift of finite type need not be a subshift of finite type. It may have an unbounded memory. Sofic systems are the symbolic systems that arise as continuous images of subshifts of finite type. There are three equivalent characterizations of these systems. The characterizations are explained and then we investigate some of the basic dynamical properties of sofic systems. The second section contains a discussion of Markov measures. These are the measures which have a finite memory. We define the measures, compute their measure-theoretic entropy, characterize the measures using conditional entropy and then prove that for a fixed subshift of finite type there is a unique Markov measure whose entropy is greater than the entropy of any other measure on the subshift of finite type. The third section investigates symbolic systems that have a group structure. These are the Markov subgroups. We show that any symbolic system with a group structure is a subshift of finite type. Then we classify them up to topological conjugacy. The fourth section contains a very brief introduction to cellular automata. The point is to see how they fit into the framework we have developed. The final section discusses channel codes as illustrated in Example 1.2.8. We describe a class of codes and develope an algorithm to construct them.KeywordsCellular AutomatonFinite TypeTopological EntropyMaximal MeasureBorel Probability MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.