On the Stochastic and Topological Structure of Markov Chains

The classification theory for shifts of finite type (topological Markov chains) in the strictest sense, that is to say, up to topological conjugacy, was initiated by Williams ([17]). Additions to this theory are represented by [1], [4], [6], [7], [11], [12] and [14]. Williams's work can be viewed either as topological or as a combined topological and measure-theoretic theory, where invariant measures are restricted to those of 'maximal' type. In a subsequent paper ([14]), Williams and the first-named author above developed a partial generalisation; a classification of all finite state stationary Markov chains up to block-isomorphism (i.e. topological and measure-theoretic equivalence). The complete invariant constructed was two parallel series of strongshift equivalences; one expressing the topological character and the other the stochastic character of the problem. Associated with this extremely complicated invariant are the topological and stochastic zeta functions. A defect in this theory is the unfortunate dislocation of invariants into topological and stochastic aspects. This defect is not merely aesthetic; it prevented the authors of [14] from formulating an appropriate notion of shift equivalence and an allied zeta function. Although it is not known whether shift equivalence implies strong shift equivalence (the converse is true), shift equivalence is a relationship which is much more manageable and in the topological case its decidability, for certain cases, has been proved ([5]). In this paper we introduce the notion of shift equivalence for stochastic matrices and a corresponding zeta function which subsumes both the topological and stochastic zeta functions. (Our construction could be deduced from Ruelle's work [15]. We should also like to point out that a more complicated version of the same zeta function was discovered by Roger Butler. A somewhat different zeta function was introduced by Martin in [8] for general ergodic theory. Readers may wish to compare Martin's zeta function with ours and to note the comments at the end of [8].) The basic new concepts used in this paper are adaptations of the so-called /̂ -function of [16], the one-parameterisation of Markov measures and the idea of restricting matrices to those whose entries are non-negative integral combinations of exponential functions (cf. [13]). With these notions and restrictions we are able to extend the results of [17], [14] and [11] and to establish the chain of implications shown on the next page. Since the techniques we require have already been elucidated elsewhere, we hope that our brief indications will suffice to convince the reader of the validity of our results. To avoid what seems to us unnecessary repetition we provide references to the results whose generalisations appear here and specify the additional ingredients required to provide complete proofs. We take this opportunity to record our gratitude to Roger Butler and Wolfgang Krieger for useful discussions related to the contents of this paper.