Celebratio Mathematica — Williams

An Autobiography

Works connected to William Parry

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W. Parry and R. F. Williams: “Block coding and a zeta function for finite Markov chains,” Proc. London Math. Soc. (3) 35 : 3 (1977), pp. 483–495. MR 466490 Zbl 0383.94011 article

For an irreducible Markov chain, movement from one state to another is specified stochastically by transition probabilities depending only on the present. In this way possible (positive probability) finite sequential movements can be specified. The topological or qualitative version of this type of behaviour motivates the concept of a topological Markov chain (or shift of finite type or intrinsic Markov chain [3]).

The basic classification problem of ergodic theory is to decide when two ergodic measure-preserving transformations of probability spaces are isomorphic. This problem specialises to a coding problem for stationary Markov chains. In this paper we are concerned with block codes or block isomorphisms between two such chains. In other words, codes shall have bounded memory and bounded anticipation. Without going into the (simple) details. a block isomorphism is a homeomorphism of the essential spaces on which the Markov chains aro supported, which preserves measure. In fact we shall take this as a definition. We see then that part of the problem of block code classification is the topological classification problem which was investigated in [4].