Constructing finitary isomorphisms with finite expected coding times
Irreducible Markov shifts with the same entropy were shown to be finitarily isomorphic by \textit{M. Keane} and \textit{M. Smorodinsky} [Isr. J. Math. 34, 281-286 (1979; Zbl 0431.28015)]. In light of their result, it is natural to ask for additional computable invariants that guarantee the more natural property of having an isomorphism with finite expected coding length (a measure of how far the code must look into the past and future). Early work showed that additional invariants are certainly needed, and eventually the natural hope seemed to be that the invariants \(\beta,\Delta,c\Delta\) formed a complete set of invariants for finitary isomorphism with finite expected coding length. This central hope remains an open question -- an attractive survey is the paper of \textit{W. Parry} [Bull. Lond. Math. Soc. 23, 1-33 (1991; Zbl 0808.28013)]. Here the authors construct a finitary isomorphism with finite expected coding length under an additional positivity assumption. In particular, they show that Markov chains are finitarily isomorphic with finite expected coding time if there is a matrix (over a suitable ring) intertwining the matrices defining the Markov chains that has a positive row and a positive column.
scientific article; zbMATH DE number 1874403
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