Constructing finitary isomorphisms with finite expected coding times
type: Item
instance of: scholarly article
title: Constructing finitary isomorphisms with finite expected coding times
zbMATH Open document ID: 1015.37007
DOI: 10.1007/BF02784521
author: Robert Mouat
author: Selim Tuncel
published in: Israel Journal of Mathematics
publication date: 2003-07-10T00:00:00Z
review text: 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.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1874403
zbMATH Keywords: finitary isomorphism
zbMATH Keywords: Markov chains
zbMATH Keywords: coding times
MaRDI profile type: Publication
cites work: Topological entropy and equivalence of dynamical systems
cites work: Optical phase conjugation: principles, techniques, and applications
cites work: When does a submodule of \((\mathbb{R}[x_1,\dots,x_k])^n\) contain a positive element?
cites work: On the finitary isomorphisms of markov shifts that have finite expected coding time
cites work: Bernoulli schemes of the same entropy are finitarily isomorphic
cites work: Finitary isomorphisms of irreducible Markov shifts
cites work: The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains
cites work: Finitary Isomorphisms with Finite Expected Code Lengths
cites work: Notes on Coding Problems for Finite State Processes
cites work: Q4744504
cites work: Natural coefficients and invariants for Markov-shifts
cites work: Q3945330
cites work: On the Stochastic and Topological Structure of Markov Chains
cites work: Invariants for finitary isomorphisms with finite expected code lengths
cites work: Hyperbolic structure preserving isomorphisms of Markov shifts
cites work: Hyperbolic structure preserving isomorphisms of Markov shifts. II
cites work: Conditional pressure and coding
cites work: Q4037022
description: scientific article; zbMATH DE number 1874403