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Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts

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Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts

Brian Marcus and Selim Tuncel

Abstract.
We give necessary and sufficient conditions for a Markov chain to factor onto a Bernoulli shift (i) as an eventual right-closing factor, (ii) by a right-closing factor map, (iii) by a one-to-one a.e. right-closing factor map, and (iv) by a regular isomorphism. We pass to the setting of polynomials in several variables to represent the Bernoulli shift by a nonnegative polynomial \$p\$ in several variables and the Markov chain by a matrix \$A\$ of such polynomials. The necessary and sufficient conditions for each of (i)--(iv) involve only an eigenvector \$r\$ of \$A\$ and basic invariants obtained from weights of periodic orbits. The characterizations of (ii)--(iv) are deduced from (i). We formulate (i) as a combinatorial problem, reducing it to certain state-splittings (partitions) of paths of length \$n\$. In terms of positive polynomial masses associated with paths, the aim then becomes the construction of partitions so that the masses of the paths in each partition element sum to a multiple of \$p^n\$, the multiple being prescribed by \$r\$. The construction, which we sketch, relies on a description of the terms of \$p^n\$ and on estimates of the relative sizes of the coefficients of \$p^n\$.

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 91-101
• Publisher Identifier: S 1079-6762(99)00066-9
• 1991 Mathematics Subject Classification. Primary 28D20; Secondary 11C08, 05A10
• Key words and phrases.
• Received by the editors January 21, 1999
• Posted on June 30, 1999
• Communicated by Klaus Schmidt

Brian Marcus