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
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Brian Marcus

IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120

E-mail address: marcus@almaden.ibm.com

Selim Tuncel

Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195

E-mail address: tuncel@math.washington.edu

Partially supported by NSF Grant DMS-9622866

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