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

Authors: Brian Marcus and Selim Tuncel
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 91-101
MSC (1991): Primary 28D20; Secondary 11C08, 05A10
Published electronically: June 30, 1999
MathSciNet review: 1696825
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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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Additional Information

Brian Marcus
Affiliation: IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120

Selim Tuncel
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195

Received by editor(s): January 21, 1999
Published electronically: June 30, 1999
Additional Notes: Partially supported by NSF Grant DMS–9622866
Communicated by: Klaus Schmidt
Article copyright: © Copyright 1999 American Mathematical Society