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

DOI:
https://doi.org/10.1090/S1079-6762-99-00066-9

Published electronically:
June 30, 1999

MathSciNet review:
1696825

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Abstract | References | Similar Articles | Additional Information

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 in several variables and the Markov chain by a matrix of such polynomials. The necessary and sufficient conditions for each of (i)-(iv) involve only an eigenvector of 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 . 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 , the multiple being prescribed by . The construction, which we sketch, relies on a description of the terms of and on estimates of the relative sizes of the coefficients of .

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Additional Information

**Brian Marcus**

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

Email:
marcus@almaden.ibm.com

**Selim Tuncel**

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

Email:
tuncel@math.washington.edu

DOI:
https://doi.org/10.1090/S1079-6762-99-00066-9

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