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), 91101
MSC (1991):
Primary 28D20; Secondary 11C08, 05A10
Published electronically:
June 30, 1999
MathSciNet review:
1696825
Fulltext PDF Free Access
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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 rightclosing factor, (ii) by a rightclosing factor map, (iii) by a onetoone a.e. rightclosing 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 statesplittings (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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 M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Mem. Amer. Math. Soc. 377 (1987). MR 89c:28019
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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:
http://dx.doi.org/10.1090/S1079676299000669
PII:
S 10796762(99)000669
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
