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
Full-text 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 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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- M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Mem. Amer. Math. Soc. 377 (1987).
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- S. Tuncel, Faces of Markov chains and matrices of polynomials, Contemp. Math., Vol. 135, Amer. Math. Soc., Providence, 1992, pp. 391–422.
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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
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