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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 $p^n$.

*Copyright 1999 American Mathematical Society*

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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
- Comments (When Available)

**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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