# Resolving Markov Chains onto Bernoulli Shifts via Positive Polynomials

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*Brian Marcus; Selim Tuncel*

The two parts of this Memoir contain two separate but related papers.
The longer paper in Part A obtains necessary and sufficient conditions for
several types of codings of Markov chains onto Bernoulli shifts. It proceeds
by replacing the defining stochastic matrix of each Markov chain by a matrix
whose entries are polynomials with positive coefficients in several variables;
a Bernoulli shift is represented by a single polynomial with positive
coefficients, \(p\). This transforms jointly topological and
measure-theoretic coding problems into combinatorial ones. In solving the
combinatorial problems in Part A, we state and make use of facts from Part B
concerning \(p^n\) and its coefficients.

Part B contains the shorter paper on \(p^n\) and its coefficients, and is
independent of Part A.

An announcement describing the contents of this Memoir may be found in the
Electronic Research Announcements of the AMS at the following Web address:

#### Table of Contents

# Table of Contents

## Resolving Markov Chains onto Bernoulli Shifts via Positive Polynomials

- Contents vii8 free
- Preface ix10 free
- Part A. Resolving Markov Chains onto Bernoulli Shifts 112 free
- 1. Introduction 314
- 2. Weighted graphs and polynomial matrices 617
- 3. The main results 1122
- 4. Markov chains and regular isomorphism 1829
- 5. Necessity of the conditions 2536
- 6. Totally conforming eigenvectors and the one-variable case 3142
- 7. Splitting the conforming eigenvector in the one-variable case 4051
- 8. Totally conforming eigenvectors for the general case 5364
- 9. Splitting the conforming eigenvector in the general case 5869
- Bibliography 7283

- Part B. On Large Powers of Positive Polynomials in Several Variables 7586