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Resolving Markov Chains onto Bernoulli Shifts via Positive Polynomials
Brian Marcus, IBM Almaden Research Center, San Jose, CA, and Selim Tuncel, University of Washington, Seattle, WA
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Memoirs of the American Mathematical Society
2001; 98 pp; softcover
Volume: 150
ISBN-10: 0-8218-2646-8
ISBN-13: 978-0-8218-2646-1
List Price: US$51
Individual Members: US$30.60
Institutional Members: US$40.80
Order Code: MEMO/150/710
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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: www.ams.org/era/

Readership

Graduate students and research mathematicians working in measure and integration.

Table of Contents

Part A. Resolving Markov Chains onto Bernoulli Shifts
  • Introduction
  • Weighted graphs and polynomial matrices
  • The main results
  • Markov chains and regular isomorphism
  • Necessity of the conditions
  • Totally conforming eigenvectors and the one-variable case
  • Splitting the conforming eigenvector in the one-variable case
  • Totally conforming eigenvectors for the general case
  • Splitting the conforming eigenvector in the general case
  • Bibliography
Part B. On Large Powers of Positive Polynomials in Several Variables
  • Introduction
  • Structure of \({\mathbf{Log}}({\mathbf(p}^{\mathbf n})\)
  • Entropy and equilibrium distributions for \({\mathbf w}\in {\mathbf W}({\mathbf p})\)
  • Equilibrium distributions and coefficients of \({\mathbf(p}^{\mathbf n})\)
  • Proofs of the estimates
  • Bibliography
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