Memoirs of the American Mathematical Society 2001; 98 pp; softcover Volume: 150 ISBN10: 0821826468 ISBN13: 9780821826461 List Price: US$54 Individual Members: US$32.40 Institutional Members: US$43.20 Order Code: MEMO/150/710
 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 measuretheoretic 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 onevariable case
 Splitting the conforming eigenvector in the onevariable 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
