Powers of positive polynomials and codings of markov chains onto bernoulli shifts

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 pn, the multiple being prescribed by r. The construction, which we sketch, relies on a description of the terms of pn and on estimates of the relative sizes of the coefficients of pn. © 1999 American Mathematical Society.