Summary: Ergod. Th. & Dynam. Sys. (2003), 23, 14851504 c 2003 Cambridge University Press
DOI: 10.1017/S0143385702001700 Printed in the United Kingdom
Positive K-theory for finitary isomorphisms
of Markov chains
RICARDO G ´OMEZ
Instituto de Matem´aticas, Area de la Investigaci´on Cient´ifica, Circuito Exterior,
Ciudad Universitaria, M´exico DF 04510, M´exico
(Received 10 September 2001 and accepted in revised form 6 February 2003)
Abstract. We use matrices over formal power series to represent irreducible and positive
recurrent Markov chains and identify a natural class of good finitary isomorphisms
(magic word isomorphisms) as those arising from elementary matrix operations.
This extends the positive K-theory framework which has been used for other classification
problems in symbolic dynamics.
One of the significant recent developments in symbolic dynamics is the `positive K-theory'
approach to classification and isomorphism , inaugurated in [13, 14] following earlier
applications of polynomial matrices [2, 3, 20, 31]. In the most important example, very
roughly, a shift of finite type is represented by a matrix A whose entries are polynomials
with Z+ coefficients; and multiplication of I - A by an elementary matrix satisfying a