Eigenvectors for infinite Markov chains and dimension groups.

Abstract

This thesis relates the theory of dimension groups to the study of nonnegative infinite matrices. Given a nonnegative matrix P = Pg,hg,h∈ G (Gamma countable and infinite), we obtain information concerning the nonnegative eigenvectors of P by studying the associated dimension groups and their trace (state) spaces. For a particular class of countable discrete Markov chains, we exhibit affine homeomophisms between nonnegative eigenvector spaces and certain subspaces of related trace spaces. This thesis also establishes some necessary conditions for the weak ergodicity of sequences of 2 x 2 real matrices.