Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

Let [Pi] be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation [small tilde] such that (A,[small tilde]) is the regular tree with d +1 edges meeting at each vertex. Let ([pi])be the simplex of corresponding Markov random fields. We show that if [Pi] satisfies a 'boundedness' condition then ([pi]).We further study the structure of ([pi]) when [Pi] is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, [small tilde]) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for ([pi]) and for ([pi])to have precisely one member.