On the Equivalence of Boundary Conditions

We show that if b and b' are two boundary conditions (b.c.) for general spin systems on Z a such that the difference in the energies of a spin configuration ~A in A c Za is uniformly bounded, [HA.~(ga) -Ha,b'(aA)[ ~< C < 0% then any infinite-volume Gibbs states p and p' obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of p and p" into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if p is extremal, p = p'. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.