One-dimensional Gibbs states and Axiom A diffeomorphisms

We study the equilibrium statistical mechanics of one-dimensional classical lattice systems with exponentially decreasing interactions.For such systems, the Fournier transforms of pair correlation functions are meromorphic in a strip, and the residues of the poles can be expressed in terms of "Gibbs distributions."The latter are defined like Gibbs states, but without the positivity condition.Using symbolic dynamics, we can apply these results to Smale's Axiom A diffeomorphisms; the Gibbs distributions then become distributions in the sense of Schwartz on the manifold.