Symmetric Gibbs measures

Abstract. We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equi-valence relations, including the orbit relation of the adic transform-ation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic prob-ability measures for the Pascal adic transformation as exactly the Bernoulli measures—a version of de Finetti’s Theorem. General-izing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, ‘canonical ’ Gibbs states, and the tri-viality of remote sigma-fields finer than the usual tail field. 1