SYMMETRICAL GIBBS MEASURES

We prove that certain Gibbs measures on subshifts of finite type are n onsingular and ergodic for certain countable equivalence relations, in cluding the orbit relation of the adic transformation (the same as equ ality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, includin g some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic tran sformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and t he shift as the Gibbs measures whose potential functions depend on onl y a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, in terval splitting procedures, 'canonical' Gibbs states, and the trivial ity of remote sigma-fields finer than the usual tail field.