Characterizations of ergodic measures on shift spaces

Abstract

We consider various characterizations of ergodic measures on the shift space (Ω, T), where Ω = X 0 ∞ {0, 1, … r − 1} r ⩽ 2 is the shift on Ω. It is shown that every ergodic measure is either purely atomic or nonatomic, and that μ is a purely ergodic measure iff μ is a periodic orbit measure. A characterization of purely atomic, ergodic Markov measures is formulated in terms of certain periodic orbit measures. We also prove that μ is nonatomic, ergodic Markov measure iff μ is the Markov measure induced by an irreducible, nonpermutation stochastic matrix, together with its stationary distribution. An example of nonatomic, non-Markov ergodic measure is constructed.