We consider g-measures for the shift on 11^=0 $ where S is a finite set. For a certain class of continuous g, two g-measures are identified; they are equal if and only if there is a unique g-measure. We prove that the natural extensions of these measures are Bernoulli. With a further restriction on g when S is a two-point set, we show that there is a unique g-measure. We also consider extensions of these results to the non-continuous case.
An example of non-unique g-measures is constructed. The function g satisfies a certain summability condition, thereby providing a counterexample to the claim in an earlier paper that all such g have a unique g-measure.
IntroductionThe concept of a g-measure was introduced by Keane [9]; an equivalent notion, chains with complete connections, was studied earlier by Doeblin and Fortet [4]. We consider g-measures for the one-sided shift on a sequence space.Let S be a finite set with the discrete topology, X = S Z + have the product topology, B denote the Borel σ -algebra on X, and T be the shift on X; that is, (T x) i = x i+1 . (Here and later, x i denotes the ith coordinate of x.) Let G X be the set of all non-negative Borel functions g on X such that y∈T −1 x g(y) = 1 for all x ∈ X, and let g ∈ G X . A probability measure µ on X is a g-measure if µ is T -invariant and µ({x ∈ X : x 0 = s} | T −1 B)(x) = g(sx 1 x 2 . . . ) for all s ∈ S, or equivalently,
We consider Gibbs states for attractive specifications on a one-dimensional lattice. If the specification is translation-invariant, there exist ergodic, translation-invariant Gibbs states v + and v" with the property that there is a unique Gibbs state if and only if v + = v". We show that v + and v~ are Bernoulli measures for the shift. DEFINITION. A probability measure /i on (X, 08) is a Gibbs state for a specification = {/AW* if fora\\yeS\xeX&nd
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