Correction to ‘A class of unique g -measures’

Abstract

Theorem 1.1 in [4] is false: the hypothesis $\sum v_n(g)<\infty$ needs to be replaced by a stronger condition. A corrected version of Theorem 1.1 is given below; a counterexample to the original result is given in [5].

Some comments on 'Unique Bernoulli $g$-measures'

“…The main effect of these changes on the generality of the results seems to be to restrict the rate of growth of the sequence which can be used in the block-variation pair. The hypotheses (1)-(3), however, are unaffected; indeed (2) is covered by an earlier result in [2] (see Section 2).…”

“…Thus, if g ∈ G is continuous, there is always at least one g-measure, and uniqueness is equivalent to the convergence (pointwise or uniform) of n −1 n−1 i=0 L i g f to a constant for every f ∈ C(X + ), whereas there is a unique g-chain if and only if L n g f converges to a constant. The following result is proved in [2].…”

Uniqueness and non-uniqueness of chains on half lines

“…(iv) On the other hand, Gibbsian uniqueness criteria can be used to show that these models have a unique invariant state at high temperatures. The only chain criterion that is temperature-sensitive is one-sided Dobrushin [11,14] which, however, is not directly applicable to the Ising chains considered here.…”