REGULARITY CONDITIONS AND BERNOULLI PROPERTIES OF EQUILIBRIUM STATES AND g-MEASURES

When T :X −→X is a one-sided topologically mixing subshift of finite type and ϕ :X −→R is a continuous function, one can define the Ruelle operator Lϕ :C(X)−→C(X) on the space C(X) of real-valued continuous functions on X. The dual operator L∗ ϕ always has a probability measure ν as an eigenvector corresponding to a positive eigenvalue (L∗ ϕν =λν with λ>0). Necessary and sufficient conditions on such an eigenmeasure ν are obtained for ϕ to belong to two important spaces of functions, W(X, T) and Bow(X, T). For example, ϕ ∈ Bow(X, T) if and only if ν is a measure with a certain approximate product structure. This is used to apply results of Bradley to show that the natural extension of the unique equilibrium state μϕ of ϕ ∈ Bow(X, T) has the weak Bernoulli property and hence is measure-theoretically isomorphic to a Bernoulli shift. It is also shown that the unique equilibrium state of a two-sided Bowen function has the weak Bernoulli property. The characterizations mentioned above are used in the case of g-measures to obtain results on the ‘reverse’ of a g-measure.