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 L φ : C ( X ) → C ( X ) on the space C ( X ) of real‐valued continuous functions on X . The dual operatorL φ *always has a probability measure ν as an eigenvector corresponding to a positive eigenvalue ( L φ * ν = 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 μ φ 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.