Abstract :
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.
