Resumen de Regularity conditions and Bernoulli properties of equilibrum states and -measures

Peter Walters

• When is a one-sided topologically mixing subshift of finite type and is a continuous function, one can define the Ruelle operator on the space of real-valued continuous functions on . The dual operator always has a probability measure as an eigenvector corresponding to a positive eigenvalue ( with ). Necessary and sufficient conditions on such an eigenmeasure are obtained for to belong to two important spaces of functions, and . For example, if and only if 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 of 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 -measures to obtain results on the 'reverse' of a -measure.