© The London Mathematical Society
Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures
Mathematics Institute, University of Warwick Coventry CV4 7AL, United Kingdom pw{at}maths.warwick.ac.uk
Received 3 July 2003. Revision received 15 May 2004.
When T : X 
X is a one-sided topologically mixing subshift of finite type and 
: X R is a continuous function, one can define the Ruelle operator L : C(X) C(X) on the space C(X) of real-valued continuous functions on X. The dual operator 
always has a probability measure 
as an eigenvector corresponding to a positive eigenvalue (
= 
with > 0). Necessary and sufficient conditions on such an eigenmeasure are obtained for to belong to two important spaces of functions, W(X, T) and Bow (X, T). For example, 
Bow(X, T) 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 µ of 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.