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

Abstract

When $T\,{:}\, X\,{\longrightarrow}\,X$ is a one-sided topologically mixing subshift of finite type and $\varphi\,{:}\,X\,{\longrightarrow}\,R$ is a continuous function, one can define the Ruelle operator ${\cal L}_\varphi\,{:}\,C(X)\,{\longrightarrow}\,C(X)$ on the space $C(X)$ of real-valued continuous functions on $X$. The dual operator ${\cal L}^*_\varphi$ always has a probability measure $\nu$ as an eigenvector corresponding to a positive eigenvalue (${\cal L}^*_\varphi\nu\,{=}\,\lambda\nu$ with $\lambda\,{>}\,0$). Necessary and sufficient conditions on such an eigenmeasure $\nu$ are obtained for $\varphi$ to belong to two important spaces of functions, $W(X,T)$ and ${\rm Bow} (X,T)$. For example, $\varphi\,{\in}\,{\rm Bow}(X,T)$ if and only if $\nu$ 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 $\mu_\varphi$ of $\varphi\,{\in}\,{\rm Bow}(X,T)$ 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.