KMS states on C^*-algebras associated with self-similar sets

Abstract

In this paper, we study KMS states for the gauge actions on C${}^*$-algebras associated with self-similar sets whose branch points are finite. If the self-similar set does not contain any branch point, the Hutchinson measure gives the unique KMS state. But if the self-similar set dose contain a branch point, there sometimes appear other KMS states which come from branch points. For this purpose we construct explicitly a basis for a Hilbert C${}^*$-module associated with a self-similar set with finite branch condition. Using this we get condition for a Borel probability measure on K to be extended to a KMS state on the C${}^*$-algebra associated with the original self-similar set. We classify KMS states for the case of dynamics of unit interval and the case of Sierpinski gasket which is related with Complex dynamical system. KMS states for these examples are unique and given by the Hutchinson measure if $\beta$ is equal to $\log N$, where $N$ is the number of contractions. They are expressed as convex combinations of KMS states given by measures supported on the orbit of the branched points if $\beta > \log N$.