KMS states and branched points

We completely classify the KMS states for the gauge action on a $C^*$-algebra associated with a rational function $R$ introduced in our previous work. The gauge action has a phase transition at $\beta = \log \deg R$. We can recover the degree of $R$, the number of branched points, the number of exceptional points and the orbits of exceptional points from the structure of the KMS states. We also classify the KMS states for $C^*$-algebras associated with some self-similar sets, including the full tent map and the Sierpinski gasket by a similar method.