A characterization of self-similar measures

For a graph directed self-similar set, we investigate a list of invariant self-similar measures. In particular, the existence of the local dimension of the associated self-similar measure is determined. Consequently, the multifractal decomposition of the graph directed self-similar set can be formulated under the open set condition. This give a positive answer to a question posed by Edgar and Mauldin, as well as Olsen.