$L^q$-spectrum of graph-directed self-similar measures that have overlaps and are essentially of finite type

For self-similar measures with overlaps, closed formulas of the $L^q$-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. $\mathbf{\text{106}}$ (2019), 56–103]. We extend the results of Ngai and the author \cite{Ngai-Xie_2019} to the graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the $L^q$-spectrum has been studied by Edgar and Mauldin \cite{Edgar-Mauldin_1992}. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures $\mu$ on ${\mathbb{R}}^d$ ($d\ge 1$), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the $L^q$-spectrum of $\mu$ for $q\ge 0$, and prove the differentiability of the $L^q$-spectrum. This framework allows us to include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimension.