A Multifractal Decomposition for Self-similar Measures with Exact Overlaps

We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\mu$, we show that there is a finite set of concave functions $\{{\tau }_1,\dots ,{\tau }_m\}$ such that the $L^q$-spectrum of $\mu$ is given by $\min \{{\tau }_1,\dots ,{\tau }_m\}$ and the multifractal spectrum of $\mu$ is given by $\max \{{\tau }_1^{\ast },\dots ,{\tau }_m^{\ast }\}$, where ${\tau }_i^{\ast }$ denotes the concave conjugate of ${\tau }_i$. In particular, the measure $\mu$ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave function. This implies that $\mu$ satisfies the multifractal formalism at values corresponding to points of differentiability of the $L^q$-spectrum. We also verify existence of the limit for the $L^q$-spectra of such measures for every $q\in \mathbb{R}$. As a direct application, we obtain many new results and simple proofs of well-known results in the multifractal analysis of self-similar measures satisfying the weak separation condition.