A Multifractal Decomposition for Self-similar Measures with Exact Overlaps

Abstract

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 $μ$, we show that there is a finite set of concave functions $\{τ_1,\ldots,τ_m\}$ such that the $L^q$-spectrum of $μ$ is given by $\min\{τ_1,\ldots,τ_m\}$ and the multifractal spectrum of $μ$ is given by $\max\{τ_1^*,\ldots,τ_m^*\}$, where $τ_i^*$ denotes the concave conjugate of $τ_i$. In particular, the measure $μ$ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave function. This implies that $μ$ 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.