In this excellent paper, the authors derive symbolic expressions for the \(L^q \)-spectrum of frequency measures associated with random substitutions under certain weak assumptions. With an additional assumption of recognisability, they establish a closed-form analytic expression for the \( L^q \)-spectrum and a variational formula, which together imply the multifractal formalism. Recognisable random substitutions display unique properties not observed in other well-understood classes of measures for the multifractal formalism: often, the unique frequency measure of maximal entropy is not a Gibbs measure with respect to the zero potential, and the corresponding subshift is not sofic. These techniques and results offer significant new insights into the geometry and dynamics of the respective measures.
Let \(\mu\) be a Borel probability measure in a compact metric space. The \( L^q \)-spectrum of \(\mu\) is defined by \[ \tau _{\mu }(q)=\underset{r\rightarrow 0}{\lim \inf }\frac{\log \sup \sum _i\mu \left(B(x_i,r)\right){}^q}{\log r}, \] where the supremum is taken over 2\(r\)-separated subsets \(\{x_i\}_i\) of the support of \(\mu\). The authors introduce the definitions of the local dimension of \(\mu\) at \(x\) and the multifractal spectrum of \(\mu\). Given a random substitution \(\vartheta_\mathbf{P}\), the authors define the Perron-Frobenius eigenvalue \(\lambda\) and the corresponding right eigenvector \(\mathbb{R}\) of the substitution matrix associated with \(\vartheta_\mathbf{P}\). In their basic results, the authors prove that for every \(q\geq 0\), the inflation word \( L^q \)-spectrum of \(\vartheta_\mathbf{P}\) and the \( L^q \)-spectrum of \(\mu_\mathbf{P}\) correspond to each other. The authors describe several fundamental characteristics of the \(L^q \)-spectrum of the measure \( \mu \), and investigate a random substitution that is primitive, compatible, and recognizable, along with its associated frequency measure \(\mu_\mathbf{P}\). Many examples, counterexamples and applications are given in the last section.