Multifractal Analysis of Measures Arising from Random Substitutions

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the $$-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the $$-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word $$-spectrum of a random substitution and show that this coincides with the $$-spectrum of the corresponding frequency measure for all $$. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.