Multifractal analysis of measures arising from random substitutions

TL;DR

This paper investigates the multifractal properties of frequency measures from random substitutions, deriving explicit formulas for the $L^q$-spectrum and confirming the multifractal formalism for a broad class of such measures.

Contribution

It introduces the inflation word $L^q$-spectrum and provides closed-form formulas for the $L^q$-spectrum of measures from random substitutions, extending multifractal analysis.

Findings

Derived a closed-form $L^q$-spectrum formula for certain measures

Proved the multifractal formalism holds for these measures

Established the inflation word $L^q$-spectrum coincides with the measure's spectrum

Abstract

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 $L^q$-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 $L^q$-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word $L^q$-spectrum of a random substitution and show that this coincides with the $L^q$-spectrum of the corresponding frequency measure for all $q \geq 0$. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.