Multifractal analysis of measures arising from random substitutions

Andrew Mitchell, Alex Rutar*

*Corresponding author for this work

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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 Lq -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 Lq -spectrum of a broad class of frequency measures. We introduce a new notion called the inflation word Lq -spectrum of a random substitution and show that this coincides with the Lq -spectrum of the corresponding frequency measure for all q ≥ 0. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.
Original languageEnglish
Article number63
Number of pages44
JournalCommunications in Mathematical Physics
Volume405
Issue number3
Early online date24 Feb 2024
DOIs
Publication statusPublished - Mar 2024

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