Multifractal analysis of measures arising from random substitutions

Andrew Mitchell; Alex Rutar

doi:10.1007/s00220-023-04895-3

Multifractal analysis of measures arising from random substitutions

Andrew Mitchell, Alex Rutar^*

^*Corresponding author for this work

Pure Mathematics

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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 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 ≥ 0. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.

Original language	English
Article number	63
Number of pages	44
Journal	Communications in Mathematical Physics
Volume	405
Issue number	3
Early online date	24 Feb 2024
DOIs	https://doi.org/10.1007/s00220-023-04895-3
Publication status	Published - Mar 2024

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title = "Multifractal analysis of measures arising from random substitutions",

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.",

author = "Andrew Mitchell and Alex Rutar",

note = "Funding: AM was supported by EPSRC DTP and the University of Birmingham. AR was supported by EPSRC Grant EP/V520123/1 and the Natural Sciences and Engineering Research Council of Canada. ",

year = "2024",

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doi = "10.1007/s00220-023-04895-3",

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T1 - Multifractal analysis of measures arising from random substitutions

AU - Mitchell, Andrew

AU - Rutar, Alex

N1 - Funding: AM was supported by EPSRC DTP and the University of Birmingham. AR was supported by EPSRC Grant EP/V520123/1 and the Natural Sciences and Engineering Research Council of Canada.

PY - 2024/3

Y1 - 2024/3

N2 - 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.

AB - 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.

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DO - 10.1007/s00220-023-04895-3

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SN - 0010-3616

VL - 405

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

M1 - 63

ER -

Multifractal analysis of measures arising from random substitutions

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