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Multifractal analysis of measures arising from random substitutions

Andrew Mitchell, Alex Rutar

TL;DR

The paper develops a comprehensive multifractal framework for frequency measures arising from primitive random substitutions, introducing the inflation word L^q-spectrum and proving its equality with the true L^q-spectrum for all q≥0. Under recognisability, the authors obtain a closed-form L^q-spectrum and establish the multifractal formalism, including a variational principle for relative local dimensions and concave, analytic spectra. The results unify entropy, topological entropy, and L^q-spectra within a single symbolic-dynamical setting, and they delineate the roles of separation conditions and recognisability in ensuring sharp bounds and exact formulas. The work yields explicit formulas under disjoint or identical-set conditions, recovers known entropy results, and provides sharp negative-q results in the recognisable regime, supported by diverse examples and counterexamples.

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.

Multifractal analysis of measures arising from random substitutions

TL;DR

The paper develops a comprehensive multifractal framework for frequency measures arising from primitive random substitutions, introducing the inflation word L^q-spectrum and proving its equality with the true L^q-spectrum for all q≥0. Under recognisability, the authors obtain a closed-form L^q-spectrum and establish the multifractal formalism, including a variational principle for relative local dimensions and concave, analytic spectra. The results unify entropy, topological entropy, and L^q-spectra within a single symbolic-dynamical setting, and they delineate the roles of separation conditions and recognisability in ensuring sharp bounds and exact formulas. The work yields explicit formulas under disjoint or identical-set conditions, recovers known entropy results, and provides sharp negative-q results in the recognisable regime, supported by diverse examples and counterexamples.

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 -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.
Paper Structure (29 sections, 28 theorems, 186 equations, 5 figures)

This paper contains 29 sections, 28 theorems, 186 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Entropy and Lq-spectra
  3. Random substitutions
  4. Statement and discussion of main results
  5. Further work
  6. Preliminaries
  7. Symbolic notation
  8. Dynamics, entropy and dimension
  9. Lq-spectra and smoothness
  10. Multifractal spectrum and multifractal formalism
  11. Random substitutions
  12. Primitive random substitutions
  13. Compatible random substitutions
  14. Frequency measures
  15. Separation conditions and recognisability
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\vartheta_{\bm{P}} = (\vartheta, \bm{P})$ be a primitive and compatible random substitution with corresponding frequency measure ${\mu_{\bm{P}}}$. Then the limits defining $\tau_{\mu_{\bm{P}}}(q)$ and $T_{\vartheta,\bm{P}}(q)$ exist and coincide for all $q\geq0$. Moreover,

Figures (5)

  • Figure 1: $L^q$-spectra
  • Figure 2: Multifractal spectra
  • Figure 4: $L^q$-spectra
  • Figure 5: Multifractal spectra
  • Figure 7: Upper and lower bounds on the $L^q$-spectrum of the frequency measure corresponding to the random Fibonacci substitution with $p=1/2$, for $k=3,5,7$. The darker shades correspond to higher values of $k$.

Theorems & Definitions (61)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Proof 1
  • Proposition 2.3
  • Proof 2
  • ...and 51 more