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Multifractal Analysis of Measures Arising from Random Substitutions.

Andrew Mitchell1, Alex Rutar2

  • 1School of Mathematics, University of Birmingham, Edgbaston, B15 2TT UK.

Communications in Mathematical Physics
|February 27, 2024
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This study introduces random substitutions to analyze frequency measures, deriving analytic formulas for the spectrum and proving multifractal formalism. These findings offer new insights into complex systems and entropy calculations.

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Area of Science:

  • Dynamical Systems and Ergodic Theory
  • Fractal Geometry
  • Probability Theory

Background:

  • Random substitutions generalize deterministic ones, creating complex frequency measures.
  • Understanding regularity properties of these measures is crucial for analyzing complex systems.
  • The multifractal formalism provides a framework for characterizing irregular measures.

Purpose of the Study:

  • To investigate the regularity properties of frequency measures generated by random substitutions.
  • To derive analytic formulas for the spectrum of these measures.
  • To establish the validity of the multifractal formalism for a new class of measures.

Main Methods:

  • Analysis of frequency measures arising from random substitutions.
  • Derivation of closed-form analytic formulas for the spectrum.
  • Introduction and analysis of the 'inflation word' spectrum.
  • Application of separation conditions to recover known results.

Main Results:

  • A closed-form analytic formula for the spectrum is derived for a natural class of random substitution measures.
  • The multifractal formalism is proven to hold for these measures.
  • Results concerning the spectrum are established for a broader class of frequency measures.
  • The 'inflation word' spectrum is shown to coincide with the spectrum of the frequency measure.

Conclusions:

  • This work introduces a novel class of measures satisfying the multifractal formalism.
  • The findings provide a deeper understanding of regularity properties in random substitution systems.
  • The derived formulas and methods have applications in calculating topological and measure theoretic entropy.