Multifractal Analysis of Measures Arising from Random Substitutions

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.