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Rauzy fractals of random substitutions

Philipp Gohlke, Andrew Mitchell, Dan Rust, Tony Samuel

TL;DR

This paper extends Rauzy fractal theory to random substitutions, proving that every compatible irreducible Pisot random substitution admits a canonical Rauzy fractal with two key constructions that are shown to coincide. It establishes that the Rauzy fractal has nonempty interior and full Hausdorff dimension while its subtiles arise as attractors of a graph directed iterated function system, albeit with overlapping that may violate the open set condition. A central contribution is the Rauzy measure, which encodes the probability data and is shown to vary continuously with the probabilities; the measure is absolutely continuous with respect to Lebesgue measure and is obtainable both as an invariant distribution limit and as a pushforward of the frequency measure. The results connect Rauzy fractals to S-adic systems, provide a framework to interpolate between deterministic Pisot substitutions and their random counterparts, and lay groundwork for diffraction and spectral analysis in random substitution contexts. Overall, the work broadens the scope of aperiodic order by embedding randomness into the Rauzy fractal paradigm and offering tools to study geometric, dynamical, and spectral properties in this richer setting.

Abstract

We develop a theory of Rauzy fractals for random substitutions, which are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. We show that a Rauzy fractal can be associated with a given random substitution in a canonical manner, under natural assumptions on the random substitution. Further, we show the existence of a natural measure supported on the Rauzy fractal, which we call the Rauzy measure, that captures geometric and dynamical information. We provide several different constructions for the Rauzy fractal and Rauzy measure, which we show coincide, and ascertain various analytic, dynamical and geometric properties. While the Rauzy fractal is independent of the choice of (non-degenerate) probabilities assigned to a given random substitution, the Rauzy measure captures the explicit choice of probabilities. Moreover, Rauzy measures vary continuously with the choice of probabilities, thus provide a natural means of interpolating between Rauzy fractals of deterministic substitutions. Additionally, we highlight connections between Rauzy fractals and Rauzy measures of random substitutions and related S-adic systems.

Rauzy fractals of random substitutions

TL;DR

This paper extends Rauzy fractal theory to random substitutions, proving that every compatible irreducible Pisot random substitution admits a canonical Rauzy fractal with two key constructions that are shown to coincide. It establishes that the Rauzy fractal has nonempty interior and full Hausdorff dimension while its subtiles arise as attractors of a graph directed iterated function system, albeit with overlapping that may violate the open set condition. A central contribution is the Rauzy measure, which encodes the probability data and is shown to vary continuously with the probabilities; the measure is absolutely continuous with respect to Lebesgue measure and is obtainable both as an invariant distribution limit and as a pushforward of the frequency measure. The results connect Rauzy fractals to S-adic systems, provide a framework to interpolate between deterministic Pisot substitutions and their random counterparts, and lay groundwork for diffraction and spectral analysis in random substitution contexts. Overall, the work broadens the scope of aperiodic order by embedding randomness into the Rauzy fractal paradigm and offering tools to study geometric, dynamical, and spectral properties in this richer setting.

Abstract

We develop a theory of Rauzy fractals for random substitutions, which are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. We show that a Rauzy fractal can be associated with a given random substitution in a canonical manner, under natural assumptions on the random substitution. Further, we show the existence of a natural measure supported on the Rauzy fractal, which we call the Rauzy measure, that captures geometric and dynamical information. We provide several different constructions for the Rauzy fractal and Rauzy measure, which we show coincide, and ascertain various analytic, dynamical and geometric properties. While the Rauzy fractal is independent of the choice of (non-degenerate) probabilities assigned to a given random substitution, the Rauzy measure captures the explicit choice of probabilities. Moreover, Rauzy measures vary continuously with the choice of probabilities, thus provide a natural means of interpolating between Rauzy fractals of deterministic substitutions. Additionally, we highlight connections between Rauzy fractals and Rauzy measures of random substitutions and related S-adic systems.
Paper Structure (32 sections, 48 theorems, 185 equations, 4 figures)

This paper contains 32 sections, 48 theorems, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Substitutions
  3. Rauzy fractals of substitutions
  4. Random substitutions
  5. Outline and overview of main results
  6. Preliminaries
  7. Random substitutions
  8. Frequency measures
  9. Rauzy fractals of C-balanced sequences
  10. Rauzy fractals of C-balanced sequences
  11. Analytic properties
  12. Rauzy fractals as generic factors
  13. Measures and the Lebesgue covering property
  14. Rauzy measures as pushforward measures
  15. Rauzy fractals of random substitutions
  16. ...and 17 more sections

Key Result

Proposition 1.1

Let $\theta$ be a unimodular, irreducible Pisot substitution over a $d$-letter alphabet, where $d \geq 2$. The Rauzy fractal $\mathcal{R}_{\theta}$ is a compact subset of $\mathbb{H}$ with non-empty interior, and thus has full Hausdorff dimension. Moreover, $\mathcal{R}_{\theta}$ is the closure of i

Figures (4)

  • Figure 1: Rauzy fractals for the tribonacci (left) and twisted tribonacci (center) substitutions. Broken line and projection (right).
  • Figure 2: Computer simulations of the distribution functions for the measure $\mu_{a,p}$ for random Fibonacci, for $p=1/16$ (left), $p=1/2$ (middle) and $p=15/16$ (right).
  • Figure 3: Evolution of Rauzy measures for the random tribonacci substitution as $p$ goes from $1$ to $0$. From top-left to bottom-right, the values of $p$ are $1$, $15/16$, $1/2$, $1/16$ and $0$ respectively. Points were generated experimentally (justified by \ref{['PROP:as-convergence-markov']}) according to the generating probabilities in such a way that point-densities approximate the mass distributions of each measure.
  • Figure 4: Rauzy fractal of the random substitution $\vartheta \colon 1 \mapsto \{12\}, 2 \mapsto \{13,31\}, 3 \mapsto \{1\}$ and positions of the Rauzy fractals of the marginals.

Theorems & Definitions (108)

  • Proposition 1.1: Siegel-Thuswaldner
  • Definition 2.1
  • Example 2.2: Random tribonacci
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: rust-spindeler
  • Theorem 2.8: miro-et-al
  • Lemma 2.9: MT-entropy
  • ...and 98 more