Diffraction of compatible random substitutions in one dimension

TL;DR

This paper analyzes the diffraction patterns of one-dimensional random substitutions, providing explicit formulas and extending the approach to various substitution families, including Fibonacci and noble means, with implications for understanding their spectral components.

Contribution

It introduces a novel method using self-similar measures to explicitly compute diffraction spectra of random substitutions, including pure point and absolutely continuous parts, and extends the analysis to broader classes.

Findings

Explicit formulas for diffraction measures of Fibonacci substitutions

Proof of absence of singular continuous components in these models

Extension of the approach to noble means and period doubling chains

Abstract

As a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our revised approach yields explicit formulas for the pure point and the absolutely continuous parts, as well as a proof for the absence of singular continuous components. This approach is then extended to the family of random noble means substitutions and, as an example with an underlying 2-adic structure, to a locally randomised version of the period doubling chain. As a first step towards a more general approach, we interpret our findings in terms of a disintegration over the Kronecker factor, which is the maximal equicontinuous factor of a covering model set.