The singular continuous diffraction measure of the Thue-Morse chain

TL;DR

This paper derives an explicit form of the singular continuous diffraction measure for the Thue-Morse chain, deepening understanding of its spectral properties in symbolic dynamics and mathematical diffraction.

Contribution

It provides a new functional equation and explicit representation of the diffraction measure, connecting it to the Riesz product for the Thue-Morse sequence.

Findings

Explicit functional equation for the diffraction measure

Representation of the measure as a Riesz product

Enhanced understanding of singular continuous spectra

Abstract

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.