Spectral and topological properties of a family of generalised Thue-Morse sequences

Abstract

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is know as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

Dedicated to Robert V. Moody on the occasion of his 70th birthday; revised and improved version

References in corpus (10)

• Pure Point Dynamical and Diffraction Spectra
• Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
• Asymptotic structure in substitution tiling spaces
• Kinematic Diffraction from a Mathematical Viewpoint
• Diffraction of limit periodic point sets
• The singular continuous diffraction measure of the Thue-Morse chain
• A note on the dynamical zeta function of general toral endomorphisms
• Quotient Cohomology for Tiling Spaces
• Surprises in aperiodic diffraction
• Algorithm for determining pure pointedness of self-affine tilings

Cited by in corpus (16)

• Mathematical diffraction of aperiodic structures
• Dynamical versus diffraction spectrum for structures with finite local complexity
• Aperiodic crystals and beyond
• Local symmetry theory of resonator structures for the real-space control of edge states in binary aperiodic chains
• Squirals and beyond: Substitution tilings with singular continuous spectrum
• Hexagonal inflation tilings and planar monotiles
• Singular substitutions of constant length
• Examples of substitution systems and their factors
• Scaling of diffraction intensities near the origin: Some rigorous results
• Scaling of the Thue-Morse diffraction measure
• A comment on the relation between diffraction and entropy
• On long arithmetic progressions in binary Morse-like words
• A note on measures vanishing at infinity
• Squiral diffraction
• Quotient cohomology of certain 1- and 2-dimensional substitution tiling spaces
• Recent progress in mathematical diffraction