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

TL;DR

This paper explores a class of singular continuous measures with full support and increasing distribution functions, analyzing their spectral, topological, and dynamical properties within mathematical diffraction and tiling spaces.

Contribution

It introduces a new family of measures with unique support and distribution properties, linking them to dynamical systems and cohomological invariants.

Findings

Supports of measures have full Lebesgue measure

Distribution functions are strictly increasing

Dynamical zeta functions are explicitly calculated

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.