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

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

2012-01-06

arxiv.org/abs/1201.1423v2

Journal of Mathematical Physics 53 (2012) 032701

Physics

Mathematical Physics

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

Scientific paper

10.1063/1.3688337

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.

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