  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.
