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1. CMB 2002 (vol 45 pp. 697)
Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type We consider two dynamical systems associated with a substitution of
Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and
the $\mathbb{R}$-action, which can be defined as a tiling dynamical
system or as a suspension flow. We describe procedures for checking
when these systems have pure discrete spectrum (the ``balanced
pairs algorithm'' and the ``overlap algorithm'') and study the
relation between them. In particular, we show that pure discrete
spectrum for the $\mathbb{R}$-action implies pure discrete spectrum
for the $\mathbb{Z}$-action, and obtain a partial result in the
other direction. As a corollary, we prove pure discrete spectrum
for every $\mathbb{R}$-action associated with a two-symbol
substitution of Pisot type (this is conjectured for an arbitrary
number of symbols).
Categories:37A30, 52C23, 37B10 |
2. CMB 2002 (vol 45 pp. 123)
Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |