32 research outputs found
Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems
There is a growing body of results in the theory of discrete point sets and
tiling systems giving conditions under which such systems are pure point
diffractive. Here we look at the opposite direction: what can we infer about a
discrete point set or tiling, defined through a primitive substitution system,
given that it is pure point diffractive? Our basic objects are Delone multisets
and tilings, which are self-replicating under a primitive substitution system
of affine mappings with a common expansive map . Our first result gives a
partial answer to a question of Lagarias and Wang: we characterize repetitive
substitution Delone multisets that can be represented by substitution tilings
using a concept of "legal cluster". This allows us to move freely between both
types of objects. Our main result is that for lattice substitution multiset
systems (in arbitrary dimensions) being a regular model set is not only
sufficient for having pure point spectrum--a known fact--but is also necessary.
This completes a circle of equivalences relating pure point dynamical and
diffraction spectra, modular coincidence, and model sets for lattice
substitution systems begun by the first two authors of this paper.Comment: 36 page
Central sets and substitutive dynamical systems
In this paper we establish a new connection between central sets and the
strong coincidence conjecture for fixed points of irreducible primitive
substitutions of Pisot type. Central sets, first introduced by Furstenberg
using notions from topological dynamics, constitute a special class of subsets
of \nats possessing strong combinatorial properties: Each central set
contains arbitrarily long arithmetic progressions, and solutions to all
partition regular systems of homogeneous linear equations. We give an
equivalent reformulation of the strong coincidence condition in terms of
central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech
compactification \beta \nats . This provides a new arithmetical approach to
an outstanding conjecture in tiling theory, the Pisot substitution conjecture.
The results in this paper rely on interactions between different areas of
mathematics, some of which had not previously been directly linked: They
include the general theory of combinatorics on words, abstract numeration
systems, tilings, topological dynamics and the algebraic/topological properties
of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225,
arXiv:1301.511