188 research outputs found
Substitution Delone Sets
This paper addresses the problem of describing aperiodic discrete structures
that have a self-similar or self-affine structure. Substitution Delone set
families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an
inflation functional equation under the action of an expanding integer matrix
in R^d. This paper studies such functional equation in which each X_i is a
discrete multiset (a set whose elements are counted with a finite
multiplicity). It gives necessary conditions on the coefficients of the
functional equation for discrete solutions to exist. It treats the case where
the equation has Delone set solutions. Finally, it gives a large set of
examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems
reformulate
Beta-expansions, natural extensions and multiple tilings associated with Pisot units
From the works of Rauzy and Thurston, we know how to construct (multiple)
tilings of some Euclidean space using the conjugates of a Pisot unit
and the greedy -transformation. In this paper, we consider different
transformations generating expansions in base , including cases where
the associated subshift is not sofic. Under certain mild conditions, we show
that they give multiple tilings. We also give a necessary and sufficient
condition for the tiling property, generalizing the weak finiteness property
(W) for greedy -expansions. Remarkably, the symmetric
-transformation does not satisfy this condition when is the
smallest Pisot number or the Tribonacci number. This means that the Pisot
conjecture on tilings cannot be extended to the symmetric
-transformation. Closely related to these (multiple) tilings are natural
extensions of the transformations, which have many nice properties: they are
invariant under the Lebesgue measure; under certain conditions, they provide
Markov partitions of the torus; they characterize the numbers with purely
periodic expansion, and they allow determining any digit in an expansion
without knowing the other digits
On substitution tilings and Delone sets without finite local complexity
We consider substitution tilings and Delone sets without the assumption of
finite local complexity (FLC). We first give a sufficient condition for tiling
dynamical systems to be uniquely ergodic and a formula for the measure of
cylinder sets. We then obtain several results on their ergodic-theoretic
properties, notably absence of strong mixing and conditions for existence of
eigenvalues, which have number-theoretic consequences. In particular, if the
set of eigenvalues of the expansion matrix is totally non-Pisot, then the
tiling dynamical system is weakly mixing. Further, we define the notion of
rigidity for substitution tilings and demonstrate that the result of
[Lee-Solomyak (2012)] on the equivalence of four properties: relatively dense
discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set
property, extends to the non-FLC case, if we assume rigidity instead.Comment: 36 pages, 3 figures; revision after the referee report, to appear in
the Journal of Discrete and Continuous Dynamical Systems. Results unchanged,
but substantial changes in organization of the paper; details and references
adde
The geometry of non-unit Pisot substitutions
Let be a non-unit Pisot substitution and let be the
associated Pisot number. It is known that one can associate certain fractal
tiles, so-called \emph{Rauzy fractals}, with . In our setting, these
fractals are subsets of a certain open subring of the ad\`ele ring
. We present several approaches on how to
define Rauzy fractals and discuss the relations between them. In particular, we
consider Rauzy fractals as the natural geometric objects of certain numeration
systems, define them in terms of the one-dimensional realization of
and its dual (in the spirit of Arnoux and Ito), and view them as the dual of
multi-component model sets for particular cut and project schemes. We also
define stepped surfaces suited for non-unit Pisot substitutions. We provide
basic topological and geometric properties of Rauzy fractals associated with
non-unit Pisot substitutions, prove some tiling results for them, and provide
relations to subshifts defined in terms of the periodic points of , to
adic transformations, and a domain exchange. We illustrate our results by
examples on two and three letter substitutions.Comment: 29 page
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
Decidability Problems for Self-induced Systems Generated by a Substitution
International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems
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