13 research outputs found
Computing modular coincidences for substitution tilings and point sets
Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in , consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the number of iterations needed. The main tool is a simple algorithm for computing modular coincidences, which is essentially a generalization of the Dekking coincidence to more than one dimension, and the proof of equivalence of this generalized Dekking coincidence and modular coincidence. As a consequence, we also obtain some conditions for the existence of modular coincidences. In a separate section, and throughout the article, a number of examples are given
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
Algorithm for determining pure pointedness of self-affine tilings
Overlap coincidence in a self-affine tiling in is equivalent to pure
point dynamical spectrum of the tiling dynamical system. We interpret the
overlap coincidence in the setting of substitution Delone set in and
find an efficient algorithm to check the pure point dynamical spectrum. This
algorithm is easy to implement into a computer program. We give the program and
apply it to several examples. In the course the proof of the algorithm, we show
a variant of the conjecture of Urba\'nski (Solomyak \cite{Solomyak:08}) on the
Hausdorff dimension of the boundaries of fractal tiles.Comment: 21 pages, 3 figure
Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets
The paper establishes an equivalence between pure point diffraction and
certain types of model sets, called inter model sets, in the context of
substitution point sets and substitution tilings. The key ingredients are a new
type of coincidence condition in substitution point sets, which we call
algebraic coincidence, and the use of a recent characterization of model sets
through dynamical systems associated with the point sets or tilings.Comment: 29pages; revised version with update
On the interplay between Babai and Černý’s conjectures
Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n2 -6n + 5 and can attain the value (Formula presented). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter (formula presented). © Springer International Publishing AG 2017
Squirals and beyond: Substitution tilings with singular continuous spectrum
The squiral inflation rule is equivalent to a bijective block substitution
rule and leads to an interesting lattice dynamical system under the action of
. In particular, its balanced version has purely singular
continuous diffraction. The dynamical spectrum is of mixed type, with pure
point and singular continuous components. We present a constructive proof that
admits a generalisation to bijective block substitutions of trivial height on
.Comment: 23 pages, 7 figure