×

Nested quasicrystalline discretisations of the line. (English) Zbl 1135.37301

Nyssen, Louise (ed.), Physics and number theory. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-028-0/pbk). IRMA Lectures in Mathematics and Theoretical Physics 10, 79-131 (2006).
Summary: One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and selfsimilarity with algebraic scaling factor \(\theta\). We explain the relation of the cut-and-project sets to nonstandard numeration systems based on \(\theta\). We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences.
For the entire collection see [Zbl 1098.11003].

MSC:

37B10 Symbolic dynamics
68R15 Combinatorics on words
82D25 Statistical mechanics of crystals
52A10 Convex sets in \(2\) dimensions (including convex curves)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

Software:

KK
PDFBibTeX XMLCite
Full Text: arXiv