# Nested quasicrystalline discretisations of the line

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1 APC - UMR 7164 - AstroParticule et Cosmologie

Abstract : 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 self-similarity with algebraic scaling factor $\theta$. We explain the relation of the cut-and-project sets to non-standard 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.

Keywords : Multiresolution wavelet Pisot number cut-and-project set quasicrystal self-similarity substitution combinatorics on words

Autor: J.-P. Gazeau - Z. Masakova E. Pelantova

Fuente: https://hal.archives-ouvertes.fr/

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