Atomic surfaces, tilings and coincidences II. Reducible case

Balanced pair algorithm for a class of cubic substitutions

“…In the reducible case, the discrete line may have other expanding directions, but the projection of the discrete line by π s still provides a bounded set; for more details, we refer to [8]. …”

Substitutions, abstract number systems and the space filling property

“…[5,Sections 3 and 4]) in a geometric setting. We also refer to [21] where a similar construction (also using the geometric language) was made for primitive unimodular substitutions with a Pisot number as dominant root and where the characteristic polynomial is allowed to be reducible. Theorems 3.3 and 3.4 describe the structure of the projected word w (n) .…”

“…Since then many researchers have studied the corresponding structures for other substitutions, with quite diverse outcomes (cf. [2,5,41,47,16,4,20,45,26,21,48]), but a general rule for deciding when a substitution leads to simple tiling of a space is still wanted, especially, because these structures turned out to be useful in the mathematical theory of quasicrystals (for details see [44,7,27,10]). …”

Purely periodic 𝛽-expansions with Pisot unit base

“…X i is a subset of the contractive subspace P and is called the atomic surface by [11,8]. (It is called a Rauzy fractal by [6,7,19] and is called a central tile by [20,3,4].)…”