On substitution invariant Sturmian words: an application of Rauzy fractals

Valérie Berthé, Hiromi Ei, Shunji Ito, Hui Rao
2007 RAIRO - Theoretical Informatics and Applications
Sturmian words are infinite words that have exactly n + 1 factors of length n for every positive integer n. A Sturmian word s α,ρ is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation R α : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give
more » ... alternative geometric proof of Yasutomi's characterization of all pairs (α, ρ) such that s α,ρ is a fixed point of some non-trivial substitution. Applying the linear transformation φ (see (8)) , we then get a tiling J of the real line: Tiling J is a tiling with two prototiles. Indeed We label the tiles of J on the right side of the origin by the sequence T 0 , T 1 , T 2 , . . . , where T n+1 is the rightside neighbour of T n . Likewise we label the tiles of J on the left side of the origin by T −1 , T −2 , . . . . One has J = {T k ; k ∈ Z}. We furthermore define the two-sided sequence (g k ) k∈Z as the sequence of left endpoints of tiles T k (one has g 0 = 0). An arithmetic description of the sequence (g k ) k∈Z is given in Section 5. Example 4. We continue Example 3. One has g −2 = 2(α − 1), g −1 = α − 1, g 0 = 0, g 1 = α, g 2 = 1. 3.4. Set equations of connected Rauzy fractals. According to Corollary 2, if σ is a primitive invertible substitution, then there exists a real number h such that X 1 = [−1 + α + h, h], X 2 = [h, h + α], that is, where J 1 = [−1 + α, 0] and J 2 = [0, α] are the two prototiles of tiling J . Let (a, i) ∈ D 1 . There exists an element [z, i * ] ∈ S such that φ • π(z) = a by Lemma 4. Let k ∈ Z such that φ • π[z, i * ] = T k ; then X i + a = J i + h + a = T k + h.