Sturmian Sequences and the Lexicographic World

In this paper, we give a complete description for the lexicographic world L = {(x, y) ∈ Σ × Σ : Σxy 6= ∅} = {(x, y) : y ≥ φ(x)}, where Σ = {0, 1}N, Σab = {x ∈ Σ : a ≤ σi(x) ≤ b, for all i ≥ 0}, φ : Σ→ Σ is defined by φ(a) = inf{b : Σab 6= ∅} and the order ≤ is the lexicographic order on Σ. The main result is that b = φ(a) for some a = 0x if and only if b is the Sturmian sequence b such that Orb(b) ⊂ [0x, 1x] and σi(b) ≤ b for all i ≥ 0. At the same time, a new description of Sturmian minimal sets is given. A minimal set M is a Sturmian minimal set if and only if, for some x ∈ Σ, M ⊂ [0x, 1x]. Moreover, for any x ∈ Σ, there exists a unique Sturmian minimal set in [0x, 1x].