Sturmian sequences and the lexicographic world
Shaobo Gan · Proceedings of the American Mathematical Society · 2000
In this paper, we give a complete description for the lexicographic world L = { ( x , y ) ∈ Σ × Σ : Σ x y ≠ ∅ } = { ( x , y ) : y ≥ ϕ ( x ) } {\mathcal {L}}=\{(x,y)\in \Sigma \times \Sigma : \Sigma _{xy} ot =\emptyset \} =\{(x,y):y\ge \phi (x)\} , where Σ = { 0 , 1 } N \Sigma =\{0,1\}^{\mathbf {N}} , Σ a b = { x ∈ Σ : a ≤ σ i ( x ) ≤ b , for\ all\ i ≥ 0 } \Sigma _{ab}=\{x\in \Sigma : a\le \sigma ^i(x)\le b,\text {for\ all\ }i\ge 0\} , ϕ : Σ → Σ \phi :\Sigma \to \Sigma is defined by