Sturmian sequences and the lexicographic world

In this paper, we give a complete description for the lexicographic world ${\mathcal{L}}=\{(x,y)\in\Sigma\times \Sigma: \Sigma_{xy}\not=\emptyset\} =\{(x,y):y\ge\phi(x)\},$ where $\Sigma=\{0,1\}^{\mathbf{N}},$ $\Sigma_{ab}=\{x\in\Sigma: a\le \sigma^i(x)\le b,\text{for all }i\ge 0\}$, $\phi:\Sigma\to\Sigma$ is defined by $\phi(a)=\inf\{b:\Sigma_{ab}\not=\emptyset\}$and the order $\le$ is the lexicographic order on $\Sigma$. The main result is that $b=\phi(a)$ for some $a=0x$if and only if $b$ is the Sturmian sequence $b$ such that $\operatorname{Orb}(b)\subset [0x,1x]$ and $\sigma^i(b)\le b$ for all $i\ge 0$. 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 $x\in\Sigma$, $M\subset[0x,1x]$. Moreover, for any $x\in\Sigma$, there exists a unique Sturmian minimal set in $[0x,1x]$.