Let $x=(x_{n})_{n=0}^{\infty}$ be a sequence in $\{0,1\}^{\mathbb{N}_{0}}$. Let $X=\overline{\{T^{n}x\ :\ n\in\mathbb{N}_{0}\}}$, where $T$ is a shift map on $\{0,1\}^{\mathbb{N}_{0}}$. $T=T|_{X}$.
Question :$\forall x\in X$, $(X,T)$ is minimal if and only if any word $\omega=\omega_{1}\cdots\omega_{k}$ occurs in $x$ with bounded gaps for any word $\omega$ occuring in $x$.
I tried to make the right statement into mathematical one, but it is not easy. Can anyone help me? (After editing) I know the definition of minimality is $X=\overline{\left\{ T^{n}x\ :\ n\in\mathbb{N}_{0}\right\} }$
Simply: $X=\overline{\left\{ T^{n}x\ :\ n\in\mathbb{N}_{0}\right\} }$ if and only if every finite word occurs in $x$. Gaps are irrelevant.