A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by $\sigma(\{x_i\}_{i\in G}) = \{x_{i+1}\}_{i\in G}$ (where $G = \mathbb{N}$ or $\mathbb{Z}$).

The entropy of a subshift is $\lim_{n\rightarrow\infty} \frac{\log(|B_n|)}{n}$ where $B_n\subseteq A^n$ is the set of length-$n$ strings which appear in some element of the subshift.

Is there a one-dimensional subshift $X$ of positive entropy s, all of whose sub-subshifts $Y\subseteq X$ also have entropy s?