# Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

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?

Yes. If $X$ is minimal (every orbit is dense) then the only subshift $Y\subset X$ is $X$ itself. The Jewett-Krieger theorem allows the construction of minimal subshifts with positive entropy, which therefore have the property you desire (albeit somewhat vacuously). Googling "minimal subshifts with positive entropy" brings up this paper by Henk Bruin, which includes such a construction and also gives two references to earlier constructions: