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?



Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts 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](http://www.mat.univie.ac.at/~bruin/papers/recdim.pdf) by Henk Bruin, which includes such a construction and also gives two references to earlier constructions:


1. Hahn and Katznelson, "On entropy of uniquely ergodic transformations", Trans. AMS **126** (1967) 335-360
2. C. Grillenberger, "Construction of strictly ergodic systems I. Given entropy", Z. Wahrscheinlichkeitstheorie **25** (1972/1973) 323-334


