Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be *entropy-minimal* if every subshift $Y \subsetneq X$ satisfies that $$h\_{\mathrm{top}}(Y) < h\_{\mathrm{top}}(X).$$ Equivalently, $X$ is entropy-minimal if for every word $\omega \in \mathcal{L}(X)$ the subshift $$ X\_{\omega} = \left\{x \in X : \omega \text{ is not a factor of } x \right\}$$ satisfies that $h\_{\mathrm{top}}(X\_{\omega}) < h\_{\mathrm{top}}(X)$. 


It is a "folklore" result that irreducible subshifts of finite type of are entropy-minimal as well as subshifts with the specification property. Weaker notions of the specification property and entropy-minimality have been studied recently by Climenhaga, García-Ramos and Pavlov. 


What I am looking for is a concrete example of a (one-dimensional) subshift $X$ that is non-entropy-minimal, transitive, and has positive topological entropy. It will be interesting if such example is a binary subshift.



Entropy-minimal subshifts
Let $f$ be a sublinear function that tends to infinity, such as $f(n) = \sqrt{n}$. Define $X \subset \{0,1,2\}^{\mathbb{N}}$ by forbidding all long enough words $w$ with more than $f(|w|)$ occurrences of $2$. Then $X$ has entropy $\log 2$ and is mixing, and properly contains the binary full shift, which likewise has entropy $\log 2$.


If you need a binary example, just replace the alphabet with three suitable binary words, like $01$, $001$ and $0001$.


