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.