Let $X$ be a subshift on a finite alphabet. I'm interested in the following property: there exist words $s,t\in\mathcal L(X)$ (the language of $X$) such that $\{s,t\}^*\subset \mathcal L(X)$. That is, $s^{k_1}t^{m_1}\dots s^{k_n}t^{m_n}\in \mathcal L(X)$ for all $n\ge1$ and all non-negative $k_j$ and $m_j$.

**Question.** What natural property of $X$ will guarantee the existence of such a free semigroup? (Apart from positive topological entropy, which $X$ must have, of course.)

For instance, if $X$ is an irreducible sofic subshift, does it always hold?