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? 



Subshifts with a free semigroup
For an irreducible sofic shift which is not periodic you will have this property. The Fischer cover gives a strongly connected deterministic partial automaton with all states initial and final recognizing the $\mathcal L(X)$. For any vertex v, the set of words labeling a loop at v is a free monoid on the words labeling a loop at v that do not visit v except at the beginning and end. If the strongly connected graph is not a cycle you have at least two such words. 


More generally you just need that in the Krieger cover some strongly connected component is not a cycle. 


