Extender sets in symbolic dynamics are a generalization of follower sets; the extender set of a finite word \(w\) in a subshift \(X\) is just the set of ways to legally complete \(w\) to a point of \(X\). Meyerovitch showed that if two words \(v\), \(w\) of the same size have the same extender set in \(X\), then \(\mu(v) = \mu(w)\) for every measure of maximal entropy \(\mu\) on \(X\).

I will describe recent joint work with Felipe Garcia-Ramos in which we generalize Meyerovitch's result above in two ways, yielding some new results and new proofs of existing results about the class of synchronizing subshifts.