The famous Morse-Thue sequence has the "no BBb" property: it contains no block of the form
One hundred years ago Axel Thue showed that the "friends" of the Morse-Thue sequence, i.e. the members of the Morse Minimal Set, the closure of the orbit of the doubly infinite Morse-Thue sequence under the shift homeomorphism, are precisely the doubly infinite sequences on two symbols having the no BBb property.
What if the sequence is wearing a disguise? Here "wearing a disguise" means that the names of the symbols have been changed using some unknown local rule, local as in cellular automata. How do you determine whether or not the undisguised sequence is a member of the Morse Minimal Set?
I will tell you more than you want to know about the Morse-Thue sequence, answer the question above, and perhaps others about substitution minimal sets.
This is joint work with Mike Keane (Wesleyan) and Michelle LeMasurier (Hamilton College).