Aernout C.D. van Enter. On the set of pure states for some systems with non-periodic long-range order. (35K, LATEX) ABSTRACT. We discuss the set of pure states of some models which are ordered in a non-periodic way. This includes both some results based on a random model and some results based on the Thue-Morse system. We discuss the possible nature of non-periodic long-range order, and also the overlap distribution between the pure states.