We consider the dynamical system (M,S) where M is the orbit closure of a nonperiodic recurrent sequence of O\u27s and 1\u27s (for example, the Morse sequence) and S is the shift map. The enveloping semigroup is E(M) = {Sⁿ : n ∈ Z} where the closure is taken in the topology of pointwise convergence. H. Furstenberg was the first to establish the existence of relationships between recurrence, IP sets, and idempotents in the enveloping semigroup, and the first author has proven that the closure of the set of idempotents coincides with the IP cluster points. In this paper the authors compute this set for (M,S) and shed light on other combinatorial properties of generalized Morse sequences
