Suppose ƒ : X* → X* is a morphism and u,v ∈ X*. For every nonnegative integer n, let z n be the longest commonprefix of ƒn(u) and ƒn(v), and let un,vn ∈ X* be words suchthat ƒn(u) = znun and ƒn(v) = znvn . We prove that there is a positiveinteger q such that for any positive integer p, the prefixes of u n (resp. v n ) of length p form an ultimately periodic sequence having periodq. Further, there is a value of q which works for all words u,v ∈ X*.