More on morphisms and almost-periodicity

Abstract We give an effective criterion, using oriented graphs, to decide whether a morphic word on a finite alphabet is (effectively) almost-periodic. We show that a word W obtained as fixed point of a morphism ϕ is almost-periodic if and only if either no growing factor occurs infinitely often in W or if the set of non-growing factors of W is finite and the graph of the incidence relation induced by the morphism on the Λ+2 factors is strongly connected (where Λ is the maximum of the lengths of the non-growing factors of W ). We also show that the problem of the finiteness of the set of non-growing factors of W is decidable.