International audienceIn a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the subshift Ω of the potential of a discrete Schrödinger operator which guarantees purely singular continuous spectrum on a generic subset of Ω. In part, this condition requires that the subshift Ω be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms f : A * → B * of the form a → pq a with p and q a palindromes, and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form a → q a p. In this paper we give a partial affirmative answer to the question of Hof, Knill and Simon: we show that every rich primitive substitutive subshift is generated by at most two morphisms each of which is conjugate to a morphism of class P. More precisely, we show that every rich (or almost rich in the sense of finite defect) primitive morphic word y ∈ B ω is of the form y = f (x) where f : A * → B * is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution g : A * → A * conjugate to one in class P
