On a question of Hof, Knill and Simon on palindromic substitutive systems

Abstract

In a 1995 paper, Hof et al. obtain a sufficient combinatorial criterion on the subshift \(\Omega \) of the potential of a discrete Schrodinger operator which guarantees purely singular continuous spectrum on a generic subset of \(\Omega .\) In part, this condition requires that the subshift \(\Omega \) be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms \(f:A^*\rightarrow B^*\) of the form \(a\mapsto pq_a\) 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 \(a\mapsto q_ap.\) In this paper we give a partial affirmative answer to the question of Hof et al.: 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 \(y\in B^\omega \) is of the form \(y=f(x)\) where \(f:A^*\rightarrow B^*\) is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution \(g:A^*\rightarrow A^*\) conjugate to one in class P.