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

Abstract

In a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the hull $\Omega$ of the potential of a discrete Schr\"odinger operator which guarantees purely singular continuous spectrum on a generic subset of $\Omega.$ In part, this condition requires the existence of infinitely many palindromic factors. In this same paper, they introduce the class P of morphisms $f:A^*\rightarrow B^*$ of the form $a\mapsto pq_a$ and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form $a\mapsto q_ap$ where again $p$ and $q_a$ are palindromes. 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 $y\in B^\omega$ 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 $g:A^*\rightarrow A^*$ of class P.