A Counterexample to a Question of Hof, Knill and Simon

Abstract

In this article, we give a negative answer to a question of Hof, Knill and Simon (1995) concerning purely morphic sequences obtained from primitive morphism containing an infinite number of palindromes. Their conjecture states that such palindromic sequences arise from substitutions that are in class $\mathcal{P}$. The conjecture was proven for the binary alphabet by B. Tan in 2007. We give here a counterexample for a ternary alphabet.