On Abelian 2-avoidable binary patterns

Abstract

We give the first example of a binary pattern which is Abelian 2-avoidable, but which contains no Abelian fourth power. We introduce a family $$\{f_n\}_{n=1}^\infty$$ of binary morphisms which offer a common generalization of the Fibonacci morphism and the Abelian fourth-power-free morphism of Dekking. We show that the Fibonacci word begins with arbitrarily high Abelian powers, but for n ≥ 2, the fixed point of f n avoids x n+2 in the Abelian sense. The sets of patterns avoided in the Abelian sense by the fixed points of f n and f n+1 are mutually incomparable for n ≥ 2.