Abelian squares are avoidable on 4 letters

Abstract

An abelian square is a non-empty word of the form P₁P₂, where P₂ is a permutation of P₁ that is, the number of occurrences of each letter is the same in P₁ and P₂. A word is called abelian 2-repetition free, or in short a-2-free, if it does not contain any abelian square as a subword. Let Σ be the four letter alphabet a,b,c,d. We give an example of a uniformly growing endo-morphism g:Σ^*→Σ^*; with ¦g(a)¦=85; such that the iteration of g yields an a-2-free ω-word. In fact, it is proved that the morphism g itself is a-2-free, that is, g(w) is a-2-free for every a-2-free word w in Σ^*. When proving that this property holds for g, one is in practice obliged to use a computer for certain tests. We performed these tests many times by using different computing environments and methods. It is explained how rechecking can be efficiently accomplished.

The morphism g is of the form g(σ(x)) = σ(g(x)) for all x in Σ, where a performs a cyclic permutation of letters in Σ. As regards morphisms of this form, one may check by using a computer/computers that, for our example morphism g, the image word g(a) is of minimal length. This checking becomes feasible by the reason that one may restrict the study to only relatively few long prefixes and suffixes of g(a).