Finite test sets for morphisms which are square-free on some of Thue's square-free ternary words

Abstract

Let $S$ be one of $\{aba,bcb\}$ and $\{aba, aca\}$, and let $w$ be an infinite square-free word over $\Sigma=\{a,b,c\}$ with no factor in $S$. Suppose that $f:\Sigma\rightarrow T^*$ is a non-erasing morphism. Word $f(w)$ is square-free if and only if $f$ is square-free on factors of $w$ of length 8 or less.