New results on pseudosquare avoidance

Abstract

We start by considering binary words containing the minimum possible numbers of squares and antisquares (where an antisquare is a word of the form $x\overline{x}$), and we completely classify which possibilities can occur. We consider avoiding $xp(x)$, where $p$ is any permutation of the underlying alphabet, and $xt(x)$, where $t$ is any transformation of the underlying alphabet. Finally, we prove the existence of an infinite binary word simultaneously avoiding all occurrences of $xh(x)$ for every nonerasing morphism $h$ and all sufficiently large words $x$.