On generalized highly potential words

Abstract

The number of palindromic factors of a given finite word is bounded above by its length increased by 1. The difference between this upper bound and the actual number of palindromic factors of a given word is called the palindromic defect (or only defect) of a given word (by definition, the defect is always nonnegative). Though the definition of defect fundamentally relies on finiteness of a given word, it can be naturally extended to infinite words. There are many results in the literature about words of defect 0, but there are significantly less results about infinite words of finite positive defect. In this article we construct a new family of infinite words whose defect is finite, and in many cases positive (with fully characterized cases when the defect is 0). All the words from our family have the set of factors closed under reversal, and each of them is either periodic (which is a less interesting case, and explicitly characterized), or recurrent but not uniformly recurrent. The fact that they are not uniformly recurrent (unless they are periodic) is of a particular significance since: first, there are some results and examples here and there featuring uniformly recurrent words of finite defect, while next to nothing is known about aperiodic words that are not uniformly recurrent; second, it is known that any uniformly recurrent word of finite defect is a morphic image of some word of zero defect, which suggests that uniformly recurrent words are in a way pretty “tame,” and that those that are not uniformly recurrent are an unexplored territory that deserves a closer look.