Palindromic defect and highly potential words

Abstract

The \emph{palindromic defect} of a finite word $w$ has been introduced by Brlek et al.\ as the difference between the length of $w$ increased by one and the number of palindromic factors of $w$ (by an earlier result of Droubay, Justin and Pirillo, this difference is always non-negative). A natural extension of this definition to infinite words has also been introduced. In this talk we present a construction of a class of infinite words, called \emph{highly potential words} because of their seeming high potential of being a good supply of examples and counterexamples regarding various problems on words, particularly the ones related to the palindromic defect and related notions. One of the most interesting properties of highly potential words is the fact that they are all aperiodic words of a finite positive defect, having the set of factors closed under reversal; words satisfying this combination of conditions have been sought after in some recent works, but not a single example is found so far.