Some Remarks on Palindromic Periodicities

We say a finite word $x$ is a palindromic periodicity if there exist two palindromes $p$ and $s$ such that $\mathrm{\mid }x\mathrm{\mid }\ge \mathrm{\mid }ps\mathrm{\mid }$ and $x$ is a prefix of the word $(ps{)}^{\omega }=pspsps\cdots$. In this paper we examine the palindromic periodicities occurring in some classical infinite words, such as Sturmian words, episturmian words, the Thue-Morse word, the period-doubling word, the Rudin-Shapiro word, the paperfolding word, and the Tribonacci word, and prove a number of results about them. We also prove results about words with the smallest number of palindromic periodicities.