Some Remarks on Palindromic Periodicities

TL;DR

This paper investigates palindromic periodicities in various classical infinite words, analyzing their properties and identifying words with minimal palindromic periodicities.

Contribution

It provides new results on the occurrence and structure of palindromic periodicities in well-known infinite words and characterizes words with minimal such periodicities.

Findings

Identified palindromic periodicities in Sturmian, episturmian, Thue-Morse, and other words.

Proved structural properties of words with minimal palindromic periodicities.

Established connections between palindromic periodicities and classical infinite words.

Abstract

We say a finite word $x$ is a palindromic periodicity if there exist two palindromes $p$ and $s$ such that $|x| \geq |ps|$ 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.