Some Remarks on Palindromic Periodicities

TL;DR

This work investigates palindromic periodicities: finite words that prefix a concatenation of two palindromes, $(ps)^\omega$, with $p,s$ palindromes. It establishes that every nonempty factor of a Sturmian word is a palindromic periodicity, and extends this to trapezoidal words and prefixes of standard episturmian words; it also analyzes the phenomenon in several famous infinite words via Walnut automata, deriving exact counts and asymptotic bounds for Thue-Morse, Rudin-Shapiro, period-doubling, regular paperfolding, and Tribonacci words. The authors further study words with few palindromic periodicities, proving minimal bounds (e.g., at least 6 for alphabets of size at least 3, and 30 for infinite binary words) and providing structural characterizations and automata-based analyses that yield explicit forms and counts. Collectively, the paper links palindromic periodicities with symmetric word-Periods, Sturmian/trapezoidal episturmian structure, and automatic-word decidability, offering concrete enumeration formulas and posing open questions about asymptotics and extreme-case constructions.

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 $(ps)^ω= pspsps\cdots$. 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.

Figures (4)

• Figure 1: The inclusion hierarchy.
• Figure 2: Rich words, trapezoidal words, and palindromic periodicities.
• Figure 3: Synchronized automaton for pdpp; accepts the base-$2$ representation of $i$ and $n$ if ${\bf pd}[i..i+n-1]$ is a palindromic periodicity.
• Figure 4: Synchronized automaton for countpdpp; accepts the base-$2$ representation of $i$ and $n$ if ${\bf pd}[i..i+n-1]$ is novel and also a palindromic periodicity.

Theorems & Definitions (39)

• Remark 1
• Theorem 2
• proof
• Example 3
• Theorem 4
• Proposition 5
• Proposition 6
• proof : Proof of Theorem \ref{['thm:trap']}
• Example 7
• Proposition 8