We say a finite word x is a palindromic periodicity if there exist two palindromes p and s such that |x| ≥ |ps| and x is a prefix of the infinite periodic word (ps)^ω = pspsps⋯. 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 distinct palindromic periodicities.
@InProceedings{fici_et_al:LIPIcs.CPM.2025.11, author = {Fici, Gabriele and Shallit, Jeffrey and Simpson, Jamie}, title = {{On Palindromic Periodicities}}, booktitle = {36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-369-0}, ISSN = {1868-8969}, year = {2025}, volume = {331}, editor = {Bonizzoni, Paola and M\"{a}kinen, Veli}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.11}, URN = {urn:nbn:de:0030-drops-231051}, doi = {10.4230/LIPIcs.CPM.2025.11}, annote = {Keywords: Combinatorics on words, Palindrome, Symmetric word, Palindromic periodicity, Walnut, Thue-Morse word, Sturmian word, Episturmian word} }
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