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Palindromic Prefixes and Episturmian Words

Stéphane Fischler
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Abstract

Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$ such that $n_{i+1} \leq 2 n_i + 1$ for any $i$, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity $\limsup n_{i+1}/n_i$, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.

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Stéphane Fischler. 2005-11-28. Palindromic Prefixes and Episturmian Words. https://arxiv.org/abs/math/0501420

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