Palindromic Prefixes and Episturmian Words

Details Palindromic Prefixes and Episturmian Words Palindromic Prefixes and Episturmian Words

2005-01-24

arxiv.org/abs/math/0501420v3

Journal of Combinatorial Theory, Series A, 113.7 (2006), 1281-1304

Mathematics

Combinatorics

27 pages; many minor changes; to appear in J. Comb. Th. Series A

Scientific paper

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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