Some extremal properties of the Fibonacci word

TL;DR

This paper investigates unique extremal properties of the Fibonacci word among characteristic Sturmian words, focusing on palindromic prefix length, minimal period, and letter occurrence counts, providing characterizations of the Fibonacci word.

Contribution

It establishes three extremal properties that uniquely characterize the Fibonacci word among characteristic Sturmian words, including length, period, and letter occurrence patterns.

Findings

Fibonacci word's palindromic prefix length is extremal

Minimal period of Fibonacci's palindromic prefixes is extremal

Number of 'b' occurrences in Fibonacci's palindromic prefixes is extremal

Abstract

We prove that the Fibonacci word $f$ satisfies among all characteristic Sturmian words, three interesting extremal properties. The first concerns the length and the second the minimal period of its palindromic prefixes. Each of these two properties characterizes $f$ up to a renaming of its letters. A third property concerns the number of occurrences of the letter $b$ in its palindromic prefixes. It characterizes uniquely $f$ among all characteristic Sturmian words having the prefix $abaa$.