Some extremal properties of the Fibonacci word

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 $abaa$.

💡 Research Summary

The paper investigates three extremal properties of the Fibonacci word f within the class of characteristic Sturmian words and shows that these properties uniquely identify f (up to the trivial exchange of the two letters).

First, the authors recall that a characteristic Sturmian word is determined by an irrational slope α and a cutting point ρ; the Fibonacci word corresponds to the golden‑ratio slope α = (√5 − 1)/2, whose continued‑fraction expansion is