Open and Closed Prefixes of Sturmian Words

TL;DR

This paper investigates the sequence of open and closed prefixes in Sturmian words, showing it uniquely characterizes these words and revealing their structural properties, including a novel factorization into squares of reversed standard words.

Contribution

It introduces a characterization of Sturmian words via their open and closed prefix sequences and describes their structural decomposition, advancing combinatorial understanding of these words.

Findings

Sequence of open and closed prefixes characterizes Sturmian words

Standard Sturmian words can be decomposed into squares of reversed standard words

The sequence provides a unique combinatorial signature for Sturmian words

Abstract

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.