Relation between powers of factors and recurrence function characterizing Sturmian words

TL;DR

This paper characterizes Sturmian words by exploring the relationship between the powers of factors and the recurrence function, providing new insights and a novel proof related to their index and continued fraction expansion.

Contribution

It offers a new characterization of Sturmian words using the relation between factor powers and recurrence functions, and presents a new proof of the index theorem.

Findings

Characterization of Sturmian words via recurrence functions

New proof of the index theorem for Sturmian words

Connection between continued fraction expansion and word index

Abstract

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.