Characterization of repetitions in Sturmian words: A new proof

TL;DR

This paper introduces a new dynamical approach using Diophantine approximation to analyze repetitions in Sturmian words, providing a shorter proof of existing characterizations and deriving a formula for their fractional index.

Contribution

It offers an alternative, shorter proof for characterizing powers in Sturmian words and derives a formula for their fractional index based on continued fractions.

Findings

New dynamical method for studying repetitions in Sturmian words

Shorter proof of existing characterization of powers in Sturmian words

Formula for fractional index based on continued fraction expansion

Abstract

We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [Powers in Sturmian sequences, Eur. J. Combin. 24 (2003), 377--390]. Further, as a consequence, we obtain a previously known formula for the fractional index of a Sturmian word based on the continued fraction expansion of its slope.