# Quelques m\'ethodes pour les mots sturmiens

## TL;DR

This chapter explores various techniques for analyzing Sturmian words, including geometric dual methods, palindrome factors, and Ostrowski numeration, providing new insights and proofs in the combinatorics of these minimal complexity infinite words.

## Contribution

It introduces several novel techniques for studying Sturmian words, including the geometric dual method and new proofs related to palindromic length.

## Key findings

- Link between palindrome factors and Ostrowski numeration systems
- Proof of a conjecture on palindromic length in Sturmian words
- Discussion of geometric dual method by Berstel and Pocchiola

## Abstract

In this book chapter, written in French, we consider the classical family of Sturmian words, defined as the aperiodic infinite words containing only $n+1$ factors of a length $n$, which is the minimal possible value. We will discuss several techniques for Sturmian words which have not been described in monographs. In particular, we will consider the geometric dual method by Berstel and Pocchiola and the link between palindrome factors and Ostrowski numeration systems. We will also discuss a conjecture on palindromic length which was proved on Sturmian words by two completely different methods for two distinct cases.