Occurrences of palindromes in characteristic Sturmian words

TL;DR

This paper investigates the distribution of palindromic factors in characteristic Sturmian words, providing a detailed description of their occurrences linked to the continued fraction expansion of the slope.

Contribution

It offers a complete characterization of where palindromes occur in characteristic Sturmian words based on their continued fraction expansion.

Findings

Palindromes occur infinitely often with bounded gaps in $c_\alpha$.

A decomposition of $c_\alpha$ relative to palindromes is established.

Occurrences of palindromes are explicitly related to the continued fraction of $\alpha$.

Abstract

This paper is concerned with palindromes occurring in characteristic Sturmian words $c_\alpha$ of slope $\alpha$, where $\alpha \in (0,1)$ is an irrational. As $c_\alpha$ is a uniformly recurrent infinite word, any (palindromic) factor of $c_\alpha$ occurs infinitely many times in $c_\alpha$ with bounded gaps. Our aim is to completely describe where palindromes occur in $c_\alpha$. In particular, given any palindromic factor $u$ of $c_\alpha$, we shall establish a decomposition of $c_\alpha$ with respect to the occurrences of $u$. Such a decomposition shows precisely where $u$ occurs in $c_\alpha$, and this is directly related to the continued fraction expansion of $\alpha$.