Occurrences of palindromes in characteristic Sturmian words

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 $\alpha$.

💡 Research Summary

The paper investigates the distribution of palindromic factors within characteristic Sturmian words cα, where the slope α∈(0,1) is irrational. Sturmian words are binary infinite sequences of minimal complexity (p(n)=n+1) and can be uniquely described by the continued‑fraction expansion of their slope. The characteristic word cα corresponds to the rotation coding with intercept 1−α and enjoys the property of uniform recurrence: every finite factor appears infinitely often and the gaps between consecutive occurrences are bounded.

The authors focus on a given palindrome u that occurs in cα and aim to pinpoint every occurrence of u. Their main contribution is a decomposition theorem that expresses cα as an infinite concatenation of blocks built from u and two auxiliary words X and Y: \