# Quasiperiods of biinfinite Sturmian words

## TL;DR

This paper investigates the quasiperiodic structures of biinfinite Sturmian words, revealing that they possess infinitely many quasiperiods and providing a detailed characterization of these quasiperiods, which enhances understanding beyond the right infinite case.

## Contribution

It extends the study of quasiperiodicity to biinfinite Sturmian words, showing they have infinitely many quasiperiods and characterizing these quasiperiods in detail.

## Key findings

- Biinfinite Sturmian words have infinitely many quasiperiods.
- Multiple quasiperiods of the same length can exist in biinfinite words.
- A characterization of quasiperiods in biinfinite Sturmian words is provided.

## Abstract

We study the notion of quasiperiodicity, in the sense of "coverability", for biinfinite words. All previous work about quasiperiodicity focused on right infinite words, but the passage to the biinfinite case could help to prove stronger results about quasiperiods of Sturmian words. We demonstrate this by showing that all biinfinite Sturmian words have infinitely many quasiperiods, which is not quite (but almost) true in the right infinite case, and giving a characterization of those quasiperiods. The main difference between right infinite and the biinfinite words is that, in the latter case, we might have several quasiperiods of the same length. This is not possible with right infinite words because a quasiperiod has to be a prefix of the word. We study in depth the relations between quasiperiods of the same length in a given biinfinite quasiperiodic word. This study gives enough information to allow to determine the set of quasiperiods of an arbitrary word.