Bifix codes and Sturmian words

TL;DR

This paper explores the properties of bifix codes within Sturmian words, revealing their structure, counting formulas, and their connection to free group subgroups, advancing understanding in combinatorics on words.

Contribution

It generalizes properties of bifix codes to Sturmian sets and establishes their relation to free group subgroups, providing new formulas and characterizations.

Findings

An $F$-maximal bifix code of degree $d$ in a Sturmian set has $(k-1)d+1$ elements.

If a word has a finite maximal bifix code with limited factors, it is ultimately periodic.

Any $F$-maximal bifix code of degree $d$ forms a basis of a subgroup of index $d$ in the free group.

Abstract

We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of words of $F$ of length $d$. An $F$-maximal bifix code of degree $d$ in a Sturmian set of words on an alphabet with $k$ letters has $(k-1)d+1$ elements. This generalizes the fact that a Sturmian set contains $(k-1)d+1$ words of length $d$. Moreover, given an infinite word $x$, if there is a finite maximal bifix code $X$ of degree $d$ such that $x$ has at most $d$ factors of length $d$ in $X$, then $x$ is ultimately periodic. Our main result states that any $F$-maximal bifix code of degree $d$ on the alphabet $A$ is the basis of a subgroup of index $d$ of the free group on~$A$.