Bifix codes and interval exchanges

We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval echange transformations is closed under maximal bifix decoding.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1308.539

Specular sets

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352

On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$

Embedding a θ\theta-invariant code into a complete one

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short) that is, languages L such that $\theta$ (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete $\theta$-invariant codes. Moreover, we establish a formula which allows to embed any non-complete $\theta$-invariant code into a complete one. As a consequence, in the family of the so-called thin $\theta$--invariant codes, maximality and completeness are two equivalent notions.Comment: arXiv admin note: text overlap with arXiv:1705.0556

The finite index basis property

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on