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.
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.
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincare maps of measured foliations a suitable definition of return words which yields that the set of first return words
to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$.
We prove in this note that, for an alphabet with three letters, the set of first return to a given word in a set satisfying the tree condition is a basis of the free group.
We introduce a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the pos
sible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words.
Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by t
he property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a basis of the subgroup that it generates.
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 s
ets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namemly the class of tree sets. We prove as a main result that for a uniformly recurrent tree set $F$, a finite bifix code $X$ on the alphabet $A$ is $F$-maximal of $F$-degree $d$ if and only if it is the basis of a subgroup of index $d$ of the free group on $A$.
In this paper, we give a necessary condition for an infinite word defined by a non-degenerate interval exchange on three intervals (3iet word) to be invariant by a substitution: a natural parameter associated to this word must be a Sturm number. We d
educe some algebraic consequences from this condition concerning the incidence matrix of the associated substitution. As a by-product of our proof, we give a combinatorial characterization of 3iet words.
