Acyclic, connected and tree sets

Abstract

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 the 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.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1011.5369, Monatsh. Math. (2015

Eventually Dendric Shifts

“…For example, what can we say about the subgroup of the free group generated by return words to a given word? In [5] it is proved that for minimal dendric sets, every set of return words is a basis of the free group, while in the case of specular sets, the set of return word to a fixed word is a basis of a particular subgroup called the even subgroup (see [4]).…”

“…The languages of dendric shifts were introduced in [5] under the name of tree sets. An important example of dendric shifts is formed by episturmian shifts (also called Arnoux-Rauzy shifts), which are by definition such that L(X) is closed by reversal and such that for every n there exists a unique w n ∈ L n (X) such that Card(R 1 (w n )) = Card(A) and such that for every w ∈ L n (X) \ {w n } one has Card(R 1 (w)) = 1 (see [5]). Example 2.1 Let X be the Fibonacci shift, which is generated by the morphism a → ab, b → a.…”

Derived sequences of complementary symmetric Rote sequences

“…So we can conclude that w (6) has the same type as w (0) , which is SU (1). Similarly, w (7) has the same type as w (1) etc. Now we use Corollary 23 to describe the derivated sequences of the Rote sequence g = 001110011100011000110001110011100011 · · · associated with the Fibonacci sequence f .…”

Enumeration Formulæ in Neutral Sets

“…One has S ∩ A 2 = {ab, ac, bc, ca, cd, da} and thus m(ε) = −1. It is shown in [5] that every nonempty word is neutral. Thus S is neutral of characteristic 2.…”