Specular Sets

Abstract

Abstract. We introduce specular sets. These are subsets of groups which 7 form a natural generalization of free groups. These sets are an abstract 8 generalization of the natural codings of interval exchanges and of linear 9 involutions. We prove several results concerning the subgroups generated 10 by return words and by maximal bifix codes in these sets.

Return words of linear involutions and fundamental groups

“…A specular set is then defined as a laminary set such that the extension graph of any nonempty word is a tree and the extension graph of the empty word has two connected components which are trees. Extensions of Theorem 6.4 and 6.9 are proved to hold in this context in [2]. This paper is organized as follows.…”

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 graph E 3 (a) is shown on the right. The tree sets of characteristic c ≥ 1 introduced in [4,11] give an example of eventually dendric shifts. Example 2.2 Let X be the shift generated by the morphism a → ab, b → cda, c → cd, d → abc.…”

Enumeration Formulæ in Neutral Sets

“…Note that we recover, as a particular case of Theorem 2 applied to the set X of words of length n in S, the fact that for a set S satisfying the hypotheses of the theorem, the factor complexity is p 0 = 1 and p n = n(Card(A) − χ(S)) + χ(S). Note that Theorem 2 has a converse (see [4]).…”