Neutral and tree sets of arbitrary characteristic

Abstract

International audienceWe study classes of minimal sets defined by restrictions on the possible extensions of the words. These sets generalize the previously studied classes of neutral and tree sets by relaxing the condition imposed on the empty word and measured by an integer called the characteristic of the set. We present several enumeration results holding in these sets of words. These formulae concern return words and bifix codes. They generalize formulae previously known for Sturmian sets or more generally for tree sets. We also give two geometric examples of this class of sets, namely the natural coding of some interval exchange transformations and the natural coding of some linear involutions

Specular Sets

“…In [19] it is proved that the converse is true for neutral sets. As all sets we will deal with are neutral, we usually omit the term "uniformly" and just mention whenever we suppose them to be recurrent.…”

Eventually Dendric Shifts

“…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.…”

“…Note that the proof shows that Theorem 7.5 holds for the more general class of shift spaces which are eventually neutral, in the sense that there is an integer m such that every word of length at least m is neutral. This class includes the shift spaces X such that L(X) is neutral with characteristic c introduced in [11] and for which Theorem 7.5 is proved in [11] with a similar proof.…”

Derived sequences of complementary symmetric Rote sequences

“…By our Corollary 8, the language L of a Rote sequence is a neutral set with the characteristic χ(L) = 0. As the language L is uniformly recurrent, we can apply Corollary 5.4 of [15] to deduce that any non-empty factor of a Rote sequence has exactly three return words.…”

“…acyclic, neutral and tree sequences introduced in [7]. Return words in sequences coding linear involutions were studied in [8] and the number of return words for more general neutral sequences was determined in [15]. In [9] the properties of return words and derivated sequences were exploited for the characterization of substitutive tree sequences.…”