Enumeration Formulæ in Neutral Sets

Abstract

Abstract. We present several enumeration results holding in sets of words called neutral and which satisfy restrictive conditions on the set of possible extensions of nonempty 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 a geometric example of this class of sets, namely the natural coding of some interval exchange transformations.

Neutral and tree sets of arbitrary characteristic

“…Acknowledgement.. This paper is an extended version of a conference paper [13]. This work was supported by grants from RégionÎle-de-France and ANR project Eqinocs ANR-13-BS02-004.…”

Codes and Automata in Minimal Sets

“…The next example shows that the prefix code can have strictly more than d X (S)(Card(A) − c) + 1 elements. If X is bifix, then it has d X (S)(Card(A) − c) + 1 elements by a result of [10]. The following example shows that an S-maximal prefix code can have d X (S)(Card(A) − c) + 1 elements without being bifix.…”

“…The complexity of a Sturmian set is p n = n(Card(A) − 1) + 1. The following result (see [10]) shows that a neutral set has linear complexity.…”

“…Given a set X ⊂ S, we denote λ S (X) = x∈X λ S (x) where λ S is the map defined by λ S (x) = e S (x) − r S (x). The following result is [10,Proposition 4]. Example 4.1.…”

Specular Sets

“…18 Let S be a uniformly recurrent laminary set containing the alphabet A. If the graph E(ε) is acyclic and if any finite S-maximal bifix code of S-degree d has d(Card(A) − 2) + 2 elements, then S is specular.…”