Bifix codes and interval exchanges

Abstract

International audienceWe investigate the relation between bifix codes and interval exchange 5 transformations. We prove that the class of natural codings of regular 6 interval echange transformations is closed under maximal bifix decoding

A geometric interpretation of the Schützenberger group of a minimal subshift

“…If X is a minimal non-periodic subshift then lim n→∞ min{|r| : r ∈ R n (x)} = ∞ for every x ∈ X . 3 2 What we call return words is sometimes in the literature designated first return words, as is the case of the article [13], which is cited later in this paper. The terminology that we adopt appears for instance in [20,21,12].…”

“…The subshift X satisfies the tree condition if G w is a tree for every w ∈ L(X ) ∪ {1}. The class of subshifts satisfying the tree condition contains two classes that have deserved strong attention in the literature: the class of Arnoux-Rauzy subshifts 4 (see the survey [23]), and the class of subshifts defined by regular interval exchange transformations (see [13,14]).…”

“…It is shown in [13,Theorem 4.5] that if the minimal subshift X satisfies the tree condition, then, for every w ∈ L(X ), the set of return words R(w) is a basis of the free group generated by the set of letters occurring in X . This result is called the Return Theorem in [13]. Combining the Return 4 The Arnoux-Rauzy subshifts over two-letter alphabets are the extensively studied Sturmian subshifts, but we warn that in [13] the Arnoux-Rauzy subshifts are called Sturmian.…”

“…This result is called the Return Theorem in [13]. Combining the Return 4 The Arnoux-Rauzy subshifts over two-letter alphabets are the extensively studied Sturmian subshifts, but we warn that in [13] the Arnoux-Rauzy subshifts are called Sturmian.…”

“…Indeed, it is shown in [5] that if X is the subshift defined by a weakly primitive substitution ϕ which is group invertible, then G(X ) is a free profinite group. The weakly primitive substitution ϕ(a) = ab, ϕ(b) = cda, ϕ(c) = cd, ϕ(d) = abc is group invertible, but the minimal subshift defined by X is a subshift that fails the tree condition [13,Example 3.4].…”