Theoretical Informatics and Applications
Theoret. Informatics Appl.36(2002) 385–388 DOI: 10.1051/ita:2003005
ON A CHARACTERISTIC PROPERTY OF ARNOUX–RAUZY SEQUENCES
Jacques Justin
1and Giuseppe Pirillo
2Abstract. Here we give a characterization of Arnoux–Rauzy sequences by the way of the lexicographic orderings of their alphabet.
Mathematics Subject Classification. 68R15.
Introduction
Letsbe a standard Sturmian infinite word onA={a, b}. If we lexicographically orderA∗bya < bthen, for any non-negative integern, the prefix of lengthnofas is the smallest factor of lengthnofsand the prefix of lengthnofbsis the greatest factor of lengthn of s, see Borel and Laubie [3] and Exercise 2.2.13 in [7]. The converse of this property is true also as was remarked and proved by the second author [8].
Here we extend the property and its converse to Arnoux–Rauzy sequences on finite alphabets [1, 9] (or strict Episturmian words in the terminology of [4, 6]), which generalize Sturmian words. This is:
Theorem 0.1. For any infinite wordson a finite alphabet Athe following prop- erties are equivalent
i) s is a standard Arnoux–Rauzy sequence;
ii) for any x∈Aif <is a lexicographic order of A∗ satisfyingx= minA, then for any positive integer n the prefix ofxs of lengthn is the smallest among the factors of s of lengthn.
Keywords and phrases: Lexicographic order, Arnoux–Rauzy sequence, Episturmian word, Sturmian word.
1 LIAFA, Universit´e Paris VII, Case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France;
e-mail:justin@liafa.jussieu.fr
2 IAMI, NCR, Viale Morgagni 67/a, 50134 Firenze, Italy;
e-mail:pirillo@udini.math.unifi.it
c
EDP Sciences 2003
386 J. JUSTIN AND G. PIRILLO
This may have some interest as many properties of Sturmian words extend to Arnoux–Rauzy sequences but few are characteristic.
Remark. In the formulation of the theorem we do not consider maxA nor the greatest factor of each length. Indeed, for any two wordsu,v of the same length, u > vis equivalent tovuwhere is the lexicographic order “opposite” to<.
1. Preliminaries
Let Abe a finite alphabet |A| =k, A∗ (resp. Aω) denotes the set of all finite (resp. infinite) words on A. The common terminology for words can be found in [7] for instance.
Fors∈Aω,sn will denote the prefix ofsof lengthn. The set of the factors of sof lengthnisFn(s) and the set of all factors ofsisF(s) =S
n≥0Fn(s). A factor uofs isright-(resp. left-)specialif, for at least two lettersx6=y, ux, uy∈F(s) (resp. xu, yu∈F(s)). An infinite wordsis recurrentif each of its factors occurs at least two (hence infinitely many) times in it.
Thereversalof a wordu=x1x2· · ·xn, xi∈Ais ˜u=xn· · ·x2x1.
An Arnoux–Rauzy sequence onAis a recurrent infinite words∈Aωsuch that for any non-negative integer n, s has exactly one right-special factor, un say, of lengthnwithunx∈F(s) for allx∈A, and exactly one left-special factor,vnsay, such thatyvn∈F(s) for ally∈A[1, 9]. These sequences are the Sturmian words when |A|= 2. They were studied in [1] for|A|= 3 with a hint of generalization to any finite alphabet. In their paper the authors assumed the infinite word to beuniformly recurrent(orminimalin the terminology of dynamical systems used there), but indeed the assumption of recurrence is sufficient and leads to exactly the same infinite words.
An Arnoux–Rauzy sequence isstandard(orcharacteristic) if all its prefixes are left-special.
In [4–6] a slightly more general kind of words is studied under the name of
“Episturmian words” and it is shown that Arnoux–Rauzy sequences onk letters are exactly the k-strict Episturmian words. Episturmian words which are not strict correspond to some “degenerate cases”, for instance when k = 2 they are the so called “periodic Sturmian words”.
2. Proof of the theorem
Proof. i) ⇒ ii). Given the lexicographic order < with x = minA as in the theorem, if ii) is false letn be minimal such thatxsn−1 is not minimal inFn(s).
Clearly n > 1. Then xsn−2 is minimal in Fn−1. Let sn−1 = sn−2y, y ∈ A.
As xsn−2y is not minimal in Fn(s), there exists a letterz 6=y such thatxsn−2z
∈ Fn(s) and thatxsn−2z < xsn−2y, whence z < y. Thus xsn−2 is right-special.
Thus as F(s) is closed under reversal [4, 6, 9], s]n−2x is left-special, hence it is
CHARACTERIZATION OF A-R SEQUENCES 387 sn−2y by the definition of Arnoux–Rauzy sequences. Thusx=y, whencez < x, a contradiction. Consequently, for all n,xsn = minF(s).
ii)⇒i). Asashas its prefixes inF(s) for alla∈A, any prefixuofsis left-special and satisfies Au⊂F(s). Suppose that, for some positive integern,shas at least two left-special factors of lengthn, its prefixsn and another onev and supposen minimal with this property. Setsn =sn−1x,v=v0y,x, y∈A. Thensn−1 andv0 are left-special, hencev0 =sn−1 andv =sn−1y,y 6=x. Then there exist letters, z6=t, r6=msuch thatzsn, tsn, rv, mv∈Fn+1(s).
Suppose for instance r 6= x (otherwise, m 6= x) and consider a lexicographic order such that r= minAand y < x. Thenrv =rsn−1y < rsn−1x=rsn, thus rsn is not minimal inFn+1, a contradiction.
Thus, for alln,shas exactly one left-special factor of lengthnand thenshas for allnat least one right-special factor of lengthn. Now supposeshas, for somen, two right-special factors of length n,uandv say, and letn be minimal with this property. Then we have u = xw, v = yw, x 6= y ∈ A where w is the unique right-special factor of length n−1. Clearly wis left-special and w= sn−1. We then havesn=wafor some letterawhence, asAsn⊂F(s),xwa, ywa∈Fn+1(s).
As u, v are right-special, there exist letters, t 6= a, m 6= a such that xwt, ywm∈F(s). Suppose for instancey6=a. Consider a lexicographic order<such that y = minA and m < a, we get ywa > ywm, thus ysn is not minimal in Fn+1(s), a contradiction.
Thus s has for all n exactly one right-special factor un of length n and as
|Fn+1(s)| − |Fn(s)|=|A| −1,unA⊂F(s).
As clearly sis recurrent it satisfies the definition of Arnoux–Rauzy sequences and as its prefixes are left-special it is a standard one.
Added in proof. The result of [8] appeared under other form in S. Gan, Sturmian sequences and the lexicographic world,Proc. Amer. Math. Soc. 129(2001) 1453- 1457. It is also related to a common work of Zamboni (private communication), Kruger, Schmeling and Winkler.
References
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388 J. JUSTIN AND G. PIRILLO
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Communicated by Ch. Choffrut.
Received April, 2002. Accepted November, 2002.
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