CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Theoretical Computer Science 255 (2001) 323–344 www.elsevier.com/locate/tcs Substitutions andinterval exchange transformations of rotation class Luis-Miguel Lopez a, Philippe Narbel b;∗ a IGM, Universiteà Marne-La-Vallee.à 2, rue de la Butte Verte, 93166 Noisy-le-Grand, France b LABRI, Universiteà Bordeaux I. 351, CNRS URA 1304, Cours de la Liberationà 33405 Talence, France ReceivedSeptember 1998; revisedJune 1999 Communicatedby M. Nivat Abstract We propose a methodfor obtaining the symbolic orbits of interval exchange transformations of rotation class over n intervals by composing a ÿnite set of basic substitutions, i.e. by doing simple parallel rewriting. Basedon surface theory, this methodis shown to be closely relatedto Rauzy induction. Sturmian objects are known to correspond to interval exchange transformations over 2 intervals. In this respect, our n intervals case is shown to be also relatedto continued fractions andthe obtainedwordshave also linear complexity. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Substitutions; Symbolic dynamics; Interval exchanges; Continued fractions; Sturmian words; Dehn twists 1. Introduction Substitutions are simple parallel rewriting processes on words which are usual endo- morphisms when appliedto ÿnite words.A substitution  is deÿned by the image words of each single involvedletter. For instance, over the alphabet {a; b}, let Â(a)=ab and Â(b)=a, then Â(aba)=Â(a)Â(b)Â(a)=abaab. Studying iterations of such processes has been extensively done in formal language theory, mostly under the name of D0L- systems (see for instance [27, 28]), as well as in symbolic dynamics [9, 24, 14]. Interval exchange transformations are simple piecewise isometric maps acting on an interval of the real line, say [0; 1), whose e@ect is to permute a ÿnite number of semi-open subintervals which makes a partition of it. For instance, let a ∈ (0; 1), This paper writing has been endedwhilethe ÿrst author was invitedat the Tokyo Institute of Technology. ∗ Corresponding author. E-mail addresses: [email protected] (L.-M. Lopez), [email protected] (P. Narbel). 0304-3975/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S0304-3975(99)00290-X 324 L.-M. Lopez, P. Narbel / Theoretical Computer Science 255 (2001) 323–344 and T(x)=x +(1− a)ifx ∈ [0;a);T(x)=x − a if x ∈ [a; 1): this deÿnes an interval exchange transformation over 2 intervals. Such a transformation is fully characterized by the length of the involvedsubintervals andthe permutation which shuFes them. When iterated, interval exchange transformations lead to generic examples of dynamical systems (see for instance [11, 30, 31, 21, 18]). Iterations of an interval exchange transformation can easily be transformedinto sym- bolic information following a traditional operation in dynamical system theory (see for instance [11]): one assigns a di@erent letter to each of the intervals so that the iterates of the interval exchange transformation, the orbits, are translatedinto words. The relationship between substitutions andinterval exchange transformations over 2 intervals, hence over alphabets of 2 letters has been already extensively deciphered, mostly under the name of Sturmian words and Sturmian substitutions (see [22, 7, 3, 4]). The main result we prove here is about a set of interval exchange transforma- tions over n intervals. This set consists of the interval exchange of rotation class (see [31, 23]) which are those which have at most two discontinuities, and for which unique ergodicity is ensured. Of course these include all the interval exchanges over 2 and3 intervals, but also many over n-intervals where n¿4. The theorem we prove here is the following: Theorem. Consider an irreducible and irrational interval exchange transformation of rotation class over n intervals, n¿2(resp. n =2). The set of its symbolicorbits can be constructively described from compositions of an explicit set of n +1 substitutions (resp. 2 substitutions) over an alphabet of n letters. The proof is basedon the so-called Rauzy induction (see mainly [26, 31, 13, 33, 23]), which is here geometrically interpretedon surfaces with a structure of non-crossing parallel curves, i.e. singular foliations. It comes out that the considered foliations can be summedup by interval exchange transformations, deÿnedby ÿrst return maps describing how curves hit some ÿnite transverse segments (see e.g. [18, p.199]). The shorter the segments, the longer the pieces of leaves between successive hits of the transverse segments. In this respect, Rauzy induction can be seen as a mecanism for systematically shrinking a set of transverse segments, andtherefore lengthening the pieces of leaves between hits (see e.g. [33]). Using the symbolic translation of the orbits of an interval exchange transformation, the idea of the proof is that this lengthening can be interpretedas the application of substitutions. Since Sturmian words are the symbolic translation of the case of interval exchange transformations over 2 intervals, we show next, ÿrst, how some of the properties of these wordsare recoveredfrom our framework, andsecondhow these properties hold in the general case over n intervals: • The complexity of an inÿnite word w is deÿned as a map N∗ → N which gives, for each m ∈ N∗ the number of subwords of length m that occur in w (see e.g. [1, 17]). This is an usual measure for the structure of inÿnite words. Sturmian words are known to have linear complexity m + 1 [22, 7, 4]. Symbolic orbits of interval ex- L.-M. Lopez, P. Narbel / Theoretical Computer Science 255 (2001) 323–344 325 change transformations over n intervals can be provedto have linear complexity (n − 1)m + 1, andtherefore: Proposition. All the inÿnite words generated by the above theorem have complexity (n − 1)m +1. • Sturmian wordsare known to be closely relatedto the classical continuedfrac- tion algorithm (see e.g. [19, 10, 3, 4]). The basic process to generate composition of substitutions in the theorem, i.e. Rauzy induction, can be seen to be this classical algorithm in the 2 intervals case, andgives a multidimensionalcontinuedfraction algorithm [25, 13, 33, 23] in the general case. We discuss the convergence property of this generalization: Proposition. For every interval exchange transformation of rotation class over n inter- vals, n¿2; Rauzy induction leads to a weakly convergent multidimensional continued fraction algorithm. We also show how this multidimensional continued fraction algorithm ÿts into the general framework of Szekeres [29, 5]. • Sturmian substitutions are those which leave stable Sturmian words. They can be generatedby sets of essentially two substitutions. We show how our framework allows one to recover one of the results about them: Proposition. Consider an interval exchange transformation over 2 intervals. Then the two substitutions given by the above theorem are the Sturmian basicones used in [10] to relate Sturmian words and classical continued fractions. 2. Interval exchange transformations Let =( 1;:::; n)beann-dimensional positive vector (n¿2) such that j j =1, calleda length vector, andlet be a permutation of {1;:::;n}.Aninterval ex- change transformation (see e.g. [31, 21, 18]) is a function T :[0; 1) → [0; 1) whose ; domain is decomposed according to b = 0 and b = i , for i =1;:::;n, i.e. as 0 i j=1 j n I where I =[b ;b), andwhose range is decomposedaccordingto the length i=1 i i i−1 i i n vector ( −1(1);:::; −1(n)) with b0 = 0 and bi = j=1 −1( j), i.e. as i=1 Ji where Ji =[bi−1;bi ). The complete expression of T ; is then given as T ; (x)=x − bi−1 + b(i)−1 for all x ∈ Ii;i=1;:::;n. One can see that the intervals Ji’s are just the intervals Ii’s gluedtogether accordingto the permutation , where the interval Ii which is in the ith place, is sent to the (i)th place. An interval exchange transformation is saidto be irreducible when its permutation does not ÿx (setwise) any strict subset {1;:::;k}⊂{1;:::;n}. The simplest non- trivial example of an irreducible T ; is given by a decomposition of [0; 1) into two 326 L.-M. Lopez, P. Narbel / Theoretical Computer Science 255 (2001) 323–344 Fig. 1. intervals (see Fig. 1(i)): =(a; 1−a), where 0¡a¡1, and (1)=2;(2) = 1. Hence, b0 =0;b1 = a; b2 =1;I1 =[0;a);I2 =[a; 1), and T ; (x)=x+(1−a)onI1;T ; (x)=x − a on I2. Another interval exchange over 3 intervals is shown in Fig. 1(ii) where 1 7 1 (1)=3;(2)=2;(3) = 1, and =(6 ; 12 ; 4 ). An interval exchange transformation is saidto be of rotation class i@ it has either one or two discontinuities [23]. For instance, over 4 intervals, if is deÿned as (1)=4;(2)=3;(3)=1;(4)=2, then T ; (x) has only two discontinuities. Let the positive orbit (resp. orbit) of a point x ∈ [0; 1) be O+(x)={T i (x);i∈ N} ; i n−1 (resp. O(x)={T ; (x);i∈ Z}), andlet I be [0; 1)\ i=1 O(bi). Then the pair (I; T ; )isadynamical system, i.e. a pair (X; T) such that X is a metric space and T : X → X is continuous. Such a system is saidto be minimal i@ for Y ⊂ X; Closure (T(Y )) = Y implies Y = X or Y = ∅. The system (I;T ; ) is minimal i@ for each x ∈ [0; 1), the orbit O(x) is dense in [0; 1) [11]. Moreover, if T ; is irreducible and irrational, i.e. the only rational relations between the i’s are multiples of 1 + ···+ n = 1, then T ; is minimal [11].