STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY

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1 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI ABSTRACT. In this paper we describe a ¾-dimensional generalization of the Euclidean algorithm which stems from the dynamics of -interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of É Résumé Dans cet article nous décrivons une généralisation à la dimension ¾ de l algorithme d Euclide, qui provient de la dynamique des échanges de intervalles. Nous examinons diverses propriétés diophantiennes de cet algorithme, en particulier la qualité de l approximation simultanée. Nous montrons qu il vérifie un théorème de type Lagrange : l algorithme est ultimement périodique si et seulement si les paramètres sont dans la même extension quadratique de É. 1. INTRODUCTION A fundamental problem in arithmetic concerns the extent to which an irrational number is approximated (in a suitable sense) by rational numbers ÔÕ. Such questions are intimately related to the underlying algebraic nature of the parameter The problem of minimizing the quantity Õ Ô leads naturally to the regular continued fraction expansion of : Ò Ò ¾ Ò Ò The expansion is obtained by iterating the Gauss map Ë µ µ given by Ë Üµ Ü where Ü denotes the fractional part of Ü. The transformation Ë may be iterated indefinitely Ë µ for all µ if and only if ¾ µ is irrational. This leads to the continued fraction expression for given above. The positive integers Ò µ are called the partial quotients. By truncating the expansion of at the th level one obtains a rational number Ô Õ 1991 Mathematics Subject Classification. Primary: 11J70, 11J13. Secondary: 37A05. Key words and phrases. Generalized continued fractions, interval exchange transformations, fractions continues généralisées, échanges d intervalles. 1

2 2 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI (called the th convergent of µ which gives the best rational approximation of in the following sense: and Õ Õ Ô Õ Õ for all Õ Õ where denotes the difference (taken positively) between and the nearest integer. This best approximation property of the convergents, coupled with the entirely arbitrary nature of the sequence of partial quotients is what distinguishes the classical continued fraction algorithm from all other known continued fraction type algorithms. The following theorem, due to Lagrange [27], constitutes a fundamental result in the theory of regular continued fractions: Theorem 1.1 (Lagrange (1769)). The partial quotients in the continued fraction expansion of an irrational number are ultimately periodic if and only if is algebraic of degree ¾. The problem of simultaneously approximating an n-tuple of real numbers ¾ Ò µ by rational numbers Ô Õ Ô ¾ Õ Ô Ò Õµ (with the same denominator) has been and continues to be an important area of investigation with a wide range of applications to different areas of mathematics. The question dates back to Hermite in [21] where he suggests finding a generalization of the continued fraction algorithm which reflects the algebraic nature of the parameter(s). As a response to this problem Jacobi developed a special case of what is now called the Jacobi-Perron algorithm (see [6, 32, 37]). Since then, a number of other multidimensional division algorithms have been studied including [2, 5, 8, 10, 18, 19, 24, 25, 26, 28, 29, 33, 43, 44] to name just a few. (See [9, 38, 39] for nice surveys on multidimensional continued fractions). It is known that for each n-tuple of irrationals ¾ Ò µ the system of inequalities Ô Õ Õ Ò for ¾ Ò has infinitely many solutions [20]. Moreover the exponent is optimal in the sense that for any Ò there exist Ò ¾ Ò µ for which the system of inequalities Ô Õ Õ has only finitely many solutions. However, unlike the 1-dimensional case where the convergents of the regular continued fraction yield the best rational approximations in the sense mentioned above, in dimension greater than one most of the usual continued fraction algorithms stop short of producing optimal simultaneous approximations of ¾ Ò µ Moreover it is not known whether many of the higher dimensional division algorithms cited above (including Jacobi-Perron) satisfy a full Lagrange type theorem : The algorithm is ultimately periodic if and only if all the parameters lie in the same algebraic extension of É Generally one can prove the only if part while the if part is only conjectural. There are a few exceptions: One is a geometric construction due to Korkina in [25] based on ideas originally due to Klein [24] and later modified by Arnold [2]. Another is a 2-dimensional multiplicative algorithm due to Hara-Mimachi and Ito in [19]; still another is a recent Ò-dimensional algorithm due to Garrity in [18], where the author considers purely periodic expansions of period length one. A fourth example concerns a recent paper of Boshernitzan and Carroll [8] in which they show that a certain family of vectorial division algorithms, when applied to quadratic vector spaces, yields sequences of remainders which are ultimately periodic. However, in this case the converse is false: periodicity does not imply that the parameters are quadratic. The division algorithm of Boshernitzan-Carroll is based on a renormalization process associated with interval exchange transformations.

3 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 3 In this paper we describe a new 2-dimensional division algorithm we call the negative slope algorithm which stems from the dynamics of a three interval exchange transformation on We consider a generalization of the Gauss map given by the following transformation µ µ µ µ defined by Ü Ýµ Ò Ò Ý Ü Ýµ Ý Ü Ýµ Ó Ó Ò Ò Ü Ü Ýµ Ü Ü Ýµ Ó Ó if Ü Ýµ if Ü Ýµ where again Ü denotes the fractional part of Ü. The map may be iterated indefinitely (the algorithm does not stop) on a pair Þ «µ if and only if Þ lies off a special set of rational lines with negative slopes (see Proposition 2.3). Thus if «and are not both rational, then may be iterated indefinitely on either «µ or «µ or both. Iterating on a point «µ leads to an expansion of the form Ò Ñ µ where and Ò Ñ are positive integers. The quantity records which of the two defining rules for is used at stage of the iteration, while Ò and Ñ record the integer parts in each coordinate. The sequence Ò Ñ µ is analogous to the sequence of partial quotients in the regular continued fraction algorithm. Geometrically this iteration produces a sequence of nested quadrilaterals É in the plane converging to the point «µ At stage of the iteration, the quadrilateral É is partitioned according to a family of quadrilaterals whose main diagonal has slope, and whose boundary is made up of lines of negative rational slope. This partition into quadrilaterals is constructed via a twodimensional Farey series derived from the coordinates of the vertices of É This geometric interpretation is analogous to the partitioning of the unit interval according to Ñ-Farey fractions. In this paper we study various diophantine properties of this algorithm including its approximation qualities. It is convenient to make a change of coordinates «µ ««µ As in the regular continued fraction algorithm, truncating the iteration of at stage gives rise to a pair of rational numbers Ô Õ Ö Õ µ with the same denominator, and with Ô Õ approximating «and Ö Õ approximating «Geometrically Ô Õ Ö Õ µ corresponds to the barycenter of the Farey quadrilateral É We show that «µ Ô Õ Õ ¾ for infinitely many In this privileged direction, we obtain a so-called semi-regular continued fraction giving an approximation which is about as good as the regular continued fraction. But it is shown that the approximation of «depends on the values of both «and ; and hence the negative slope algorithm is not a skew product of the regular continued fraction, or for that matter other higher-dimensional continued fractions found in the literature. In contrast the approximation of «is only linear in Õ. As a consequence of the approximation qualities mentioned above, we show that the transformation satisfies a full Lagrange type theorem: The iteration «µ (or equivalently the expansion Ñ Ò µµ is ultimately periodic if and only if «and belong to the same quadratic extension of É The negative slope algorithm is intimately connected with the dynamics of -interval exchange transformations; in fact, it may be independently reformulated in terms of the dynamical and symbolic properties of -interval exchange transformations. Though these connections are not necessary to the understanding of the present paper, the last section is devoted to a short presentation of these ideas. In the sequels [16, 17] we use the symbolic/combinatorial/arithmetic interaction to

4 4 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI solve several long standing problems on the symbolic, spectral and ergodic properties of -interval exchanges. In particular, in [17] the diophantine properties developed in the present paper are used to characterize possible eigenvalues of a -interval exchange and to obtain necessary and sufficient conditions for weak mixing, solving two questions posed by Veech in [42]. In a forthcoming fourth paper we apply our methods to study the joinings of -interval exchanges. ACKNOWLEDGEMENTS The authors were supported in part by a joint NSF/CNRS Cooperative Research Grant between the U.S. and France. The third author is also partially supported by a grant from the Texas Advanced Research Program. We are very grateful to V. Berthé and A. Fisher for many fruitful conversations. We also wish to thank the referee for his many useful comments and suggestions. 2. THE NEGATIVE SLOPE ALGORITHM Let Á denote the interval. We assume that «µ ¾ Á Á. We define by recursion a sequence «Æ µ à with à finite or Ã, with For set «and Æ min«æ «µ If either «or Æ is equal to then the algorithm stops. In this case we take à Otherwise if Æ, put while if Æ, put ««and ««and It is readily verified that «and Æ, if they are defined, satisfy the above inequalities. Having defined «Æ µ with the required properties, we write (2.1) where Ò Ñ are each positive integers. Set (2.2) «Ò Æ «with «Æ Ñ Æ with Æ Æ Æ «µ If either «or Æ is equal to the algorithm stops and we set à Otherwise if Æ, put (2.3) while if Æ, put (2.4) «and ««Æ and Æ «It is readily verified that «and Æ satisfy the required inequalities. The above arithmetic algorithm determines a sequence (2.5) «µ Ò Ñ µ Ã

5 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 5 we call the negative slope expansion of «µ where sgn Æ µ We put à It also defines the sequence «Æ Æ µ à with values in Á Á where for we take Æ «and «As we shall see in what follows, the sequence «Æ Æ µ à plays an important role while «Æ µ à is mainly an auxiliary sequence. A geometric interpretation of this algorithm is given in the last section of the paper in connection with interval exchange transformations. The referee suggested the following alternative geometric description: at stage we consider the interval Æ and the two subintervals and Æ «Æ, of respective lengths and «. Except in the cases in which «or or «Æ (in which case the process stops), this naturally defines a partition of Æ into three subintervals. The middle interval is of length Æ If the first and third intervals do not intersect, i.e., Æ, then we replace their lengths by the lengths of the complementary intervals, otherwise we keep the lengths «and This defines the auxiliary quantities «µ Finally we subtract the length of the middle interval as many times as possible from the lengths of the first and last intervals (or their complementaries if Æ µ by construction we can always make the subtraction at least once. What is left over from the first interval has length «and what is left over from the third interval has length After renormalization, we recover the formula for the map given in the introduction. For each à we define the quantities Ë «««Æ Æ «Æ The following inequalities are easily verified: Æ (2.6) (2.7) (2.8) Ë ¾Æ and hence ¾ Æ Æ and hence Ë Æ Æ and hence Lemma 2.1. For each à we have «µ «Æ Æ µ where Á Á Á Á is defined by Ü Ýµ Ò Ò Ý Ü Ýµ Ý Ü Ýµ where denotes the fractional part of Ó Ó Ò Ò Ü Ü Ýµ Ü Ü Ýµ Ó Ó if Ü Ýµ if Ü Ýµ A geometric motivation for the name given to this algorithm is described in the following section.

6 6 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI Proof. We will show that for each à equation (2.2) we have we have «Æ Æ µ «Æ Æ µ By If Æ then (2.1) and (2.3) give Æ Æ «Æ Æ «Ò Æ «with «Æ «Ñ Æ with Æ whence Thus and similarly «Æ Æ Æ «Æ Ò «Æ Ñ Æ Æ Æ Æ Æ «Æ Æ µ Æ «Æ «Æ Æ µ as required. On the other hand if Æ then (2.1) and (2.4) yield whence and similarly «Æ Ò Æ «with «Æ Æ «Ñ Æ with Æ «Æ Æ Æ Æ Æ Æ Æ Æ «Æ Æ µ «Æ «Æ Æ µ The lemma shows that the negative slope expansion «µ in (2.5) may be obtained directly from the sequence «µµ à «Æ Æ µ The quantity codes which of the two defining rules for is used at stage of the iteration, while Ò and Ñ record the integer parts in each coordinate.

7 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 7 Lemma 2.2. For each à set Then (2.9) Ñ Ò Ò Ñ µ Ñ Ò Ë Æ Ë Æ Proof. We express the quantities «and Æ in terms of «and Æ. If then (2.1), (2.2) and (2.4) imply whence «Æ Ò Æ «Æ «Ñ Æ Æ Æ «µ (2.10) and «Ò µæ «Ñ µæ (2.11) Æ Æ «µ Ò Ñ µæ «µ If then (2.1), (2.2) and (2.3) imply and «Ò Æ ««Ñ Æ Æ Æ «µ Ò Ñ µæ «µ The lemma now follows immediately from the above expressions. We next look at the improper case when the algorithm stops (this situation is called Stőrungen by Perron and others). Proposition 2.3. The algorithm stops (à ) if and only if «µ lies in the intersection of Á Á and one of the following rational lines: Õ«Õ Ô Õ«Õ µ Ô Õ µ«õ Ô for any integers Ô ¾Õ. Proof. The algorithm stops if and only if Æ Ã or «Ã or à the triple Ë Ã Ã Æ Ã µ satisfies one of the relations, hence if and only if

8 8 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI Ë Ã Æ Ã Ë Ã Ã Ë Ã Ã Suppose Ë Ã Ã with ; then, by applying the matrix formula in Lemma 2.2, we get that for each à there exist integers and such that (2.12) Ë Æ where We note that since «and are nonnegative, and can each be assumed to be nonnegative. By writing each of Ë and Æ in terms of Ë Æ (using Lemma 2.2), we find that Ñ Ò µ Ò Ñ µ and Looking at the possible parities of Ñ and Ò, we see by induction on that is an odd positive integer while is a nonnegative even integer. Moreover since we obtain Æ Ë µë Ë µæ µë Ë Æ µë Ë Æ µë Æ ¾ µæ Æ where the last two inequalities follow from (2.8) and (2.6) respectively. Hence and thus (2.13) ¾ µ ¾ µ ¾ µ ¾ ¾ Equation (2.12) when gives a linear relation of the form Õ«Õ µ Ô or Õ µ«õ Ô with Ô ¾Õ In fact, Õ and Ô ¾ ¾ Conversely, if «and satisfy such a relation, we can write it in the form Ë Æ, where is an odd positive integer, is a nonnegative even integer, and. Suppose for we have Ë Æ, where is an odd positive integer, is a nonnegative even integer, and If then we must have as Ë, and hence the algorithm stops. On the other hand if, then we apply the matrix formula in Lemma 2.2 as above to get Ë Æ, with the same parity properties, and with Since ¾ by (2.13), we obtain that Hence by this process we obtain a à for which either à or à If à we saw that the algorithm stops. While if à either «Ã Ã Æ Ã or Ã Ã Æ Ã, and in either case this implies à since à is even and «Ã and are each less than Æ Ã Ã Thus in all cases the algorithm stops.

9 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 9 A similar reasoning takes care of the case Ë Ã Æ Ã, which corresponds to the rational values of «and relations Ë Æ Lemma 2.4. If the algorithm does not stop (Ã ), then Ò µ µ for infinitely many and Ñ µ µ for infinitely many. Proof. Suppose the algorithm does not stop and Ò µ µ for ; then it follows from (2.10) that ««for some constant and for all ; but (2.11) implies that Æ Æ «Since Æ it follows that a contradiction. In terms of -interval exchange transformations, Proposition 2.3 gives necessary and sufficient conditions for the corresponding -interval exchange to have periodic orbits, while Lemma 2.4 implies that in the absence of periodic orbits the exchange is minimal (each point has a dense orbit). 3. FAREY QUADRILATERALS We begin by recalling a well-known fact about regular continued fractions. For an arbitrary sequence of positive integers Ò µ the set Á of numbers in µ whose first partial quotients are Ò Ò ¾ Ò is a half-open interval having endpoints with Indeed, the Á are given by, and inductively, Ò µ Á Ò Ò µ Ò if is odd (3.1) Ò if is even. Ò Ò µ Ò µ The endpoints of each interval are successive Farey approximations of the numbers in that interval. For a sequence Ò Ñ µ with Ò Ñ and ¾ we shall formulate a similar description for the set É of points «µ ¾ Á Á having Ò Ñ µ ¾ Ò ¾ Ñ ¾ µ Ò Ñ µ as their first negative slope coefficients. It is useful to regard «Æ «and as functions of the variables «µ defined inductively by setting ««Æ and and, for all (3.2) Æ Æ «if «Æ if «if Æ «if ««Ò Æ Ñ Æ We record a few simple facts, the proofs of which are elementary. Lemma 3.1. If «µ begins with Ò Ñ µ Æ then the values at «µ of the functions ««Æ for Æ are exactly those of their constant namesakes in the negative slope algorithm of Section 2.

10 10 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI «Æ Æ ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ Æ ««ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ Æ Æ «Æ FIGURE 1. The quadrilateral É The two configurations correspond to the possible values of ¾ Lemma 3.2. The functions «Æ are of the form «Ô «µ «Ô «µ Ô µ µ«ô µ Æ where ¾ and Ô «µ Õ «µ Ô µ Ô Æµ ÑÜ Ô «µ Ô µ µ and Õ Æµ ÑÜ Õ «µ µ Õ «µ Õ µ Ô Æµ «Ô Ƶ Õ Æµ Õ µ Ô Æµ Õ Æµ Õ µ µ are nonnegative integers such that Lemma 3.3. The lines «and «Æ have negative slope while the lines and Æ have slope and the line Æ has slope The intersection of the lines «and Æ lies on the line Æ as does the intersection of the lines and «Æ Corollary 3.4. The four lines ««Æ and Æ bound a quadrilateral Geometric formulation. We claim that É is the semi-open ¾ quadrilateral of Corollary 3.4, excluding the lines «Æ and Æ and that the positions of the bounding lines are as shown in Figure 1. This holds trivially in the case Assume the assertion holds for some It follows from Lemma 3.1 that É is the subset of É on which (3.3) Ò Æ «Ò µ Æ Ñ Æ Ñ µ Æ ¾ If Ò or Ñ is then we must also exclude the lines «and from É in which case É is actually an open quadrilateral. This is (partly) due to our convention of stopping the negative slope algorithm when «or

11 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 11 Ä «Æ ÄÄÄÄ Ä ÄÄÄÄ «Ä Ò Æ ÄÄÄÄ Ä ÄÄÄÄ «Ä Ò µæ ÄÄÄÄ Æ Ä ÄÄÄÄ Ä ÄÄÄÄ Ä ÄÄÄÄ Ä ÄÄÄÄ Ä ÄÄÄÄ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ Æ «Ä ÄÄÄÄ Ñ Æ Ä Ñ µæ FIGURE 2. The position of É in É when and The horizontal aspect is exaggerated to exhibit greater detail. The lines «Ò µæ «Ò Æ Ñ µæ Ñ Æ are the lines ««Æ Æ respectively, which bound the quadrilateral É We first consider the case Equations (3.2) give «Æ «and Æ whence inequalities (3.3) may be rewritten as Ò µæ «Ò Æ Ñ µæ Ñ Æ Each of the lines «ÒÆ Ò contains the intersection of the lines Æ and «and their slopes strictly decrease to as Ò Similarly, each of the lines ÑÆ contains the intersection of the lines Æ and and their slopes (strictly) increase to as Ñ Figure 2 shows how the four lines bound É in the desired manner. Now let us assume that Using (3.2) we may rewrite the inequalities (3.3) as Ò Æ Æ «Ò µæ Æ Ñ Æ Æ Ñ µæ Æ Each of the lines «ÒÆ Æ Ò contains the intersection of the lines Æ and «Æ and their slopes (strictly) decrease to as Ò (To see this last part, use (3.2) to write the equation of the line as Ò µæ ) Similarly the slopes of the lines ÑÆ Æ (strictly) increase to and each line contains the intersection of the lines Æ and Æ Figure 3 shows that these lines bound É as claimed. This completes the induction.

12 12 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI Æ Ñ Æ Æ Ñ µæ «ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ Æ Æ Æ «Æ Ò µæ «Æ Ò Æ «FIGURE 3. Location of É in É in the case and The horizontal direction is exaggerated; the slopes of all the lines are typically quite close to The lines «Æ Ò µæ «Æ Ò Æ Æ Ñ µæ Æ Ñ Æ are the lines ««Æ Æ respectively, which bound the quadrilateral É Remark 3.5. The line Æ divides the quadrilateral É into two triangular regions, corresponding to the possible values of If then É is contained in the triangle bounded by Æ «Æ and Æ while if then É is contained in the triangle bounded by Æ «and 3.2. Arithmetic formulation. We shall give a formula similar in flavor to (3.1) for the vertices of É First we need a specialized result. Lemma 3.6. Let ¾ ¾ ¾ be nonnegative (or nonpositive)integers such that and are not both zero, ¾ Suppose that the lines Ü Ýµ Ü Ý are mutually nonparallel, that the slope of lies between the slopes of and ¾ and that each intersection lies in the closure of the first quadrant. Let Ò Ñ and set Ò Ò Ò ¾ Ñ ¾ Ñ ¾ Ñ Ü Ýµ Ü Ý Ü Ýµ Ü Ý Then (3.4)

13 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 13 and ¾ ¾ Ò ¾ ¾ Ñ ¾ ¾ Ò ¾ ¾ Ñ ¾ ¾ Ò ¾ ¾ Ñ Ò Ó whence ¾ ¾ Ò ¾ ¾ Ñ ¾ ¾ Ò ¾ ¾ Ñ ¾ ¾ Ò ¾ ¾ Ñ ¾ ¾ Ò ¾ ¾ Ñ Note that formula (3.4) for the intersection gives the same numerators and denominator as the last line of the lemma Equations for bounding lines. We must be careful how we write our lines if we wish to apply Lemma 3.6. We write the bounding lines of É as ««Æ and Æ while the diagonal is written Æ The coefficients of «and in each expression are given by Lemma 3.2. For example, the line «Æ becomes Ô Æµ Ô µ µ«ô Ƶ Ô µ µ Õ Æµ µ Õ µ µ and the coefficient of «is Ô Æµ Ô µ µ Labeling vertices. The vertices of É are the intersections of one of the lines «and «Æ with one of the lines and Æ Let È be the point of intersection of the lines «and Æ and label the remaining vertices È ¾ È and È in a clockwise manner. Let Ô È Ö (3.5) Õ Õ be the vertices of É as obtained from (3.4) using the coefficients of and without reducing the fractions. We see from Figure 1 that if then È and È are the lower right and upper left vertices of É respectively, and if then these roles are reversed. Remark 3.7. Two of the vertices could be obtained using formula (3.4) in other ways, as the intersection of Æ with any one of the four lines bounding É is either È or È However, it follows from the first equation of (3.2) that formula (3.4) gives the same expression for the vertices in each case Application of Lemma 3.6. We are finally ready to show how the lemma may be used to find the vertices of É from those of É The details depend on Let ¾ and be the lines «and Æ respectively. These (with the coefficients of 3.2.1) satisfy the hypotheses of Lemma 3.6. The vertices of É are the intersections of the bounding lines «ÒÆ and ÑÆ Ò ¾ Ò Ô ¾ See Ò Ñ ¾ Ñ Ñ given by Lemma 3.6 as ÒÔ ÑÔ Ö¾ ÒÖ Õ ÒÕ ¾ ÑÖ ÑÕ Õ ¾ ÒÕ ÑÕ Figure 4. Let ¾ and be the lines «Æ Æ and Æ respectively. As before, these satisfy the hypotheses of Lemma 3.6. The vertices of É are the intersections of the lines «Æ ÒÆ and Æ ÑÆ Ò ¾ Ò Ò Ñ ¾ Ñ Ñ given by Lemma 3.6 as Ô ÒÔ ÑÔ Õ ÒÕ ÑÕ Ö ÒÖ ÑÖ Õ ÒÕ ÑÕ

14 14 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI Ô Õ Ö Õ Æ «ÒÆ Ô ¾ ÒÔ ÑÔ Õ ÒÕ ¾ ÑÕ Ô ¾ Ö¾ Õ ¾ «Õ ¾ Ö¾ ÒÖ ÑÖ Õ ÒÕ ¾ ÑÕ ÑÆ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À FIGURE 4. Formula for a vertex of É in the case The other cases are similar. Ô Õ Ö Õ Note that the expressions for the bounding lines are the same as those of in the first case, while in the second case they differ from by sign. In either case, the expressions obtained for the vertices of É are identical to those given by This is important because it means we can iterate the process Algorithm. Assembling our observations thus far leads to the following recursive formula for the vertices of the É Put Ô Õ Ö Ô¾ Õ¾ Ö Ô Õ Ö Ô Õ Ö These are the numerators and denominators from in the case We see from Subsection that if then Ô Õ Ö µ Ô¾ Õ ¾ Ö ¾ µ Ò µ Ô Õ Ö µ Ñ Ô Õ Ö µ Ô ¾ Õ ¾ Ö ¾ µ Ô¾ Õ ¾ Ö ¾ µ Ò µ Ô Õ Ö µ Ñ µ Ô Õ Ö µ Ô Õ Ö µ Ô¾ Õ¾ Ö¾ µ Ò Ô Õ Ö µ Ñ µ Ô Õ Ö µ Ô Õ Ö µ Ô¾ Õ¾ Ö¾ µ Ò Ô Õ Ö µ Ñ Ô Õ Ö µ

15 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 15 while if then Ô Õ Ö µ Ô Õ Ö µ Ò µ Ô Õ Ö µ Ñ Ô Õ Ö µ Ô ¾ Õ ¾ Ö ¾ µ Ô Õ Ö µ Ò µ Ô Õ Ö µ Ñ µ Ô Õ Ö µ Ô Õ Ö µ Ô Õ Ö µ Ò Ô Õ Ö µ Ñ µ Ô Õ Ö µ Ô Õ Ö µ Ô Õ Ö µ Ò Ô Õ Ö µ Ñ Ô Õ Ö µ One easily verifies Lemma 3.8. For each and In summary we have proved Ô Õ Ö µ Ô Õ Ö µ µ Ô Õ Ö µ Ô Õ Ö µ Ô¾ Õ¾ Ö¾ µ Ô Õ Ö µ Proposition 3.9. The set of points whose negative slope expansion begins with a given finite sequence Ò Ñ µ Ò Ñ µ is a nonempty semi-open (or open, if ¾ Ò Ñ ) quadrilateral É as shown in Figure 1. The vertices of É are given by Algorithm 3.3. Now we can prove a converse of Lemma 2.4: Corollary If Ò Ñ µ is such that Ò µ µ for infinitely many and Ñ µ µ for infinitely many then there is a unique point «µ ¾ Á Á with negative slope expansion Ò Ñ µ Proof. For ¾ let É be the quadrilateral determined by Ò Ñ µ Ò Ñ µ Obviously, É É holds for all We just need to show that ÔÉ consists of a single point. The geometry of É and Lemma 3.8 together imply Ñ É µ ¾Õ It is evident from our algorithm 3.3 that Õ increases as increases, whence the intersection É contains at most one point. An easy geometric proof shows that if we have two indices such that Ò µ µ for two distinct ¾ µ and Ñ µ µ for two distinct ¾ µ then É É It follows from this that É Allowing the coefficients Ò Ñ µ Ò Ñ µ to vary yields a collection É of disjoint quadrilaterals. The union of É is the set of initial points for which the negative slope algorithm does not terminate within the first steps. It follows from Lemma 2.1 that if «µ Ò Ñ µ then «µµ Ò Ñ µ This, together with Corollary 3.10 gives Corollary For ¾ and É ¾ É the map of Lemma 2.1 restricts to a homeomorphism É Éµ ¾ É Remark A similar statement holds for Let É ¾ É If É is open then É is a homeomorphism É µ µ and if É is semi-open then É is a homeomorphism É Á Á We thus have an alternate characterization for the quadrilaterals:

16 16 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI ¾ ¾ ¾ Â Ë É À ÉÉÉ ÀÀÀÀ Ë Â Â ËË Ë Â É À ÉÉÉ ÀÀÀÀ Ë Â ÂË Ë Â Â ËË É À Ë Â Ë ÀÀÀÀ  ÂË ÉÉÉ Ë Â Â ËË Ë ÂÀ É ÀÀÀÀ ÉÉÉ Ë Â ÂË É Ë Â ÂÂËË É À Ë É ÀÀÀÀ É É Ë ÉÉÉ Â Â Ë É À Ë Â É ÂÂËË É ÀÀÀ É Ë Â À Ë Â ÉÉÉ À É Â Ë À É Â À É ÂÂ Ë Â À ÂÂËË Ë Â É À É ÉÉÉ À É Â À É Ë Â À Â Ë Ë À É ÂÂ Ë É Â À ÂÂ É É ËË Ë Â À É À Ë Â À É Ë É ÂÀ Ë É É Â Ë Ë Â Â À É É É ÂÂ Ë À ËË À É É Ë Ë Â À É ÂÉ Ë Â Â Ë À É À Ë Â Â ËË À É É Ë ÂÂ É Ë À É É Ë Ë Â ÀÉ É À Ë Â Â Ë À É Â Ë É Ë ÀÉ ÂÂ É É Ë Ë Â Â Ë À É É Ë ÀÉ Ë Â ¾ ¾ ¾ FIGURE 5. The unit square É is divided into quadrilaterals by the lines lines «ÒÆ and ÑÆ Ò Ñ ¾ The figure shows these lines for Ò Ñ Also shown are the first few lines subdividing the lower-leftmost quadrilateral (É when Ò Ñ ). Corollary The interior of each quadrilateral of É is a maximal domain of continuity for Figure 5 illustrates the quadrilaterals which constitute the natural Markov partition into the sets of continuity of the map ; each of them is in one-to-one correspondence with the entire set É As was pointed out by the referee, this explains why there is no restriction of finite type on the possible negative slope expansions, which is not the case, for example, for the Jacobi-Perron algorithm. The figure also shows that µ is an indifferent fixed point (the map can be extended there by continuity), which explains why the sequence µ plays such a special role. There are other fixed points on the boundary, corresponding to forbidden infinite sequences ѵ or Ò µ, but they are not indifferent. 4. SIMULTANEOUS RATIONAL APPROXIMATIONS For the regular continued fraction approximation of an irrational «, there are general bounds on the distance between «and its convergents, and precise estimates of the quality of approximation in terms of the partial quotients. We explore analogous results for the negative slope algorithm; it is convenient to make the change of coordinates «µ ««µ.

17 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 17 Proposition 4.1. For à set Å ¾ Then Ô Ô Ô Å Ö Ö Ö Õ Õ Õ where Ô, Õ and Ö are integers defined recursively by the relations Ô Ñ Ò µô Ô Õ Ñ Ò µõ Õ Ö Ñ Ò µö Ö Ò Ñ µ starting from Ô, Õ, Ö, Ô, Õ, Ö. Moreover detå Ô Õ Ô Õ and (4.1) Õ Õ Õ Õ Õ Õ ««Å Ë Æ Proof. We start from Æ, «, «, so the matrix formula is a straightforward consequence of Lemma 2.2. The recursive relations for Ô, Õ and Ö follow from Å and Å Å Since det for all it follows that detå which in turn implies Ô Õ Ô Õ Finally the last inequality follows from the recursive relation for Õ and the fact that Ñ and Ò for each The matrix formula (4.1) above allows us to approximate the quantities «and «(and hence «and ) by two rationals with the same denominator. Namely one approximates and by integers, which may be, or ¾ in the case of, and, or in the case of If we use the central values,, we obtain the rationals Ô Õ and Ö Õ ; other values would involve the integers Ô Ô, Õ Õ, Ö Ö, with and. The next proposition and corollary concern the rational approximation of «Proposition 4.2. Let Ò Ñ µ à be the negative slope expansion of «µ Then we have the following semi-regular continued fraction expansion of «(see [1, 26, 32]) (4.2) «Ñ Ò ¾ Ñ ¾ Ò ¾ Ñ Ò Ñ Ò

18 18 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI In case à this formulas stops with Ñ Ã, Ò Ã and à otherwise it is infinite. Moreover for à we have (4.3) where «Ô Õ Õ Õ µõ (4.4) and (4.5) Ñ Ò ¾ Ñ ¾ Ò ¾ Proof. The matrix equation (2.9) of Lemma 2.2 implies Ë Ñ Ò µæ Ë Ñ Ò µæ which in turn yields (4.4) and (4.2) since «. Using the matrix relation (4.1) of Proposition 4.1 we obtain hence «Ô Ë Ô Ô µæ Õ Ë Õ Õ µæ Ô Ô Ô Õ Õ Õ Ô «µ Ô Õ Ô Õ µ Õ Õ Õ µõ µ From Proposition 4.1 we deduce that Ô «Õ Õ Õ Õ Õ Õ Ñ Ò µ Õ Õ Ñ Ò µõ µõ Õ Õ Õ µõ where the last equality is a consequence of the recursive relation for Õ Corollary 4.3. If à and Ñ Ò µ µ then «Ô (4.6) (4.7) and (4.8) Õ Õ ¾ «Ô Õ ¾ Õ Õ «Ô Õ ¾ Õ ¾

19 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 19 If every string of consecutive for which Ñ Ò µ µ has length at most Å then for all à «Ô Å ¾ Õ Suppose the negative slope expansion of «µ does not stop. If the lengths of strings of consecutive for which Ñ Ò µ µ are unbounded, then there exists a sequence  of integers such that for ¾ Â, Õ ¾ «Ô (but then we shall see later that Ô Ô Õ Õ Õ Õ ¾ provides a better approximation). Proof. If Ñ Ò µ µ, then either, or equivalently or Ñ Ò in which case by Proposition 4.1 we have that Õ ¾Õ which in turn implies Õ Õ Õ Õ and Õ Õ ¾ In either case, (4.6) and (4.7) are readily verified using equation (4.3) of Proposition 4.2. The estimate (4.8) stems from the intermediate equality «Ô Õ Õ Õ µ Õ in the proof of Proposition 4.2. If Ñ Ô Ò Ô Ô µ µ for every Ô but Ñ Ò µ µ, then the recursive relation for Õ in Proposition 4.1 yields Õ Õ Õ Õ Õ Õ µõ µõ Õ Õ with Õ ¾Õ, by taking either or ; The fourth claim now follows from (4.3) of Proposition 4.2. Suppose now Ñ Ô Ò Ô Ô µ µ for every Ô Å Æ. Let Æ, È Õ, É Õ. Then the recursion formulas give Õ ÆÉ Æ µè, Õ Æ µé ÆÈ, while the semi-regular continued fraction expansion of begins in ¾ hence Å Å µ, where Å denotes the Å-th iterate of the function ܵ ¾ ¾ ¾ Å ¾µ Å Ü ; hence Ü By Proposition 4.2 Õ ¾ «Ô Õ Õ Õ µõ ÆÉ Æ µè É È ÆÉ Æ µè µ Å

20 20 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI which is large whenever Æ is large and Æ is small, independently of È and É The last claim now Å follows. Remark : The semi-regular expansion of «is in general not unique, but in our case it is completely determined by the negative slope algorithm; in fact the expansion depends on both «and and not only on the value of «In particular, the negative slope expansion may stop because Õ«Õ µ Ô, while «admits an infinite semi-regular expansion. If for each we have Ñ Ò ¾, that is if Ñ, Ò, and never occurs, then the semi-regular expansion of «is the nearest integer continued fraction expansion defined by Hurwitz in [22]. In this case the Ô Õ are a subsequence of the convergents of «([32], part I, chapter 5, paragraph 40). We next look at the induced rational approximation of «Proposition 4.4. Suppose Ã. Then (4.9) where (4.10) and «Proof. By Proposition 4.1 we have Define «Ö Õ µí Õ Õ Õ µõ Í Õ Õ Ñ Ò µ «Ö Ë Ö Ö µæ Õ Ë Õ Õ µæ Ö Õ µ Õ Ö Õ Ö µ Õ Õ Õ µõ Í Õ Ö Õ Ö We establish (4.10) by induction on For we have Next suppose that By Proposition 4.1 hence, if Ò Ñ, then if Ò Ñ, then as required. Í Ò Ñ Ò Ñ Õ Õ Í Õ Õ Í Í Õ Ò Ñ µ Í Í Õ Õ Õ Õ Í Ò Ñ Õ Í Ñ Ò ¾µÕ Õ Õ Õ Õ

21 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 21 Finally the last claim is a straightforward consequence of the matrix equation in Lemma 2.2. Corollary 4.5. For every, «Ö Õ Suppose the negative slope expansion of «µ does not stop. If, on a sequence  of integers, we have Ñ Ò and Ò Ñ Ñ Ò, then Ö Õ «for ¾ Â. There exist «and with an infinite expansion, and a constant, such that «Ö for every Proof. We use (4.9) of Proposition 4.4; it implies «Ö ÜÍ Õ Ý Õ Õ ÜÕ with Õ Õ Õ Õ Õ Ü Ý by the inequality (2.8). It is readily verified that this function of two variables has no extremum (except µ) inside the domain Ü Ý, hence its maximum on this domain is reached on Ü Ý If Ü Ý and Ü, then ÜÍ Õ Ý ÜÍ Õ Üµ Õ Õ ÜÕ Õ Õ where the last inequality is a consequence of (4.10) in Proposition 4.4. If Ü Ý and Ü then Õ Õ ÜÍ Õ Ý ÞÍ Õ Þµ Õ Õ ÜÕ Õ Õ ÞÕ with Þ. Again using (4.10) one verifies that the maximum of this quantity is reached either for Þ or Þ, and so is either Õ or Í Õ Õ Õ, hence at most Õ as required. The second claim follows immediately from Proposition 4.4, the matrix equation in Lemma 2.2, and the equality µ Õ Ö Õ Ö µ Õ µõ Í Õ Õ Í Õ Õ Ñ Ò µµ We can build an infinite sequence Ñ Ò µ such that the Ñ and Ò are bounded, Ò Ñ ¾ for every Ò, and and Í have opposite signs for every Ò In view of Proposition 4.4 this sequence defines «and satisfying the third claim.

22 22 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI 5. MORE DIOPHANTINE PROPERTIES In this section we express in terms of the negative slope algorithm some diophantine properties of «, and some properties of simultaneous approximation of «and «(or ). The following proposition was proved by del Junco [15] for some particular semi-regular expansions; the last part of our proof is similar to the one in del Junco s paper. Proposition 5.1. Suppose the negative slope expansion of «µ does not stop. Then the two following properties are equivalent in the negative slope expansion of «µ, the Ò Ñ are unbounded or the lengths of the strings of µ are unbounded; for every, there exist integers Ô and Õ such that «Ô Õ Õ ¾ Proof. If the Ò Ñ are unbounded, the second assertion follows from «Ô Õ Õ Ñ Ò µõ µõ Õ by putting Ô Ô, Õ Õ for suitable. If Ñ Ô Ò Ô Ô µ µ for every Ô Å and ; then, as in the proof of Corollary 4.3, we have. The relation «Ô Ô Ô Å Õ Õ Õ coming from Proposition 4.1 implies that «Ô Ô Õ Õ Õ Õ µ Õ Õ Õ µ The second assertion now follows, if Å is large enough, by taking Ô Ô Ô, Õ Õ Õ for suitable. If Ñ Ô Ò Ô Ô µ µ for every Ô Å and ; then we get ¾ Å ¾, and the second assertion follows, if Å is large enough, by taking Ô Ô Ô, Õ Õ Õ for suitable. Suppose now that «and satisfy the second assertion. For Ü, let Ü denote the distance of Ü to the nearest integer. For a given, we put Ú ¾ ¾ ¾ ¾ and apply the hypothesis on «and to get Ø ¾ for a positive integer, an integer with, and some Ú Ø Ú. Suppose that is a positive integer, is an integer with, Ø satisfies Ø and Ø ¾

23 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 23 Then either and Ø, or Ø Ø µ Ø Hence, because of the formula (4.4), which says that integer, we get ¾ Ø ¾ ¾ ¾ Ø ¾ is, up to the addition of an with,, Ø Ø, and we can continue. ¾ So this process will give some such that or, and ¾ Ø But this is possible only if is close to, which means Ñ Ò is big, or close to, which means there is a long string of µ. The following proposition is expressed in terms of «and «, but, because of the good approximation of «, the good (or bad) approximation of «is equivalent to the good (or bad) approximation of «, or. Of course, it is not irrelevant that this kind of properties is used in the theory of three-interval exchanges [23]. Proposition 5.2. Suppose the negative slope expansion of «µ does not stop. Then the two following properties are equivalent in the negative slope expansion of «µ, the lengths of the strings of µ are unbounded or for every, there exist such that either Ñ Ò Ñ Ò or Ñ Ò Ñ Ò or Ñ Ò for every, there exist integers Ô, Õ, Ö such that «Ô Õ and «Ö Õ Õ And the two following properties are equivalent in the negative slope expansion of «µ, there exist such that for every Å there exist such that Ñ Ò Å and Ñ Ò Ñ Ò Õ ¾

24 24 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI there exist such that for every, there exist integers Ô, Õ such that «Ô Õ Õ ¾ and for any integer Ö «Ö Õ Õ Proof. We prove first that the first assertion implies the second one, then that the third one implies the fourth one, then the two converses simultaneously. If Ñ Ò Ñ Ò or Ñ Ò, by Corollary 4.5 Ô Õ and Ö Õ provide the simultaneous approximation of the second assertion. If Ñ Ò Ñ Ò, the computations in Corollary 4.5 show that Ô Õ and either Ö Õ or Ö Õ provide the simultaneous approximation of the second assertion. If Ñ Ô Ò Ô Ô µ µ for every Ô Å, and ; then ; Å but also. Then if Å is large enough Ô Ô Å Õ Õ and Ö Ö Õ Õ provide the simultaneous approximation of the second assertion. If Ñ Ô Ò Ô Ô µ µ for every Ô Å, and ; then ¾ Å ¾, and. Then if Å is large enough Ô Ô Å Õ Õ and Ö Ö Õ Õ provide the simultaneous approximation of the second assertion. If Ñ Ò is large and Ñ Ò Ñ Ò, then Ô Õ gives a good approximation of «; but the computations in Corollary 4.5 show that neither Ö Õ, nor Ö Õ, nor Ö Õ provide an approximation of «better than in ¾Õ ; hence, because of the first assertion of Corollary 4.5, our third assertion implies our fourth one. Suppose now «and satisfy our second or our fourth assertion. We can then make the construction of the proof of last proposition, to get,, and, hence is close to, or ¾. Suppose that ; then, by construction, if we replace by, and replace by ; hence, with the notations of Section 4, hence is the value that formula (4.4) gives to is the value that iterations of formula (4.4) give to if we Ë Æ Å Õ Õ Æ Æ Ë Æ where But then Ñ Ò must be large, and, because of the above discussion, if the first assertion is not satisfied there is no good approximation of «with denominator Õ, while if the third assertion is not satisfied there is a good approximation of «with denominator Õ. Suppose that ; then, by construction, is the value that iterations of formula (4.4) gives to if we replace by ; and we get Õ Õ Õ

25 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 25 But then we must have and there must be a long string of Ñ Ô Ò Ô Ô µ µ starting at Ô ; hence our first assertion is satisfied, by the discussion above there is a good approximation of «with denominator Õ Õ, and we must have started from our second assertion. Suppose that in fact ¾ ; then, by construction, formula (4.4) gives to if we replace by ¾; and we get Õ Õ Õ is the value that iterations of But then we must have and there must be a long string of Ñ Ô Ò Ô Ô µ µ starting at Ô ; hence our first assertion is satisfied, by the discussion above there is a good approximation of «with denominator Õ Õ, and we must have started from our second assertion. 6. PERIODIC NEGATIVE SLOPE EXPANSIONS We show that the negative slope algorithm satisfies the following Lagrange type theorem: Theorem 6.1. Suppose «µ does not lie on one of the rational lines of Proposition 2.3. Then the sequence Ñ Ò µ is ultimately periodic (or equivalently the sequence «µ is ultimately periodic), if and only if «and are in the same quadratic extension of É. Proof. If the sequence Ñ Ò µ is ultimately periodic, it follows from the definition of the algorithm that for some we have «µ «µ, and hence,. But we have ««Æ Å and the same relation with, hence «Æ «Å Å with Æ Æ Æ. As detå, we check that Å ««Õ Õ µ Ô Ô µ Õ Ö Õ Ö Ô Õ Ô Õ Ô Ö Ô Ö Õ Ô Hence the matrix Å Å has integer coefficients, and the entries in its second column are, or, and. We deduce Æ from the third line of the above relation; the first line gives then a non-trivial algebraic relation of degree ¾ satisfied by «, from which we deduce for an integer ; then the second line implies «¾ É Ô µ «¾ É Ô µ hence «and are in É Ô µ. We suppose now that «¾ É Ôµ and ¾ É Ôµ.

26 26 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI Then «¾ É Ô µ, and there exist integers,, such that We write this relation where From Proposition 4.1 we deduce that «where These two relations imply and hence where «µ ¾ «µ «««µ «µ Æ Ô Õ Ô Õ Þ µ ¾ Ø Ü µ µ Ý Ü Ü, Ý, Þ, Ø are integers as and have integer coefficients and has determinant or ; let us show these coefficients are universally bounded on those for which Ñ Ò µ µ. For a matrix Å, we denote by Å the maximum of the absolute value of its entries. Let Ô Õ «µ Ô Õ «µ We check that hence where Á ¾ is the identity matrix. Hence Þ Ý Ø µ «µ µ «µ «µá ¾µ for universal constants and ¾. We check that ¾ ¾Õ

27 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 27 while Ô Õ «µ Ô Õ «µ Hence, by Corollary 4.3, for those Ò for which Ñ Ò µ µ, ¾ Õ and the coefficients of, which are Ü, Ý, Þ, Ø, are bounded by ¾. Now, «and being in É Ô µ and «being irrational, there exist integers, such that «µ «µ By the same reasoning as above, the relation ««Æ Å translates into where given above, an explicit computation of the second line of the previ- Using the expression of Å ous equality gives with and hence and Å Å Ü Ý Ü Ô Õ Ô Õ µ Ô Õ Ö µ Ý Ô Õ Ô Õ µ Ô Õ Ö µ Ô Õ Ô Õ µ Ô Õ Ö µ Ü Ý Ô Õ Ö Ô Õ Ö Ô Õ Ö Õ Ô Õ Ö Õ µ Õ Ô Õ «µµ Õ Ö Õ «µµ and similarly with replaced by ; it follows then from Proposition 4.2 and Corollary 4.3 that the integers Ü and Ý are universally bounded. Hence for the for which Ñ Ò µ µ, the quantity satisfies only a finite number of equations of degree two with integer coefficients, and the pair µ satisfies only a finite number of relations Ü Ý with integer coefficients; as these are infinitely many

28 28 SÉBASTIEN FERENCZI, CHARLES HOLTON, AND LUCA Q. ZAMBONI because «is irrational, there exists such that and. But because of the definition of the algorithm, this implies for every. Ñ Ò µ Ñ Ò µ Remark 6.2. In [11] Burger gives necessary and sufficient conditions on the partial quotients of two quadratic irrationals to insure that they are elements of the same quadratic number field. It would be interesting to understand these conditions in the context of the negative slope algorithm. 7. CONNECTION WITH INTERVAL EXCHANGE TRANSFORMATIONS The regular continued fraction algorithm provides a link between the arithmetic properties of an irrational number «, the ergodic and spectral properties of a circle rotation by angle «and the combinatorial properties of a class of binary sequences called the Sturmian infinite words (see [5, 14, 30, 31, 33]). A fundamental problem is to generalize and extend this rich interaction to dimension two or greater, starting either from a dynamical system or a specified class of sequences. A primary motivation is that such a generalization could yield a satisfying algorithm of simultaneous rational approximation in Ê Ò In case Ò ¾ there are a number of different dynamical systems all of which are natural candidates to play the role of a rotation in dimension one. One such system is a rotation on the ¾-torus [3, 5, 33, 34, 35]. In this case, the corresponding symbolic counterpart is a class of sequences of complexity ¾Ò introduced by Arnoux and Rauzy in [5] which are a natural generalization of the Sturmian sequences; the arithmetic component is given by a ¾-dimensional division algorithm originally defined by Arnoux and Rauzy in [5] and later studied in greater generality in [13, 43, 44] Though the resulting arithmetic/ergodic/combinatorial interaction is very satisfying in the special case of the so-called Tribonacci system, as pioneered by work of Arnoux and Rauzy [3, 5, 35], a more general canonical equivalence (through what is called a natural coding) between two-dimensional rotations and Arnoux-Rauzy sequences is not always verified (see [12]). Berthé and Vuillon [7] studied the dynamics resulting from two rotations on the circle (see also work of Arnoux, Berthé and Ito in [4]). They found in this case the symbolic counterpart is given by a family of ¾ - shifts, and that the Jacobi-Perron algorithm provides a suitable arithmetic tool for studying this class of dynamical systems. Since a circle rotation is equivalent to an exchange of two intervals on another possible generalization is the dynamical system arising from an exchange on three intervals. Associated to each «and with «is a dynamical system on the interval given by ܵ Ü «if Ü ¾ «, ܵ Ü ¾«if Ü ¾ ««, ܵ Ü «if Ü ¾ «. That is, the unit interval is partitioned according to three subintervals, ««««which are then rearranged according to the permutation ¾ µ In [16] we show that the negative slope algorithm is intimately connected with the dynamics of -interval exchanges, and in fact bears the same connection with this class of dynamical systems as regular continued fraction with circle rotations.

29 STRUCTURE OF THREE INTERVAL EXCHANGE TRANSFORMATIONS I : AN ARITHMETIC STUDY 29 The negative slope algorithms may be reformulated combinatorially in terms of the structure of the so-called bispecial words of the symbolic subshift obtained by symbolically coding the trajectories of points under a 3-interval exchange transformation according to the above partition into three subintervals. This connection relies on a generalization of a combinatorial construction originally developped in [36] to study the evolution of bispecial factors in Arnoux-Rauzy sequences. More precisely, let Ô «Õ «, Ô «µ and Õ «so that Ô Õ are the points of discontinuity of and Ô Õ the points of discontinuity of We call a subinterval Á bispecial if Á is an interval of continuity of Ò for some Ò and Á contains either Ô or Õ and its image Á Ò Áµ contains either Ô or Õ Among the bispecial intervals, we denote by Á (respectively  ) those bispecial intervals for which Ô ¾ Á Ô ¾ Á, respectively Õ ¾ Â Õ ¾ Â, ordered so that Á Á ¾, respectively   ¾ Then in [16] we show that there are infinitely many Á and  and that (under some initial conditions on the initial lengths «µ, Æ Á  for each. The sign of Æ determines on which side (left or right) the intervals Á and  are cut to produce the next bispecial intervals Í Î that is Í, respectively Î, is the largest bispecial interval properly contained in Á, respectively  It can be shown that Í contains Ô and its image Í contains Õ, while Î contains Õ and its image Î contains Ô in particular Í is not Á and Î is not  ; The quantities Ò and Ñ count the number of bispecial intervals between Á and Á and  and  respectively. If Á ¾ Ò Á are the Ò bispecial intervals between Á and Á (so Í µ then «µæ For each fixed is obtained from by cutting Æ on the same side, while Á is obtained by cutting Ò by «on the opposite side. From this point of view, the negative slope algorithm corresponds to a double renormalization process: unlike Rauzy induction ([34, 41]) or the Boshernitzan-Carroll induction process [8], we induce according to the first return map simultaneously on two subintervals. In some stages the two inductions proceed independent of one another and generate the quantities Ò Ñ (of the expansion of «µµ while in other instances the induction process depends on an inequality involving both intervals, and gives rise to the quantity This combinatorial construction gives a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point of view and which in turn gives an explicit characterization of the eigenvalues of the associated unitary operator. In [17] we obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previously known examples, allow us to exhibit new interesting examples of weakly mixing three-interval exchanges. Finally our methods provide affirmative answers to two long standing questions posed by W.A. Veech in [42] on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum. REFERENCES [1] J. AARONSON, M. KEANE, The visits to zero of some deterministic random walks, Proc. London Math. Soc. 44, no. 3 (1982), p [2] V.I. ARNOLD, -graded algebras and continued fractions, Comm. on Pure and Applied Math., XLII (1989), p [3] P. ARNOUX, Un exemple de semi-conjugaison entre un échange d intervalles et une translation sur le tore, (in French) Bull. Soc. Math. France 116 no. 4 (1988), p [4] P. ARNOUX, V. BERTHE, S. ITO, Discrete planes, ¾ -actions, Jacobi-Perron algorithm and substitutions, preprint [5] P. ARNOUX, G. RAUZY, Représentation géométrique de suites de complexité ¾Ò, Bull. Soc. Math. France 119 no. 2 (1991), p

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