A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI

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1 A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS ON 3-INTERVALS KARMA DAJANI Abstact. Wepove the conjectue that an inteval exchange tansfomation on 3-intevals with coesponding pemutation (1; 2; 3)! (3; 2; 1); and ationally independent discontinuity points, is neve measue theoetically isomophic to an iational otation. 1. Intoduction Inteval exchange tansfomations wee st intoduced by Keane in [K1], and ae dened as follows. Let I = [0; 1); n 2and = P ( 1 ; ; n ) a pobability vecto with i > 0: Dene 0 = 0 and i = i k=1 k; and set I i = [ i,1; i ): Let be a pemutation of f1; 2; ;ng; and conside the pobability vecto P =(,1 (1); ;,1 (n)): Note that,1 (i) > 0 fo all i: Let 0 =0and i = i k=1,1 (k); and set Ii =[ i,1 ; i ): Dene T : I! I by Tx= x, i,1 + (i),1 if x 2 I i :T is called an (; ) inteval exchange tansfomation on n intevals. It is clea that T is invetible, T i,1 = (i),1 and T maps I i isometically onto I (i) : Futhe, T is continuous except possibly at f 1; ; n,1g: At these points T is ight continuous. Note that T is continuous at i if and only if (i +1)= (i)+1: In othe wods, T is discontinuous at i if and only if T i,1;t i do not appea in this ode as consecutive tems in the odeed set f0 n ; ;n 2ng: We say T is in standad fom if T is discontinuous at i fo all i =1; 2; ;n, 1 o equivalently, if(i +1)6= (i) + 1 fo all i =1; 2; ;n, 1: Notice that any inteval exchange tansfomation on n intevals can be witten in standad fom as an inteval exchange tansfomation on m intevals with m n: Since if T is not in standad fom, then T is continuous at i fo some i, then(i +1)=(i) +1; and so T maps the inteval [ i,1; i+1 ) isometically onto [ (i),1 ; (i)+1 ): Thus, we can edene T on intevals with end points f 0 ; ; i,1; i+1 ; ; n g: We epeat this pocess until all the emaining 's ae discontinuity points of T: 1991 Mathematics Subject Classication. 28D05. Key wods and phases. Rotations, inteval exchange. 1

2 2 KARMA DAJANI The pemutation coesponding to T is said to be ieducible if (f1; 2; ;kg) 6= f1; 2; ;kg; fo all k =0; 1; ;n, 1: Note that if is educible, then T can be decomposed into two inteval exchange tansfomations, one on [0; k ) and the othe on [ k ; 1): We assume thoughout this pape that T is ieducible. Inteval exchange tansfomations have been studied by seveal authos. Hee we mention few of the known esults. In [K1], Keane studied the minimality of such tansfomations, and in [K2] questions concening unique egodicity wee investigated. It is easy to see that if n =2;T coesponds to a otation and if n =3, then T can be seen as an induced tansfomation of a otation. Thus, if the 's ae ationally independent, then in both cases T is uniquely egodic. Keynes and Newton [KN], and also Keane [K2] gave examples of inteval exchange tansfomations that ae not uniquely egodic. Masu [M], and independently Veech [V1, V2, V3, V4, V5] showed that almost evey minimal inteval exchange tansfomation is uniquely egodic. Late Boshenitzan [B] gave anothe poof of this esult by moe elementay means. Some of the spectal popeties wee studied by Veech in a seies of papes [V3,V4,V5]. Oseledets [O] and Goodson [G] constucted egodic inteval exchange tansfomations with simple spectum. Recently, Bethe, Chekhova and Feenczi [BCF] poved that evey egodic inteval exchange tansfomation on thee intevals has simple spectum. The st inteval exchange tansfomation with continuous spectum was given by Katok and Stepin [KS], thei example is also an exchange on thee intevals. In [BCF], the authos gave othe examples of exchanges on thee intevals with continuous spectum, and they conjectued that no non-tivial exchange on thee intevals is measue theoetically isomophic to an iational otation. In section 2 we pove this conjectue as a coollay of a ecent esult by Simin Li [S], whee he gave necessay and sucient conditions fo an inteval exchange tansfomation to be conjugate to an iational otation. 2. Non-tivial exchanges on 3-intevals Let 0 <l<m<1with 1;l;m ationally independent. Conside the inteval exchange tansfomation T given by Tx = 8 < : x +1, l x 2 [0;l); x +1, l, m x 2 [l; m); x, m x 2 [m; 1): T coesponds to the pemutation (1; 2; 3)! (3; 2; 1): Notice that T is the only inteval exchange tansfomation on 3-intevals which is ieducible and in standad fom. Moeove, by a esult of Keane [K1], T is minimal. We call T a non-tivial exchange tansfomation on 3-intevals. It is well known that T is an induced tansfomation of the inteval exchange tansfomation S dened on [0; 1, l + m) by x +1, l x 2 [0;m); Sx = x, m x 2 [m; 1, l + m): Since afte nomalization S is isomophic to an iational otation, S is minimal and uniquely egodic, and hence so is T:

3 A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS 3 Let = 1,l+m 1,l and = 1 1,l+m. In [KS], the authos poved that if has unbounded patial quotients and if fo some subsequence q n of denominatos of convegents of ; we have j, p n j <o( 1 q n qn 2 ); and j, j > c q n q n fo all and some constant c > 0; then T is not measue theoetically isomophic to an iational otation. In [BCF], it is poved that when has bounded patial quotients, and 2 K() fo some Canto set K(); then T is not measue theoetically isomophic to an iational otation. Simin Li [Li] gave ecently necessay and sucient conditions fo an inteval exchange tansfomation to be conjugate to an iational otation. Theoem 1 (Li). Let T be an inteval exchange tansfomation, and let d(t n ) be the numbe of discontinuities of T n : Then, T is conjugate to an iational otation if and only if (i)t n is minimal fo all n 1; (ii) fd(t n )g is bounded by some intege N >0 and (iii) thee exists k>0 and M 2 N 3 +3N 2 such that d(t k )= d(t 2k )== d(t Mk ): Since a non-tivial inteval exchange on 3-intevals is uniquely egodic, to show that it is not measue theoetically isomophic to an iational otation, we pove that fd(t n )g is an unbounded sequence. Theoem 2. Let T be a non-tivial inteval exchange tansfomation on 3-intevals with ationally independent discontinuity points. Let D(T n ) be the set of discontinuity points of T n ; and let d(t n ) denote the cadinality of D(T n ): Then and hence, d(t n )=2n: D(T n )=ft,i l; T,j m :0 i; j n, 1g; Poof: The poof is done by induction on n: The esult is tue fo n =1: Suppose D(T k )=ft,i l; T,j m :0 i; j k, 1g; fo k =1; 2; ;n: We pove the esult fo k = n +1: Let 0 < 1 < 2 < < 2n < 1 be the discontinuities of T n witten in inceasing ode. By the induction hypothesis, D(T n )=f i :1 i 2ng = ft,i l; T,j m :0 i; j n, 1g: Let 0 = 0 and 2n+1 =1: The undelying patition of T n is given by P(T n )=f[ i ; i+1 ):i =0; 1; 2ng: Let n be the pemutation coesponding to T n (notice that T n is an inteval exchange tansfomation). Then, T n f 0 ; 1 ; ; 2n g = f0 n ;n 1 ; ;n 2n g with 0 = n 0 =0; and T n i = n fo i =0; 1; ; 2n: Futhemoe, since n (i+1),1 1;l and m ae ationally independent, and each i n is a linea combination of 1;l and m with intege coecients, it follows that l; m =2 f n 0 ;n 1 ; ;n 2ng: Now invetibility oft implies that T n 0 ; ;Tn ;Tm;Tl ae all distinct. 2n

4 4 KARMA DAJANI Suppose l 2 (,1 n ;n ); and m 2 (s,1 n ;n s ): We conside thee cases. Case 1. If = s; then T,n l; T,n m 2 ( p,1; p )wheep =,1 n (): Since T is an ode peseving isomety on [ p,1; p ); it follows that T,n l<t,n m: The undelying patition of T n+1 is then given by P 1 (T n+1 )=f[ 0 ; 1 ); ; [ p,2; p,1); [ p,1;t,n l); [T,n l; T,n m); [T,n m; p ); [ p ; p+1 ); ; [ 2n ; 2n+1 )g: To pove the esult, we need to show that f 1 ; ; p,1;t,n l; T,n m; p ; ; 2n g is the set of discontinuity points of T n+1 : Let D 1 = f 0 ; ; p,1;t,n l; T,n m; p ; ; 2n g and E 1 = f0 n ; ;n,1 ;l;m;n ; ;2n n g; both consideed as odeed sets. Then TD 1 = E 1 ; and by discontinuity oft n at p ; we have T n p 6= n : Futhe, n i 2 (0;l)fo1i, 1; and i n 2 (m; l) fo i 2n; : Hence, T n+1 D 1 = TE 1 = ftm =0;T n ; ;T2n n ;Tl=1, m; T0 n =1, l; T1 n ; ;Tn,1 g: Hee, the elements of TE 1 ae listed in inceasing ode. We st show thatt n+1 is discontinuous at i fo i 6= p: To do this, we need to pove thatt n+1 i,1 and T n+1 i do not appea in this ode as consecutive tems in TE 1 : By assumption, T n is discontinuous at i ; hence T n i,1 and T n i do not appea as consecutive tems of the fom j n I 1 =[l; m) and I 2 =[m; 1): If T n i,1;t n i 2 I j fo some j =0; 2; then since T maps I j isometically onto TI j ; it follows that T n+1 i,1 and T n+1 i cannot appea as consecutive tems in TE 1 : If T n i,1 2 I j and T n i 2 I k fo j 6= k; then eithe T n i,1 2 I 0 and T n i 2 I 2 ; o T n i,1 2 I 2 and T n i 2 I 0 : In the ;n j+1 in E 1: Let I 0 =[0;l); st case we get T n+1 i < 1, m < T n i,1; and in the second case, we get T n+1 i,1 < 1,m <T n+1 i : Hence, T n i,1 and T n i do not appea as consecutive tems of the fom j n ;n j+1 in E 1; and so T n+1 is discontinuous at i : Now, the discontinuity oft n at p implies that T n+1 p 6= T n ; and Tm =0<T n <T n+1 p : Hence T n+1 (T,n m)=tm =0andT n+1 p do not appea as consecutive tems in TE 1 : So T n+1 is discontinuous at p : The discontinuity oft n+1 at T,n l follows fom the fact that T n+1 p,1 is an inteio point ofti 2 ; while T n+1 (T,n l)=1, m is the left end-point ofti 1 : Finally, T n+1 (T,n m)=0<t n < 1, m = T n+1 (T,n l) implies that T n+1 is discontinuous at T,n m: Theefoe, D 1 = D(T n+1 ): Case 2: If <sand p = n,1 < n,1 s = q; then T,n l 2 ( p,1; p )andt,n m 2 ( q,1; q ): The discontinuity oft n at p and q implies T n p 6= n and T n q 6= s n : The undelying patition of T n+1 is easily seen to be P 2 (T n+1 )=f[ 0 ; 1 ); ; [ p,2; p,1); [ p,1;t,n l); [T,n l; p );

5 A NOTE ON ROTATIONS AND INTERVAL EXCHANGE TRANSFORMATIONS 5 [ p ; p+1 ); [ q,1;t,n m); [T,n m; q ); [ q ; q+1 ); ; [ 2n ; 1)g: To show the discontinuity oft n+1 at 1 ; ; 2n ;T,n l; T,n m; we conside the odeed sets D 2 = f 0 ; ; p,1;t,n l; p ; ; q,1;t,n m; q ; ; 2n g and E 2 = f0 n ; ;n Then, T n D 2 = E 2 : Notice that n ae inteio points of I 1 and s n T n+1 D 2 = TE 2,1 ;l;n ; ;s,1 n ;m;n s ; ;2n n g:,1 ae inteio points of I 0; n 1 ; ;n ; ;2n n ae inteio points of I 2: Thus, = ftm =0;T n s T n ; ;T2n n ;Tl=1, m; ; ;T n s,1 ;Tn 0 =1, l; ;T n,1 g: ; ;n s,1 Hee, the elements of TE 2 ae listed in inceasing ode. We st pove that T n+1 is discontinuous at i fo i 6= p; q: If T n i,1;t n i 2 I j ; then since T n i,1;t n i do not appea as consecutive tems in E 1 and since T is an isomety on I j ; we have that T n+1 i,1 and T n+1 i ae not consecutive tems of E 2 ; and thus T n+1 is discontinuous at i : If T n i 2 I j and T n i,1 2 I k fo k 6= j; then we conside seveal cases. If T n i 2 I 2 and T n i,1 2 I 0 o I 1, then since T n i,1 6= l we have T n+1 i < 1, m<t n+1 i,1: If T n i 2 I 1 and T n i,1 2 I 2 ; then since T n i 6= l it follows that T n+1 i,1 < 1, m<t n+1 i : If T n i 2 I 1 and T n i,1 2 I 0 ; then T n+1 i <T n+1 i,1: If T n i 2 I 0 and T n i,1 2 I 1 ; then since i 6= q we have T n+1 i,1 <Ts,1 n < T0 n T n+1 i : If T n i 2 I 0 and T n i,1 2 I 2 ; then T n+1 i,1 < 1, m<t n+1 i : In all the above cases we see that T n+1 is not continuous at i : The discontinuity oft n+1 at p and q follows fom the fact that T n+1 p 6= T n and T n+1 q 6= Ts n ; so that neithe T n+1 p and T n+1 (T,n l)not n+1 q and T n+1 (T,n m) appea as consecutive tems in TE 2 : Finally, fom T n+1 (T,n l)= 1, m<1, l<t,1 n and T n+1 (T,n m)=0< 1, m<ts,1 n n+1 wehave thatt is discontinuous at T n+1 (T,n l) and T n+1 (T,n m): Thus, D 2 = D(T n+1 ): Case 3: If<sand p = n,1>,1 n+1 n s = q; then the undelying patition of T is given by P 3 (T n+1 )=f[ 0 ; 1 ); ; [ q,2; q,1); [ q,1;t,n m); [T,n m; q ); [ q ; q+1 ); ; Let and Then, [ p,2; p,1); [ p,1;t,n l); [T,n l p ); [ p ; p+1 ); ; [ 2n ; 1)g: D 3 = f 0 ; ; q,1;t,n m; q ; ; p,1;t,n l; p ; ; 2n g E 3 = f n 0 ; ;n T n+1 D 3 = TE 3,1 ;l;n = ftm =0;T n s T n ; ; n s,1 ;m;n s ; ;2n n g: ; ;Tn 2n ;Tl=1, m; ; ;T n s,1 ;Tn 0 =1, l; ;T n,1 g:

6 6 KARMA DAJANI The elements of D 3 ;E 3 and TE 3 ae listed in inceasing ode. A simila agument as in the above two cases shows that D 3 = D(T n+1 ): Thus, the theoem is poved. Theoem 3. Any non-tivial inteval exchange tansfomation on 3-intevals with ationally independent discontinuity points is not measue theoetically isomophic to an iational otation. Poof: By theoem 2 and unique egodicity, the esult follows fom Li's theoem. In [BCF], the authos poved that evey egodic inteval exchange tansfomation on thee intevals has simple spectum. Using this esult and theoem 3, we have the following coollay. Coollay 1. Evey non-tivial inteval exchange tansfomation on thee intevals with ationally independent discontinuity points has eithe ational o continuous spectum. Refeences [BCF] Bethe, V., N. Chekhova ands. Feenczi - Coveing numbes: aithmetics and dynamics fo otations and inteval exchanges, J. D'Analyse Math. 79 (1999), [B] Boshenitzan, M. - Acondition fo minimal inteval exchange maps to be uniquely egodic, Duke Math. J. 52 (1985), [G] Goodson, G.R. - Functional equations associated with the sectal popeties of compact goup extensions, Poceedings of Confeence on Egodic Theoy and its connection with Hamonic Analysis, Alexandia 1993, Cambidge Univesity Pess, 1994, [K1] Keane, M.S. - Inteval exchange tansfomations, Math. Z. 141 (1975), [K2] Keane, M.S. - Non-egodic inteval exchange tansfomations, Isael J. Math. 26 (1977), [KN] Keynes, H. and D. Newton - A minimal non-uniquely egodic inteval exchange tansfomation, Math. Z. 148 (1976), [KS] Katok, A.B. and A.M. Stepin - Appoximations in egodic theoy, Uspekhi Math. Nauk 22, 5 (1967), (Russian), tanslated in Russian Mth. Suveys 22, 5 (1967), [Li] Li, Simin - A Citeion fo an Inteval Exchange Map to be Conjugate to an Iational Rotation, J. Math. Sci. Univ. Tokyo 6 (1999), [M] Masu, H. - Inteval exchange tansfomations and measued foliations, Ann. ofmath. [O] 115 (1982), Oseledets, V.I. - On the spectum of egodic automophisms, Doklady Akad. NaukSSSR 168, 5 (1966), (in Russian), tanslated in Soviet Math. Doklady 7 (1966), [R] Rauzy, G.-Echanges d'intevalles et tansfomations induites, Acta Aith. 34 (1979), [V1] Veech, W.A. - Inteval exchange tansfomations, J. D'Analyse Math. 33 (1978), [V2] [V3] [V4] [V5] Veech, W.A. - Gauss measues fo tansfomations on the space of inteval exchange maps, Ann. of Math. 115 (1982), Veech, W.A. - The metic theoy of inteval exchange tansfomations. I Geneic spectal popeties, Ame. J. Math. 106 (1984), Veech, W.A. - The metic theoy of inteval exchange tansfomations. II Appoximation by pimitive exchange, Ame. J. Math. 106 (1984), Veech, W.A. - The metic theoy of inteval exchange tansfomations. III The Sah- Anoux-Fathi invaiant, Ame. J. Math. 106 (1984), Univesiteit Utecht, Fac. Wiskunde en Infomatica and MRI, Budapestlaan 6, P.O. Box , 3508 TA Utecht, the Nethelands addess: dajani@math.uu.nl