For each real number β>1 the β-transformation is dened by Tβx = βx(mod1). In this paper the natural extension Tβ of the ergodic system underlying Tβ is explicitly given. Furthermore, it is shown that a certain induced system of this natural extension is Bernoulli. Since Tβ is weakly mixing, due to W. Parry, it follows from a deep result of A. Saleski that the natural extension is also Bernoulli, a result previously obtained by M. Smorodinsky

Dajani, K.

Kraaikamp, C.

Solomyak, B.

NARCIS 

The natural extension of the β-transformation

Utrecht University Repository

The natural extension of the -transformationKarma Dajani, Cor Kraaikamp and Boris SolomyakAbstractFor each real number  > 1 the -transformation is dened by Tx = x(mod1). In this paperthe natural extension Tof the ergodic system underlying Tis explicitly given. Furthermore, itis shown that a certain induced system of this natural extension is Bernoulli. Since Tis weaklymixing, due to W. Parry, it follows from a deep result of A. Saleski that the natural extensionis also Bernoulli, a result previously obtained by M. Smorodinsky.AMS classication : 28D05, 11K551 IntroductionLet  > 1 be a real number, and let Tbe the transformation of the unit interval [0,1) given byTx = x(mod1): This transformation gives rise to the -expansion introduced by A. Renyi [9] :for any 0  x  1;x =1Xk=1dk k; (1)where dk= dk(x; ) = bTk 1xc; k  1 (here bc denotes the greatest integer not exceeding ).We will denote (1) by x = :d1d2: : : dn: : : or by d(x; ) = (dk(x; ))k1(we will write dkinstead ofdk(x; ) whenever there is no risk of confusion).When  is an integer, (1) is the usual digit expansion with base : In this case Tis clearly relatedto the Bernoulli-shift on  symbols; however, for  62 Z the situation is more complicated. Renyishowed that for each  > 1 the -transformation Tis ergodic, and that there exists a uniqueprobability measure ; equivalent to Lebesgue measure and invariant under T; such that for eachBorel measurable set B 2 B one has(B) =ZBh(x)dx ;where h(x) is a measurable function satisfying1 1 h(x) 11 1:Shortly afterwards, W. Parry [8] noticed that T Tcan easily be shown to be ergodic, andtherefore Tis weakly-mixing. Parry showed thath(x) =1F ()Xx<Tn(1)1n; x 2 [0; 1) ;Research of this author was supported in part by US NSF Grant 9201369.1where F () =R10(Px<Tn(1)1n)dx is a normalizing constant.The "gap" between the ergodic properties of Tfor  2 Z and  62 Z (as is well-known, Bernoul-licity is a much stronger property than weakly-mixing) was bridged by M. Smorodinsky in [12].Smorodinsky showed that for each  > 1 the system( [0; 1);B; ; T) ;where B is the collection of Borel sets of [0,1), is weakly Bernoulli. A deep result by N. Friedmanand D. S. Ornstein [5] then yields that the natural extension of ( [0; 1);B; ; T) is a Bernoulliautomorphism.In this paper we will obtain Smorodinsky's result by explicitly constructing the natural extensionof ( [0; 1);B; ; T) (section 2), and showing that a certain induced transformation of this naturalextension is isomorphic to a Bernoulli automorphism, which is completely understood (see also sec-tion 3 or [1]). In section 4 we then show that Smorodinsky's result follows from Parry's observationthat Tis weakly-mixing, and a theorem by A. Saleski [11].2 Natural Extension of the -transformation2.1 An easy exampleIn case  > 1 is the positive root of the polynomial Xm Xm 1  : : : X   1; m  2, one easilysees1that the natural extension of ( [0; 1);B; ; T) is given by (X;B; ; T), whereX:=m 1[k=0(Tm k1; Tm k 11] [0; Tk1] ; (2)and T: X! Xis dened by T(x; y) := (Tx;1(bxc + y) ) : Furthermore,  is normalizedLebesgue measure on Xwith12+23+ : : :+mm+1 1as a normalizing constant. See also gure1 (for the case m = 4).Figure 11See also [1]. For m = 2 one has that  =p5+12, which is the golden mean. For m  3 we call these 's (pseudo)golden mean numbers. See [6] for a dierent approach.22.2 Denition of the natural extension>From the example in the previous section it is obvious that for (pseudo) golden mean numbers the orbit of 1 under T:1 > T1 > T21 > : : : > Tm 11 > Tm1 = 0 ;plays a crucial role in the denition of the natural extension. In order to generalize the aboveexample to any  > 1 we recall the following well-known result, which describes the admissiblesequences of digits from f0; 1; 2; : : : ; bcg; see also [2, Prop. 2.3].Proposition 1 Let W := f0; 1; 2; : : : ; bcgNand let Dbe the set of -expansions of the realnumbers x 2 [0; 1): Furthermore, let  be the shift and <lexbe the lexicographical ordering on W:1. If d(1; ) = :b1b2: : : b`0000 : : : ; that is, the -expansion d(1; ) of 1 is nite, then s 2 Wbelongs to Dif and only if for all n  0; n(s) is lexicographically less than the periodicsequenced() = :b1b2: : : b` 1(b`  1)b1b2: : : b` 1(b`  1)b1b2: : : b` 1(b`  1) : : : :In this case we put ` := supfk 2 N : bk6= 0; bk+i= 0 ; 8i  1g:2. When the -expansion d(1; ) of 1 is not nite, the conditionn(s) <lexd(1; ) ; for all n  0is necessary and sucient for the sequence s 2 W to belong to D: In this case we writed() = d(1; ) and ` =1.Using the procedure described in [10] (see also [4], p. 239), the natural extension (H;F ; ; T)of ( [0; 1);B; ; T) is constructed in the following way. LetR0= [0; 1]2and Ri= [0; Ti1] [0;1i] ; i  1;the underlying space His obtained by stacking (as pages in a book) Ri+1on top of Ri, for each2i  0 (in case 1 has a nite -expansion of length n, only nRi's are stacked). Let Bibe the collectionof Borel sets of Ri, and let the  algebra F on Hbe the direct sum of the Bi's, i.e. F = Bi.Furthermore, the measure on Hwhich is Lebesgue measure on each rectangle Riis denoted by ,and we put  =1(H). Finally T: H! His dened as follows. Let d() = :b1b2: : : and let(x; y) 2 Ri; i  0, where x = :d1d2: : : is the -expansion of x and y = : 00 : : :0| {z }i timesci+1ci+2: : : is the-expansion of y (notice that (x; y) 2 Riimplies that d1 bi+1). In view of proposition 1, we nowdeneT(x; y) := (Tx; y) 2(R0, if d1< bi+1;Ri+1, if d1= bi+1;(3)2The index i indicates at what height one is in the stack.3wherey=8>><>>:b1+ : : :+bii+d1i+1+y= :b1: : : bidici+1ci+2: : : , if d1< bi+1;y= :000 : : :00| {z }i+1 timesci+1ci+2: : : , if d1= bi+1:Notice that in case i = 0 one hasy=8><>:1(y + d1) , if d1< b1;y, if d1= b1:Remark At rst glance it seems that for a (pseudo) golden mean number  > 1 the natural exten-sion (X;B; ; T) from section 2.1 is a far cry from the above mentioned system (H;F ; ; T).E.g. each point in the space Xis 2-dimensional, while each point in His 3-dimensional (thethird dimension being the height in the stack). We will discuss here briey for each  > 1 a system(~H;B; ~;~T), metrically isomorphic to (H;F ; ; T) (and therefore also a natural extensionof T), which coincides with (X;B; ; T) in case  is a (pseudo) golden mean number.Put~H:=` 1[i=0[0; Ti1) [1Si 1Xj=01j;1SiXj=01j) ;where ` is dened as in proposition 1 andS :=` 1Xj=01j:For (x; y) 2~H, let i = i(y) 2 N [ f0g be such, thaty 2 [1Si 1Xj=01j;1SiXj=01j) ;and dene the map 	 :~H! Hby	(x; y) :=0@x; Sy  i(y) 1Xj=01j; i(y)1A;where i(y) indicates at what height one is in the stack H. Clearly the map 	 :~H! His abijection. Let B be the collection of Borel sets of~Hand let ~be dened by~(B) := (	(B)) , for B 2 B :Notice that ~is normalized Lebesgue measure on (~H; B) and that Parry's invariant measure for Tis the projection of ~on the rst axis. Finally, dene~T:~H!~Hby~T(x; y) := 	 1(T(	(x; y))) , for (x; y) 2~H:We have the following proposition.Proposition 2 For each  > 1 the systems (H;F ; ; T) and (~H;B; ~;~T) are metrically iso-morphic. In case  is a (pseudo) golden mean number the systems (H;F ; ; T) and (X;B; ; T)are identical.43 Generalized Luroth Series and -expansionsIn [1] the so-called Generalized Luroth Series (shortly GLS) were introduced and their ergodicproperties studied. We recall here some of the basic denitions and properties of GLS-expansions,and then we will show how GLS-expansions can be used to obtain a deeper understanding of thenatural extension of the  transformation T.3.1 Generalized Luroth SeriesLet In= (`n; rn]; n 2 D  N = f 0; 1; 2; : : :g; be a nite or innite collection of disjoint intervals oflength Ln:= rn  `n; such that[n2DIn= [0; 1] andXn2DLn= 1and0 < Li Lj< 1 ; for all i; j 2 D; i > j :Furthermore, let I1:= [0; 1] nSn2DIn; L1:= 0 and dene the maps TD; SD: [0; 1]! [0; 1] byTDx :=8><>:x `nrn `n; x 2 In; n 2 D ;0 ; x 2 I1and SDx :=8><>:rn xrn `n; x 2 In; n 2 D ;0 ; x 2 I1Now let " = ("(n))n2Dbe an arbitrary, xed sequence of zeroes and ones. We dene the mapT": [0; 1]! [0; 1] byT"x := "(x)SDx + (1  "(x))TDx ; x 2 [0; 1] (4)where"(x) :=("(n) ; x 2 In; n 2 D0 ; x 2 I1:Let "n:= "(Tn 1"x) and dene for x 2  := [0; 1] n I1= [n2DIns(x) :=1rn  `nand h(x) :=`nrn  `n; in case x 2 In; n 2 D :Moreover we denesn= sn(x) :=(s(Tn 1"x) ; Tn 1"x =2 I1;1 ; Tn 1"x 2 I1;and hnis dened in a similar way. One nds thatx =h1+ "1s1+( 1)"1s1T"x =h1+ "1s1+ ( 1)"1h2+ "2s1s2+ : : : (5): : : : : :+ ( 1)"1+:::+"n 1hn+ "ns1s2: : : sn+( 1)"1+:::+"ns1s2: : :snTn"x :For each k  1 and 1  i  k one has si 1=L > 1; where L = maxn2DLn, and jTk"xj  1: Thus,x pkqk=Tk"xs1s2: : :sk Lk! 0 as k! 1 ;5wherepkqk=h1+ "1s1+ ( 1)"1h2+ "2s1s2+ ( 1)"1+"2h3+ "3s1s2s3+ : : :+ ( 1)"1+:::+"k 1hk+ "ks1s2: : : sk:Let "0:= 0; then for each x 2 [0; 1] one hasx =1Xn=1( 1)"0+:::+"n 1hn+ "ns1s2: : :sn: (6)For each x 2 [0; 1] we dene its sequence of digits an= an(x); n  1; as follows :an= k ()defTn 1"x 2 Ik;for k 2 D[f1g: The expansion (6) is called the (I, ") Generalized Luroth Series (GLS) of x:Notice that for each x 2 [0; 1]nI1one nds a unique expansion (6), and therefore a unique sequenceof digits an; n  1: Conversely, each sequence of digits an; n  1; with an2 D [ f1g; a16= 1denes a unique series expansion (6). We denote (6) byx =""1; "2; "3; : : : "n; : : :a1; a2; a3; : : : an; : : :#:Examples1. Let In:= (1n;1n 1]; n  2: In case "n= 0; n  2; one gets the classical Luroth Series, while"n= 1; n  2 yields the alternating Luroth Series.2. For n 2 N; n  2; put Ii= (in;i+1n]; i = 0; 1; : : : ; n   1: In case "(i) = 0 for all i; T"yieldsthe n-adic expansion. In case n = 2 and "(0) = 0; "(1) = 1 T"is the tent map.The ergodic properties of the GLS-transformation are given by the following theorem.Theorem 1 1. The stochastic variables a1(x); a2(x); : : : ; an(x); : : : corresponding to the (I; ")-GLS operator T"from (4) are i.i.d with respect to the Lebesgue measure , and(an= k) = Lkfor k 2 D [ f1g :Furthermore, (In)n2Dis a generating partition.2. The (I; ") GLS operator T"from (4) is measure preserving and Bernoulli.3. Let T": [0; 1] [0; 1]! [0; 1] [0; 1] be given by :T"(x; y) := T"x;h(x) + "(x)s(x)+( 1)"(x)ys(x)!; (x; y) 2 [0; 1] [0; 1] ;then the system([0; 1] [0; 1]; B  B;  ; T")is the natural extension of ([0; 1]; B; ; T") : Furthermore, ([0; 1] [0; 1]; B  B;  ; T") isBernoulli.Remark The proof of theorem 1 in [1] is straightforward; e.g. 1. follows from (5) and fromLk= 1=sk; 0  Tn"x < 1. It is also easy to see that T"is Bernoulli, and that T"is the naturalextension of T". Since T"is Bernoulli, it certainly is weakly Bernoulli, hence [5] yields the desiredresult that T"is Bernoulli. In fact it is an exercise to show that T"is Bernoulli implies that T"isBernoulli, see also [3].63.2 Inducing on the Natural Extension of the  transformationLet us return for a moment to the example of section 2.1. Again, let  > 1 be a (pseudo) goldenmean number and let Xbe as in (2). Let Y:= [0; 1]  [0; 1=] and W: Y! Ythecorresponding induced transformation under T; that isW(x; y) = Tk+1(x; y) ;where k = k(x) 2 f0; 1; : : : ; m  1g is such, that x 2 (Tm k1; Tm k 11] : The orbit of 1 under Tyields a natural partition of [0,1). Let T"be the (I, ")-GLS operator with this partition, i.e. thepartition I is given by (Tm i1; Tm i 11] ; 0  i  m  1 (see also gure 1), and let "(n) := 0 foreach digit n: Notice that for x 2 (Tm i1; Tm (i+1)1] ; 0  i  m  1; one has thatT"x = Ti+1x :In [1] the following theorem was obtained for the special case  > 1 is a (pseudo) golden meannumber.Theorem 2 Let 	 : [0; 1] [0; 1]! Ybe dened by	(x; y) := (x;y)and let the measure  on Ybe dened by (A) := (  )(	 1(A)) for each Borel set A  Y.Then one has1. 	 is a measurable bijection which satises 	  T"= W	 :2. The dynamical systems ([0; 1] [0; 1];  ; T") and ( Y; ;W) are isomorphic.In the remaining part of this section we give a generalization of theorem 2 to all  > 1.LetW: R0! R0be the induced map of Ton R0. That is,Wis given as follows; let (x; y) 2 R0,where x = :d1d2: : : is the -expansion of x and d() = :b1b2: : :, thenW(x; y) =(T(x; y) if d1< b1;Tn(x; y) if di= bifor 1  i  n   1 and dn< bn:This follows from denition (3) of Ton R0, for{ if d1< b1then T(x; y) 2 R0.{ if di= bifor 1  i  n   1 and dn< bnone has Ti(x; y) 2 Rifor 1  i  n   1 andTn(x; y) 2 R0.In other wordsW(x; y) =8><>:Tx;1(y + d1), if (x; y) 2 R10;Tkx;b1+   +bk 1k 1+dkk+yk, if (x; y) 2 Rk0; k  2:7For (x; y) 2 R0, let n(x; y) := inff k  1 ; Tk(x; y) 2 R0g and Rk0:= f (x; y) 2 R0; n(x; y) = k g.Notice that n(x; y) is the rst return time of the map Tto R0, where n 2 N. The above showsthatR10= [ 0; :b1] [0; 1] ; Rk0= [ :b1: : : bk 1; :b1: : : bk 1bk] [0; 1] ; k  2andR0=1[k=1Rk0:Let I = fIn; n  1g be the partition on [0,1) dened as follows : for each n  1, there existunique integers k  0 and 1  i  bk+1such thatn = b0+ b1+   + bk+ (i  1) ;where by denition we put b0:= 0. SetIn=b0+b1+   +bkk+i  1k+1; b0+b1+   +bkk+ik+1: (7)Note that for (x; y) 2 In [0; 1],W(x; y) = Tk+1(x; y) =Tk+1x ; b0+b1+   +bkk+i  1k+1+yk+1:We have the following proposition.Proposition 3 Let I = (In)n2Dbe the partition from (7). Then for "n= 0 for all n  1 we havethat the natural extension T"of the (I; ")-GLS expansion T"is identical to W.Proof. The proof follows from the fact that T"on Ib0++bk+(i 1) [0; 1] for 1  i  bk+1is givenbyT"(x; y) = (T"x; b0+b1+   +bkk+i  1k+1+yk+1)and that for x 2 (b0+b1+   +bkk; b0+b1+   +bk+1k+1], T"x = Tk+1x. Comparing with (7), onesees thatW= T", and hence by theorem 1 it follows that (R0;FR0; R0;W), with FR0= F \R0and R0= jR0, is a Bernoulli system.24 The natural extension of the -transformation is BernoulliIn this section we will show that from proposition 3 and a result by Saleski we at once haveSmorodinsky's result that the natural extension of the -transformation is Bernoulli. We rstrecall Saleski's result, see also [11].Theorem 3 Let (X;B; ; T ) be a non-atomic Lebesgue space with an automorphism T . Let A 2 Bbe a subset of X of positive measure and denote by TAthe induced transformation of T on A.Moreover, suppose we have that TAis Bernoulli, T is weakly mixing andHA_1i=1_1j=1TiAYjj _1i=0TiAP< 1 ;where P is a Bernoulli partition of (A; TA) and Yj= fA  [ji=1T iA; A \ [ji=1T iAg: Then T isa Bernoulli automorphism.8We have the following theorem.Theorem 4 The system (H;F ; ; T) is Bernoulli.Proof. The system (H;F ; ; T) is weakly-mixing, due to W. Parry, and for which the inducedtransformation Wis Bernoulli with Bernoulli partition P = I]:= I  [0; 1] where I is as givenin (7). The partitions Yjof R0as dened in Saleski's theorem 3 have the following form :Yj=0 ; b0+b1+   +bjj [0; 1] ;b0+b1+   +bjj; 1 [0; 1]:Clearly, Yj< I]for each j  1, i.e. I]is a renement of each Yj, so that_1i=1_1j=1WiYj< _1i=0WiI]and henceHR0_1i=1_1j=1WiYjj _1i=0WiI]= 0:Thus, by theorem 3, we have that (H;F ; ; T) is Bernoulli.2Final remark After reading a rst draft of this paper, H. Nakada [7] remarked that the abovedescribed method to prove Bernoullicity of the -transformation can easily be generalized to a classof functions L, introduced and studied by K. M. Wilkinson [13]. This class L of piecewise lineartransformations of the unit interval onto itself contains among others the -transformation and thelinear mod one transformation. Let In; n 2 D; be a nite or innite collection of intervals as insection 3.1, and let T : [0; 1)! [0; 1) for x 2 In; n 2 D; be dened byTx := n(x  `n) + n; (8)where  = infn2Dn> 1 and for each n 2 D; 0  n< 1; nrn+ n 1: Put(j1; j2;    ; jn) := Ij1\ T 1Ij2\    \ T (n 1)Ijn;where jk2 D; 1  k  n: We say that n= (j1; j2;    ; jn) is a full-interval of rank n if(Tn(n)) = 1; and non-full otherwise.LetBnbe the union of those intervals (j1; j2;    ; jn) which are full, but none of (j1; j2;    ; jk); 1 k  n  1; is full, and suppose that1Xn=1(Bn) = 1: (9)Denote by N the class of transformations of the form (8), satisfying (9). Then N contains Wilkin-son's class L (see [13], corollary 3.5). Clearly each T 2 N has a GLS as an induced transfor-mation, with GLS-partition P , consisting of all full intervals (j1; j2;    ; jn) for which none of(j1; j2;    ; jk); 1  k  n   1; is full.9References[1] Barrionuevo, Jose, Robert M. Burton, Karma Dajani and Cor Kraaikamp { Ergodic Propertiesof Generalized Luroth Series, TU Delft Report 94-105 (1994), 1-16.[2] Blanchard, F. { -Expansions and Symbolic Dynamics, Theoretical Comp. Sc., 65 (1989),131-141.[3] Brown, J. R. { Ergodic Theory and Topological Dynamics, Academic Press, New York, 1976.[4] Cornfeld, I. P., S. V. Fomin and Ya. G. Sinai { Ergodic Theory, Grundlehren der math. Wiss.245, Springer-Verlag New York, Heidelberg, Berlin (1982).[5] Friedman, N. A. and D. S. Ornstein { On isomorphisms of weak Bernoulli transformations,Adv. in Math., 5 (1970), 365-390.[6] Ito, Shunji { A Construction of Transversal Flows for Maximal Markov Automorphisms,Tokyo J. Math., 1 (1978), 305-324.[7] Nakada, Hitoshi. Personal communication with the second author, Delft. March 20, 1995.[8] Parry, W. { On the -expansion of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960),401-416.[9] Renyi, A. { Representations for real numbers and their ergodic properties, Acta Math. Acad.Sci. Hung., 8 (1957), 472-493.[10] Rohlin, V. A. { Exact endomorphisms of a Lebesgue space, Izv. Akad. Naik SSSR, Ser. Mat.,24 (1960); English AMS translation, Series 2, 39 (1969), 1-36.[11] Saleski, Alan { On induced Transformations of Bernoulli Shifts,Math. Systems Th. 7 (1973),83-93.[12] Smorodinsky, Meir { -Automorphisms are Bernoulli Shifts, Acta Math. Acad. Sci. Hung.,24 (1973), 273-278.[13] Wilkinson, Keith M. { Ergodic Properties of a Class of Piecewise Linear Transformations, Z.Wahrscheinlichkeitstheorie verw. Gebiete, 31 (1975), 303-328.10Karma Dajani Cor KraaikampUniversiteit Utrecht Technische Universiteit DelftDept. of Math. and Thomas Stieltjes Institute for MathematicsBudapestlaan 6 TWI (SSOR)P.O.Box 80.010 Mekelweg 43508 TA Utrecht 2628 CD Delftthe Netherlands the Netherlandsdajani@math.ruu.nl cork@dutiosd.twi.tudelft.nlBoris SolomyakDept. of Mathematics, GN-50University of WashingtonSeattle, Washington 98195solomyak@math.washington.edu11

http://dspace.library.uu.nl:8080/bitstream/handle/1874/2539/909.pdf?sequence=1

The natural extension of the β-transformation

Abstract

Similar works

Full text

Available Versions

NARCIS

Utrecht University Repository