Multiplicity, rank pairs

M U L T I P L I C I T Y , R A N K P A I R S

By

J. KWIATKOWSKI* AND Y'. LACROIX t

Abstract. For any pair (m, r) such that 2 < m _< r < oo, we construct an ergodic dynamical system having spectral multiplicity m and rank r. The essential range of the multiplicity function is described. I f r > 2, the pair (m, r) also has a weakly mixing realization.

O. Introduct ion

Given a dynamical system (X, #, T), one associates with it a measure-theoretic invariant, the rank r(T), and a spectral invariant, the multiplicity m(T). The pair (m(T), r(T)) is such that 1 < m(T) < r(T) < o0.

It was conjectured in [M1 ] that for any such pair of integers (or oo), there exists an ergodic system (X, #, T) realizing it. Pairs (1, 1) were constructed in [Ch], (1,2) in [dJ], (1,r) in [M1], (2,r) in [GoLe], (r,r) in JR1,2], (r, 2r) in [M2], ( p - 1,p) in [FeKw] and (1, c~) in [LeSi], [Fe]. Gaussian-Kronecker systems always realize the pair (I, oo) ([dlR]). The latest result of this series is a density theorem [FeKwMa] proving that given m, the set of r 's such that the pair (m,r) is realizable has density 1.

In this note we construct realizations of pairs (m,r) with 2 < m < r < oo. The pair (oo, oo) is realized with any ergodic system of positive entropy. Thus, together with [M1], we prove that all pairs (re, r) with 1 < m < r < oo are obtainable.

The transformations we use are continuous Morse automorphisms over a finite abelian group (see Section I for the preliminaries). These systems have partly discrete spectrum.

However, the same pairs (m,r) can be obtained within the class of weakly mixing systems. We give to this end some hints in Remark IV.I, but for the sake of simplicity, computations are only carried out with full details for the Morse automorphisms.

Our examples (see Section II) sit in the class of so-called natural factors of a compact abelian group extension ([KwJLe]).

* Supported by KBN grant no. 2 P30103107. The author acknowledges the hospitality of the Mathematics Department of Universit6 de Bretagne Occidentale, Brest, where this paper was written.

t Supported by C.A.F. Nord Finist6re.

205 JOURNAL D'ANALYSE MATHEMATIQUE, Vol. 71 (1997)

206 J. KWlATKOWSKI AND Y. LACROIX

We first construct an ergodic group extension (X x G, T~,, # | me) realizing the

pair (r, r). Next we produce a natural factor (X x H, T~,,, # | mH), where H = G/H0

is a quotient group. Using methods developed in [KwJLe], we compute the spectral multiplicities o f

the systems: those o f the natural factor decrease. But, surprisingly, the ranks o f

the natural factors remain equal to the rank of the initial system, r.

The systems have a continuous Morse shift representation which we use for the

rank computations. In Section III we prove the following for 2 < r < ~ :

T h e o r e m (r, r) The system Of x G, T~, # | mG) satisfies r(T~) = r = m(T~), and the essential range o f its multiplicity function is {d : dlr}. It is measure theoretically isomorphic to a strictly ergodic continuous Morse automorphism.

I f 2 <_ m < r < c~, passing to the natural factor Of x H, T~, M, # | mH), we prove

in Section III that m(Tr ) = m, while in Section IV we prove that its rank is r, by

proving that its covering number satisfies F* (T~o,) < 1/(r - 1 ). We obtain the

T h e o r e m (m, r) The system Of • H, T~M , # | mH) is such that r(T~M ) = r, m(Tr ) = m, and the essential range o f its multiplicity function is {1, . . . ,m}.

It is measure-theoretically isomorphic to a strictly ergodic continuous Morse automorphism.

I. Pre l iminar ies

Throughout this paper G shall denote an additive finite abelian group, and f~ the

space o f bi-infinite sequences taking values in G.

1.1. B locks and operat ions on blocks A finite sequence B = B[0] �9 �9 �9 B[k - 1], B[i] E G, k > 1, is called a block over G.

The number k is called the length o r B and denoted IBI. I f w E f~ (or w is a one-

sided sequence over G) and B is a block, then w[i,s] and B[i,s] (0 < i < s < k - 1)

denote the blocks w[i].., w[s] and B[i]... B[s] respectively. I f C = C[0] �9 �9 �9 C [ f - 1]

is another block, then the concatenation orB and C is the block

BC = B[O].- , B [ k - 1]C[O]... C [ f - 1].

Concatenation extends to more than two blocks in the obvious way. We define also

fo rq E Z, q times

. . . . . .

I f v : G ~ G is agroup automorphism, let v(B) be the block

v(B) = v ( B [ 0 ] ) . . , v(B[k- 1]).

MULTIPLICITY, RANK PAIRS 207

I fg E G, by B(g) we will denote the b lockB+g = B(g) = (B[0] +g). . . (B[k- 1] +g). Then v(B(g)) = v(B)(v(g)), g E G. Finally, we define the product B x C o r b and C as follows (ICI =sO:

B x C = B(C[0 ] ) - - .B (C[ f - 1]).

As for concatenation, this multiplication operation "x" is extended to more than two blocks, and is associative.

1.2. Occurrences, frequencies, density, d distance The block B is said to occur at place i in w (resp. in C as above (k < f ) ) if

w[i, i+ [B[- 1] = B (resp. C[i, i+ [B[- 1] = B). We shall write B < w (resp. B < C) when this happens for some position i.

The frequency of B in C (resp. in w) is the number

fr(B, C) = I c l - l # { 0 ~ i ~ ICl - IBI - 1; B occurs at place i in C},

(resp. fr(B, w) = lim fr(B, w[0, s - 1]) if this limit exists).

For a one-sided infinite subsequence of w, E = {w[n],n a I c N}, we call the density of E the corresponding density of the set I in N, and denote it by D(E, w) (if it exists).

Let/5 > 0. We say that B 6-occurs at place i in C (resp. in w) if

[l(B,C[i,i+ [BI- 1]) </5 (resp. d(B,w[i,i+ IBI- 1]) < 6),

where

d(x l - . "Xn,Yl "" "Yn) = n - l ~ : { i : x i • Yi};

is the normalized Hamming distance or d-bar distance between blocks. It has the following properties:

(1)

(a) d(B(g), C(g)) = el(B, C), g E G, (b) cl(v(B), v(C)) = d(B, C), (c) d(B x C,B x D) = d(C,D), (d) d(al .-.Ak, BI.. .Bk) = (I/k))-'~_l d(Ai,Bi)

(IAil = IBjl, 1 < i,j < k),

IAIIA21a(A1,B1) + IA21 (e) d(A1A2,BIB2) = [All-I- [All + [A2[

(I-4;I--IB;I, i = 1,2), I , I I

(f) d(A,B) > ~lld(A1,B1) if A =A1A2, B =B1B2, I - - I

(g) d(B, C) _< d(B,A) + d(A, C).

208 J. KWIATKOWSKI AND Y. LACROIX

I f IBI _> 2, we let B be the block o f length IBI - 1 defined by

(2) /~[i] = B[i + 1] - B[i], 0 < i < IBI - 2.

We shall also use the following property o f the d distance:

(3) 3(B, ~) < 33(B, c ) .

1.3. The dynamical system associated with a sequence We let S denote the left shift homeomorphism o f f~ or f~[] = (G u {D}) z. I f

w = w[0]w[1] . . . is a one-sided sequence over G, we let wD be the element o f f~n

defined by orb[n] = or[n] i fn > 0, ovt~[n] = D otherwise. We then define

12o, = {y E f~ : 3(ni), ni ~ o o , y = l i .m~/wu}. 1

The topological flow (f~,o,S) is min imal if there is no proper closed and S- invariant subset o f fro,. We say that (fL,, S) is unique ly ergodic i f there is a unique

borelian normalized S-invariant measure #~ on f~o,. (fL,,S) is said to be s tr ic t ly

ergodic i f it is both minimal and uniquely ergodic (for short, we say that o; is strictly ergodic).

Ifo; is strictly ergodic, then for each block B, and q E Z,

~ a ( [ B ] q ) : f l ' ( B , w ) ,

where [B]q = { y E ~o; : Y[q, q + ISl - 1] = B } .

1.4. R a n k and covering number o f (f~,,S, #,o)

For an ergodic dynamical system, the rank and the covering number are classical measure theoretic invariants ([d J], [Fe]). In the case o f a symbolic strictly ergodic

system (ft,o, S, #~,), we formulate their "combinatorial" definitions below.

Let .4 be a (finite) family o f blocks and B a block such that [BI ~ {I-41 : A E A}; we let

~I(B,A) = min{d(B,A): A E A, IAI = IBI}.

I f A = {A1,. . . ,Ak}, C is a block, and 6 > 0, we define

ICll + I~] + ICpl} t6 (.A, C) = t6 (.4 l, " " , Ak, C) = m a x

where the maximum is taken over all concatenations o f the form

C = ~1C1E2... cvCpEp+l

M U L T I P L I C I T Y , R A N K PAIRS 209

for which ~l(Ci, .At) < ~, 1 < i < p. Then we define, for a strictly ergodic one-sided

sequence w,

t6(A,~o) = liminft6(A,~v[0,N]) (= l imt6(A,~v[0,N])) . N---* oo

In particular, t6(A,~v) is defined for a block A. It is known ([dJ], [M2]) that in the

case under consideration the rank o f ( f ~ , S, #,~) is at most r i f for any ~5 > 0 and

any N E 1% there exists .4 ofcardinal i ty r such that IA[ > N, A E A, and

t~(.A,~o) > 1 - 6 .

Then (fL,, S, Iz~) is of rank equal to r i f it is o f rank at most r but not at most r - 1.

The rank is a measure-theoretic invariant. We denote it by r(S) or r~,. We say that the covering number F*(w) (also denoted by F*(S)) of~o is at least

a ( 0 < a < 1 ) i f

V~5 > 0, Vn>_ 1, 3A, [A[ _>n, t6(A, t o ) > a .

Then the covering number F*(o~) is the supremum of such a's. The covering

number is a measure-theoretic invariant, and

r~. F*(w) >_ 1.

1.5. Adding machines and cocycles Let T : (X,B, #) , (X,B, #) be an (nt)-adic adding machine, i.e.

A/+I = nt+l/nt > 2 for t > 0, A0 = no _> 2,

nt[nt+l,

oo

X = {x = ~ qtnt-1 : t----O

0 <_ qt <-- At -- 1,n_1 = 1}

is the group of(nt)-adic numbers and Tx = x + i, i = ( I, 0, 0,...). The space X has

the standard sequence (~t) ofT-towers. Namely,

~t = (lYO, D ' ""~ t i t - - l ) '

where

Dto = { x E X : qo . . . . = q, = 0 } , T ~ ( / ~ o ) = D t , s = O, . . . , n t - 1.

Then ~t+l refines ~, and the sequence o f partitions (~t) converges to the point

partition.


A cocycle is a measurable function ~ : X ~ G. A cocycle ~ defines an

automorphism T~ on (X x G, # | m6) by

x e X , gE G,

where mG is the Haar measure o f G. T~, is ergodic i ff for every non-trivial ,,/E

(G is the dual group), there is no measurable so lu t ion f : X ---, S 1 to the functional

equation

,~(~(x)) = f ( T x ) / f ( x ) , x E X [Pal.

1.6. Morse coeycles (M-eocyeles) We say ~ : X ~ G is an M-cocycle i f for every t > 0, qa is constant on each

level D~ for i = 0, 1, ..., nt - 2. Such a cocycle is defined by a sequence o f blocks

(At)t>_o, IAtl = nt - 1, and

~1~ =At[i], i = 0, 1,...,nt - 2.

Now we describe Morse sequences (M-sequences). Let b0, bl, ... be finite blocks

with [btl = At, bt[0] = 0, t > 0. Then we may define a one-sided sequence by

to = bo • bl • �9

Such a sequence is called a generalized Morse sequence over G if it is not periodic

and i f each o f the sequences

OJt = bt x bt+l , . . . , t _> 0,

contains every symbol in G. By grouping some of the bt's we can assume that each

block bt contains every symbol in G. When this is the case, it is known [Ma] that

(fl~, S) is strictly ergodic i f if(g, bt) = 1 / # G for every g E G and t > 0. It is not

hard to observe that the condition

fr(g, b t ) > p > 0 , for e v e r y t > 0 , g E G ,

already implies the strict ergodicity o f to. A Morse sequence to allows one to define an M-cocycle ~ = ~ o n X a s follows:

let

Bt = b o x . . . x bt, t > O.

Then choose (At)t>O = (Bt)t>_o (el. (2)): it is easy to check that it defines an

M-cocycle ~ as above.


It follows from [Kw] and [Le] that the dynamical systems (fL,,S, tzo,) and (X x G, T~,, # | mo) are measure-theoretically isomorphic i fw is strictly ergodic. We finally define

(4) qO(P)(X) = ~(X)"+" " '" + ~o(Tp- Ix) , xEX, p> 1.

Then ~o1~~ = Bt[i + p] - Bt[i], p _> 1, 0 <_ i < nt - p - 1.

1.7. Spectral multipHcity(ies) and continuous Morse sequences If w is a strictly ergodic Morse sequence over G, and qo = qo~ is the associated

M-cocycle, for ~/E (~, we define

L-r = { f | EL2( X x G , # | �9

Let Ur, be the unitary operator induced by T, onL2(Xx G, #| The subspaces L7 are Ur~-invariant and we have the spectral decomposition

LU(X x G, # | m6) = ~]~ LT.

,r60

It is shown in [KwSi] that T~ has simple spectrum on each L. r Let #-~ be the spectral measure of T~, on L. r. It follows from [Ke] that any two of those #-r are either orthogonal or equivalent.

The essential range o f the multiplicity function is then the subset of N consisting of the cardinalities of the equivalence classes of the measures #-r, "/ E G. It is a spectral invariant. The spectral multiplicity then coincides with the maximal number of this set. We denote it by m(T~).

The subspace Le (b := the trivial character) is generated by the eigenfunctions of T~ corresponding to all nt-roots of unity. A Morse sequence w is continuous

(or the M-cocycle ~oo, is weakly mixing) ifL~ contains all eigenfunctions of T~, or equivalently if each measure #7, 7 r b, is continuous. The following is proved in [IwLa]:

Proposition 1.0 For a strictly ergodic M-sequence w over G, a su :c i en t

condition for it to be continuous is that t > O.

1.8. Prel iminary results We know that the measures #~, 7 E G, are either equivalent or orthogonal. We

shall use the following criteria to know which case holds:

Proposition 1.1 ([FeKw], [GoKwLeLi]) l f o~ = box bl x . . . is a strictly ergodic

Morse sequence, where for every t, bt is o f the form

b, = a,v(a,). .. v k,-l (a,),


where v is an automorphism o f G, dt are blocks and kt are integers such that O 0 ~/=0 1 {kt < oo, then I~ ~- l~(7) fo r all 7 in 0, where f~ is the dual automorphism

tOY.

Propos i t i on 1.2 ([GoKwLeLi]) Additionally, i f for given 7, 7' E G,

/ lim 7[~(n')(x)]d# and 7 [~(n')(x)]dl z (see (4)) t--*oo

exist and differ from each other, then #7.1_ #,/ (note that T n' ~ Idx in the weak topology).

1.9. Q u o t i e n t M - e o e y e l e ( s e q u e n c e ) Let H0 be a subgroup of G, and H = G/H o be the quotient group. Let 7rH : G ~ H

be the quotient map (a group homomorphism). If me denotes the Haar measure

on H, then the map

PH : = I d x x 71H : ( X x G,T~,#| --~ ( X x H , T~u,lz| )

is a factor map (the natural factor map associated with the quotient group H).

For a block B over G, we define the block Be over H by

BH = /(B[IBI - 1]),

and for 77 E 12, let r/n be defined by

~TH[n] = 7rH(r/[n]), n E Z.

Using the obvious equality 7rn(B x C) = 7rn(B) x ~rn(C), it is not hard to see that

i f w = b0 x bl x -.. is a M-sequence over G (we let 9 be the corresponding M-

cocycle), then o) H = bon X blu x . - - and it is an M-sequence, over H. It determines

an M-cocycle qo n which, moreover, satisfies

~0H = "/I'H o ~.

We now recall some facts from [KwJLe]: define L. r,n = {f | : f E L2(X,/z) } for

7 E/-/. Then using the factor map PH, we can identify this Ur~n-invariant subspace

ofLZ(X x H, # | ran) with the Ur~,-invariant subspace L~ o f L2(X x G, lz | ma) defined in Section 1.7, where q E G is the unique element of G such that

qlH0 = e I n o and "/oTrn~---" ~.


In fact, if Vp M : L5 ~ L7,n is defined by VpM( f | ,~) = f | 7 then it provides a unitary equivalence between the pairs (L~r, Urn) and (LT,H, Ur~M). The cyclicity of the spaces LT,H under the related unitary action remains, as in Section 1.7.

So we obtain

Proposit ion 1.3 l f the assumptions o f Proposit ion 1.1 hold, and 7, 7' E [-I are

such that ;~, z~ belong to the same f-trajectory, then #7 ~- #7'.

II. The candidates for (m, r)

First of all we select G = Z~ (direct product) and r < n (r _> 2 is given). Some

conditions on n shall be specified in the proof of Theorem (m, r) (see Section IV.3). For h E G, we write h = ( h i , . . . , hr), hi E ~n. We let

ei = ( ~ , 1 , 0 , . . . , 0), 1 < i < r.

i - I times

Then h = hlel + . . . + hre~. Let v E Aut(G) be defined by

v ( h ) = ( h r , h l , . . . , h r _ l ) , h E G.

Now we start defining some blocks (the goal is to define the blocks bt, t > 0). Put, f o r t > 0,

F1 = Ft,1 = O e l ( 2 e l ) . . . ( ( l - 1 ) e l ) ,

F2 = Ft,2 = O e 2 ( 2 e 2 ) . . . ( ( l - 1 ) e 2 ) , �9 : : .

Fr = Ft,r = O e ~ ( 2 e r ) . . . ( ( l - 1 ) e r ) ,

1 = l t ,

w h e r e hilt and It ~ oo. Observe that IF, I = It, and that v(Fi) = Fi+l, 1 < i < r,

where Fr+l := F1. Let

{ 1~1 = ]~t, 1 = F1 x F2 x . . . x Fr,

/32 = fit,2 = F2 x " " x Fr x F1, �9 : : :

t~r = l~t,r = F r x F I x . . . x F r - 1 .

Then I/ ;I = r , and v(A) = / ~ / + 1 , 1 < i < r, where similarly flr+a : = i l l . Next let

/3 = / ~ t = ]~1/32 " '" ~ r (concatenation) ( = ]~1v(]~1) �9 �9 �9 ~,r-I ( i l l ) ) ,

and Pt times

A

b , = = ? . . . . . . v(/ O . . . . . .


where kt = rpt. We have [bt[ = rptl~ := At. We choose the sequence (Pt) such that

~t>O 1/pt < oo. Finally we letnt = A0... At andBt = bo x . . . x bt as in Section 1.6. Let w be the associated M-sequence over G and ~ the related M-cocycle. Notice that nrl)~t .

Now, given 2 < m < r, we define

Ho = { o y " x 7_g,-",

H = G / H o -

and let the subscript H denote the corresponding blocks, M-sequence, M-cocycle as in Section 1.9.

Proposition II. 1 The sequence w is strictly ergodic and ( f~ , S, #9) is measure- theoretically isomorphic to (X • G, T~, # | mr). Moreover, it is a continuous M-sequence.

Proof Using Section 1.6, we shall prove that for t > 0,

fr(g, bt) = 1/nL

Looking more closely at the blocks bt it is seen that for g E G, fr(g, ~t,i) = 1 / # G = I/n r, 1 < i < r. So fr(g,13t) = 1/n" and fr(g, bt) = 1In r.

S i n c e nr[At, we may apply Proposition 1.0 to obtain the continuity. []

Proposition 11.2 The sequence wH is strictly ergodic and (f~M, S, #~n) is measure-theoretically isomorphic to (X x H, T~,M, ~ | mn). Moreover, it is a continuous M-sequence.

Proof We use facts from Section 1.6 again. The strict ergodicity ofwH follows, as in the preceding proof, from the equation

fr(h, bin) = Y~ fr(g, bt) = #HOn -'7"~,( = #H1 = _~)1, h E H. 7rn(g)=h

The continuity is proved as for w. t~

The system (fl~,S,#~) is our candidate for the (r,r) pair, while the system ( f~H, S, # ~ ) is our candidate for the (m, r) pair.

III. Spectral muitiplieity(ies) of T~, T~oM, Theorem (r, r)

We first apply Propositions 1.1 and II.1, Section 1.7 and the description o f

candidates in Section II to obtain, using ~]t>_o 1/kt < oo,

Proposition III.1 m(T~,) > r.


P r o o f As indicated above, only check that the v-trajectory of el is of length r. []

Proposition III.2 m(T~. ) >_ m.

Proof Let 7-/0 = ann H0 = {'y E G : "/l~0 = el~0}. Then as in [KwJLe], using Section 1.9 and Proposition 1.3, we have

m(T~o.) E max{#(7-/o r T)},

where T runs along the set of ~-trajectories in G. As is easily computed, this max is equal to m. []

To prove that m(T~) = r, we shall use Proposition 1.2.

Proposition 111.3 m( T~) = r.

P r o o f Recall that from the block constructions in Sections II and 1.6, we have

~1~ = Bt[i + 1] - Bt[i]. F o r h = hie1 + . . . + hrer E G, let h( t ) = hi + h2lt+l + . . . + r - I r -1 hence h(t) / "~t+l ~ 0. M o r e o v e r , hrl~+ 1 . Then h(t) < 2nlt+ 1 ,

Dt+ 1 ~(h( t )n t ) = bt+l [p + h(t)] - bt+l [p], X E pn,+j'

f o r 0 < f < nt and 0 <_ p <_ ) ~ t + l - - h(t) - 2. W e w r i t e p = qrfft+ 1 + il~+ 1 + w, w h e r e

0 < q < Pt+l, 0 < i < r, 0 < w < Ft+ 1 . Next we write w = wl -Fw2lt+l + " " +Wrfft+t where0 < wj < I t+l , 1 < j < r. Assuming that for each 1 < j < r, wj < lt+l - -n- - 1, we see from the block construction that for suchp's,

_____ ~- r ) t+ l ~(h(t)n,) vi-l(t3t+l,l[W+],(t)]_3t+l,l[W]) v/-1 (h), x,_~pn,+j,"

For given h and t, the set o fp ' s satisfying the desired conditions from above has cardinality )~t+~ - o()~t+l), hence we obtain, decomposing X along each T-tower

~t+l, that for any 7 ~ G,

t l i m f x ~o ChCt)n') 1 r-i~ ~" ( 7( (x ) )du(x ) = r

Y - - 1 ^ " t l Moreover, (h(t)nt) is a rigid time for T. For "~" ~ G, let A.~,, -- ( l / r ) ~---0 ~(~' ). I f ~/, 7' E G do not belong to the same ~-trajectory, then A. r / A. t, (in g2(o, ma)). Hence since A~ ~ 0, there exists an h E G such that A.r(h ) ~ A~,(h). Then taking

the rigid time (h(t)nt), applying Proposition II.2, we obtain #-r .1_/~-r'. We conclude with Proposition III. 1. []

Proposition III.4 m( T~,,) = m.

216 J. K W I A T K O W S K I A N D Y. L A C R O I X

P r o o f Using the same notations as in Section 1.9 and Propositions 11.2, 111.2,

and 111.3, we deduce that i f 7, 7' E A r, if #~ _L #4' then #-r _L #.~, as in [KwJLe]. Hence we may deduce the equality

m(T~o,) = max{#(7-t0 n T)},

where T runs along the set o f 0-trajectories in G. []

Next it is easy to compute that the set o f lengths off- t rajectories in (7 is {d : dlr }. Hence with Section 1.7 and Proposition III.3, we conclude that the essential range o f the multiplicity function o f T~, equals {d: dlr}.

And we see that i f for 1 < s < m, Ts denotes the f,-trajectory o f the character

dual to el + ..- + es, then #(7-/0 n Ts) = s. Therefore, with Propositions 111.3 and 111.4, we have obtained:

Proposition 111.5 The system (Xx G, T,, # | mG) satisfies m( T~ ) = r, and the essential range o f its multiplicity funct ion is { d : dlr }.

For 2 <_ m < r, the system (X x H, T~,, # @ mH) satisfies m(T~, ) = m, and the essential range o f its multiplicity function is { 1 , . . . , m}.

Now we pass to proving that r(T~) = r. Since r(T~) >_ m(T~) = r, we only prove that r(T~) <_ r. Let us define

Ei : : gt , i = B t -1 x t3t, i , 1 0. For 0 < u < F, we have

(5) (a) U = U l + u 2 l + " ' + u r F - ~ , O < u i < l , l < i < r , (b) 3i[u] = v/-l(31[u]) = v/- l (ule l + . . . + Urer), 1 0, we have

31 (g)[u] = (Ul + g l , . . . ,Ur + g r ) = 31[U + u(g)],

where u(g) = gl + g 2 1 + . . . + grF -1, and u(g) s 2nF -1. Hence u(g)/]31[ s 2n / l and

a / ( ~ t , l ( g ) [ 0 , f f t - u(g) -- 1],/3t, l [ u ( g ) , f f t - - 1]) < 2 n / l t -*t O.


In a similar way we obtain, using/3i = ~-l(/31), 1 < i < r, and ( lb) , (5b), that the

above inequality is valid for/3t,i, 1 0,

te( {Et,i : 1 1 - 2n/lt ~t--,oo 1,

i f 2n/lt < 6, because as is obvious from Section 1.4, i f 0 _< 6' < 6, then

t6({Et,i : 1 t6,({Et,i : 1 <_ i < r},w). Hence we obtain, using

Section 1.4 and Proposition III.5:

Theorem (r, r) The system (Y x G, T~, # | mG) satisfies r(T~) = r = m(T~), and the essential range o f its multiplicity function is {d : dlr}. I t is measure- theoretically isomorphic to a strictly ergodic continuous Morse automorphism.

IV. The rank o f T~.: Theorem (m,r)

Using Theorem (r, r) and the natural factor PH, it is obvious that r(T~.) <_ r. So from now on, we make some computations to show that F~(w) < 1/(r - 1),

because F*(T~.o.) �9 r(T~.) _> 1. We first o f all proceed by introducing new blocks

(over the alphabet H), more convenient than those having the subscript "H".

Recall that

WH = b0. x b l . x �9 �9 �9

bt. =/~t;t, I ~ t . / ~ t , l . " " " / ~ t , r . ,

f i t , i v = F t , i . x " " x F t , . H x Ft,1. x . . . x Ft,i-l..

Identifying H with Z m = elZn -~- ' ' ' + emZn) where ~l , . . . ,~ ,n are the natural

generators o fZ~ , we get ~rH(ei) = ei for 1 < i < m and IrH(ei) = 0 for m + 1 < i < r.

Then Ft,i. = Fi. = 0~i(2ki).. . ( ( l - 1)~i) i f 1 < i < m,

Ft,i. = F i . = 0 . . - . . . 0 i f m + 1 < i < r.

It times

So from now on, we shall write e l , . . . , em, Fi = Ft,i, ~i =/3t,i, bt, w instead of

e l , . . . , em, Ft,i., ~t,iM, btM, and wH respectively. Therefore we shall make the forthcoming computations with the following new

blocks (over H):

Ft,i = Fi = Oei(2ei)... ((I - l)ei) if i < i < m,

Ft,o =Fo=Ft,i=Fi=O ...... Oifm+l<i<r,

I times


and r - - m

f l l = F1 x . . . x F m x F o x . . . x F o , r - -F r

f12 = f 2 X " " x Fm x F 0 x . . . X Fo x F1 ,

u

tim =Fro x F 0 x - . . x FoxF1 x . - - • Fm_l, r - - m

A

flm+l : ~ " 0 X " '" X F g x F 1 x . . . xFm,

r-m-1 A

fir = F o x F ] x . . . x F m X ~ ' O X " " x F o .

Then 3 = 31 " " 3r, bt = ~P', ~: = bo x bl x . . . and wt = bt x bt+l x . . . . Notice that

for t _> 0, w = Bt x wt+l.

In the sequel we shall need the formulas for the values 3i[u], 0 <_ u < F. It follows from the definition o f the F ' s and 3 's that ((5a))

(6) ill[u] = u~ i) e, + . . . + U(im ) em,

where

(7) ( u l i ) , . . . ,U(im )) = (Ur--i+2,... ,Ur, Ul , . . . ,Um_i+l) i f l < i < m,

(ul i), , U(im )) = (Ur-i+2, , . . . , Ur+m-i+l) i f m + 1 max{ 16, 8 r / m } , for every t _> O.

Let

(9) O < _ s ' < r l r - l , s ' = i ' / r + u ' , w i t h O _ m / 6 r .


P r o o f Using (3) we get

(10) ~l(/3(a),Ig,h) > �89 h).

T a k e 0 < s < rl r - 1 and represent it a s s = il r + u w i t h O < i_< r - 1 and

0 0,

and

(12) /3(a)[s] = e = - ( e l + . . . +em) i f u = 0.

Next let S = {0 < s < rF - 1 : 0 <_ ul _< l - 2, where u is o f the form (5a)}, and

H =

111 =

el "" .e l e e 2 . . . e 2 e . . . e e m . . . e m e O . . . O e . . . e O ~ , I t --1 lr-- I lr--1 lr--1 lr-- I

IIelI[s', s' + rF - 2], (llI[ = r t r - - 1).

It follows from (6) and (7) that whenever ul < l - 1,

I ~ i[u]=ei i f l 1 - 7 '

which implies

.(14) cl(~(a),11) < 2/l t .

Similarly we establish that

(15) ~l(1g, n,111) < 2/l , .

It is easy to remark that

(16) cl(11,III) > m / r (because 1 < i' < r - 2).

Now, using ( lg) , (14)-(16) we get

220 J. K W I A T K O W S K I A N D Y. L A C R O I X

m < ~r I I I ) r

< ~1(II, 13(a)) + {l(13(a), Is + a(Is III)

4 < ~ +cl(fl(al,Is

The above, (8) and (10) imply

(18)

Addit ionally ,

l ( m ~ ) m d(/3(a),Ig,h) >_ -~ r - > 6--;"

[]

m (19) cl(l~(a),Ig,h[S',S t + rl r -- 1]) > 8-~v,

f o r any a , g , h e G, where v = u ' / l r i f i ' = O a n d v = 1 - u ' / F i f i' = r - 1. Fur thermore ,

C a s e 1~ if i ' = 0 then

(20) a ' - ~ = p ( e l + . . . + e r a ) and ut < 4 l r ( 6 + l ) .

I f g - a # gt _ a' then

m (21) 3(H(a) , Ig ,h[S ' ,S ' + r r - 1]) _> (1 - u) + ~ v .

C a s e 2~ i f f = r - 1 then a t - h' = p ( e l + . . . + era) a n d u' > F(1 - 46 - 1/l).

I f h - a # h t - a t then (21) holds.

P r o o f The inequality 6 < m / 6 r , (17) and L e m m a IV. 1 imply i' = 0 or i' = r - 1. Assume that i' = 0. Using (10), (14), (15) (with a' ,g~,h ') we have

(22) {l(~(a'), Ig,h, Is', s' + rE -- 1]) _> �89 (~l(II, I I I ) -- 4 / l ) .

L e m m a IV.2 Assume that

{ m 1 1 } (17) 2l(~(a'),Ig, h,[S',s' + rl r - 1 ] ) < 6, whereO < 8 < min 6r ' 2 r ' 1 6 '

- t l r - - I a n d s' is as in (9), a ' , ~ , h' E G, u' = u l' + u~21 + . . . + Ur~ . Then either i' = 0 or i t = r - 1, a n d

- t - p m o d n f o r s o m e p E Z n . U~ ~-- �9 �9 �9 = U r


It is easy to see that i' = 0 implies ~1(11,III) > (u' - 1 ) / l r. Thus (22) gives

1 U t - 6 > g ( F 1 / )

which in turn implies that u' < 4 F ( 6 + 1/ l ) . This proves the second part o f (20).

To prove the remaining part o f the Lemma let us remark that the last inequality implies u' < �89 because 6 < �88 and 4 / l t < 41- ((8)). Then using ( l f ) we get

3(/~/(a ' )[o,r- u ' - 1], tS i (g)[Ut ,1 r - 1]) f i r - ~ t t

<_ ~ _ u , d ( ~ ( a ),&,,h,[s ,s + r r - 1]) < 2r6 < 1,

for every 1 < i < r. Thus there exists at least one u, 0 < u < F - u ' - 1 (depending

on i) such that ~i[u] + a' = ~3i[u + u'] + g ' . This equality and (6) (for u and u') give

(23) u~ (i) e l + . . . + U/m(i) em = a ' _ g l .

Then using (7) we obtain (18), which with (23) implies that a0 - g o = p ( e l + . . . + e n ) .

This completes the p roof o f (20).

Now we will prove (19) and (21). Using the same arguments as before we get

that the equality 13/[u] + a = 13i[u + u'] for every 0 < u < F - u' - 1 is equivalent to the condition u] (/)el + . . . + Utm(i)em = a - g. This equivalence and (23) then imply

f

(24) {I(II1/,1V/) = ~ 0 i f g - a = g' - a',

t 1 i f g - a # g l - a ' ,

where 111i =/3/(a)[0, l r - u' - 1], IV/=/3i(g)[u ' , g - 1], 1 4, we have d(A1,A2) _> ~.1

Let us examine the number u'. I f p > 0 ((18)) then u' > F -1 > 4. I f p = 0 and

u' > 0 then nlu ~ for 1 n > 4. Denoting

V~ = ~i (a)[F - u' , F - 1], V/i = ~i+1 (g)[O, u' - 1], 1 4.


It follows f rom (25) that

(26) 8(Vz', V/i) _> ~ for 1 r (1 - ~-) y~cl(llli,lVi)+-[;i=l i=l

m u'

- 8 r l r"

I f in addit ion g - a # g~ - a ' , then (24) and (26) imply (21). This proves Case 1 ~ . The p roo f o f Case 2 ~ is similar. []

L e m r a a IV.3 Le t ~ = q'rl r + f l r + u', l = It, 0 <_ q' < Pt -- 1, 0 < i' < r - 1,

0 < u' < l r - 1, and let

(27) Ig,,h, = bt(g ')bt(h ')[~,s~ + p t r F - 1].

A s s u m e that ~l(bt(a'),l~,h, ) < 6, where

1 . m 1 l } 0 < 6 < 3 m m { 6 r ' 2 r '

! Then u~ - . . . =_ u r - p rood n f o r some p E ~ a n d ei ther i' = 0 or i' = r - 1.

I f s~ <_ �89 then a' - g~ = hp; a n d i f ~ >_ �89 then a' - h' = hp, where

hp = p ( e l + . . . + era).

Moreover, pu t t i ng v = u ' / l r i f i' = 0 and v = 1 - u ' / F i f i' = r - 1, r = q ' / P t i f

s~ <_ �89 r a n d ( = 1 - q ' /p t i f s~ >_ �89 we have:

C a s e 1~ i f i ' = 0 then u' < F(46 + 1/l) and

m (28) cl(bt(a),l~,h) > -~ru f o r a , g , h E H,

m (29) ~l(bt(a),l~,h) > - ~ r V + ( 1 - v ) ( 1 - ( ) i f g - a # h p a n d h - a = h p ,

m (30) ~l(bt(a),l~,h) > -~r v + (1 - v ) ( i f g - a = h t, and h - a ~ hp,

m (31) ~l(bt(a),l~,h) > - ~ r V + ( 1 - v ) i f g - a r


Case 2~ /f i ' = r - 1 then u' > / r (1 - 4 6 - 1/l) and (28)- (31)hold .

Proof Using (ld) and (le) we obtain

(32)

a(bt(a ' ) , I~ ,h , ) =(1 q ' - 1 )a (3(a ' ) , lg • , [ s ' , s ' + rl r - 11) P t

+ q ' 3 ( 3 ( a ' ) , I h , , h , [ S ' , S ' + r l r - 1]) P t

+ L 3 ( 3 ( a ' ) , l g , , h , [ S ' , S ' + r l r - 1]), P t

where s' = i 'F + u'. Assume that s~ < �89 r. Then 1 - (q' - 1) /pt > ] and

using (27), (32), we obtain the inequality cl(3(a'),Ig,,g, [s', s' + rF - 1]) < 36. Then Lemma IV.1 implies i' = 0 or i' = r - 1. Now, we apply Lemma IV.2 with a ' ,g '

andh ' :=g ' . We then have u~ = . . . = U'r = p m o d n a n d a ' - g ' = hp. I f i ' = 0 then u' < F(46 + 1/l). Further, (19) and (32) imply (28). Similarly, (29), (30) and (31) are consequences o f (19), (21) and (32).

I f / ' = r - 1 then u' > / r (1 - 46 - 1/l) and the proof o f the remaining part o f the Lemma is the same. []

L e m m a IV.4 Le t C , D be blocks o v e r H s u c h that ICl = [D[ + 1, [D[ = k > 1. Le t 0 < s' < p t r l r - 1, l = It and

I = bt x D, 11= (bt x C)[s ' ,s ' + kptrl r - 1]).

A s s u m e that 3(1,11) < 6, where

1 { r o l l } O < 6 < ~ m i n 1 6 r ' 2 r ' 1 6 "

Then there exists p ~ Z~ such that

(33) [l(D(hp), C[O, k - 1]) < 6,

o r

[l(D(hp), C[1,k]) < 6,

where

(34) hp = p ( e l + . . - + e r a ) .

P r o o f We l e t / j = bt(D~]), 11j = bt (C~])bt(C~ + 1])[s',s' + p t rF - 1], for 0 _<j <_ k - 1. Then applying ( ld) we deduce that

(35) k - 1

a(i,11) = 1 <

j=O


so there exists a 0 < jo < k - 1 such that [1(/jo,I/jo) < 6. Let us suppose that s' < �89 r. Then Lemma IV.3 implies that D[jo] - C[jo] = hp for s o m e p E Zn. Let

Zo = {0 < j < k - I : D [ j ] - C[.~ # hp}, Zl = { 0 _ < j _ < k - 1 : D [ j ] - C [ j + I ] • h p } .

Then using (35), defining for shortness dj := [1(/j,I/j), 0 _<j _< k - 1, we obtain

(36) 6 > [t(1,11) = -k ~ 1 jEZo\Z, jEZ,\Zo j~ZoUZt

Using (28)-(31) we get

m d(/j,1/j)) > ~rr v + (1 - v)

m d(/j,I/j)) > ~r v + (1 - v)(

m [1(~,i/j)) _> ~ . + (1 - v)(1 - r

m [1(/y,I/j)) >_ ~r v i f j r Zo u Z1.

So (36) and the preceding four inequalities imply

m 1 - v > ~r v + T ( ( 1 - ( ) # Z o + r

(37) ( m min{#Zo, # Z 1 } )

> v ~ k

ifj E Zo n Z1,

It follows from the above calculations that

i f j E ZI \ Zo,

i f j E Zo \ ZI,

min{#Zo, #Z1 } + k

> ~ min{#Zo, #Z1 },

so since 6 < m/16r, we obtain

m min{#Zo, # e l } > 0 .

8r k

Thus with (37) we deduce that

min{#Zo, # z , } < & k


Finally, let us observe that

min{#Z0, #Z1 ) = min{cl(D(hp), C[0, k - 1]), ~l(O(hp), C[1, k])}, k

from which (33) follows. []

C o r o l l a r y IV.1 Under the assumptions o f Lemma IV.4, i f additionally we 1 assume that46k < 1, then D(hp) = C[0, k - 1 ] / f ( < �89 andO(hp) = C[1, k] t f ( > 3"

P r o o f I f follows from the definition o f v (Lemma IV.2) that 1 - v > �89 if

46 + 1~It < �89 (use also (8)). Then (37) implies

#2:0 i f f < 1 # Z I i f f > 1 6>-7-k-- and 6>-Tk--

Thus either #7-.0 = 0 or #Z1 = 0. t3

L e m m a IV.5 Let C,D be blocks over H such that k = IDI > 1 and IC[ = [D I + 1. L e t O < ~ < nt - 1 a n d 1 = B t • D, [ I = (Bt • C)[s , s + k n t - 1]). S u p p o s e t h a t

1 { m l l } (38) cl(I,[1) < 6 where 0 < 6 < ~ rain 16r' 2 r ' 16 "

Then there exists p E Zn such that

{ cl(D(hp),C[O,k- 1]) < 6,

(39) or [t(D(hp), C[1,k]) < 6,

where hp is defined by (34).

P r o o f We use an induction argument on t. For t = 0 the Lemma follows from

Lemma IV.4. So let us suppose that (39) is true for t - 1 and assume that (38)

holds for t.

We have Bt x C = B t -1 x (bt x C) , B t x D = B t - i x (bt x D ) , and s = s l n t - i +$2,

where 0 < Sl < At - 1, and 0 < sz < n t - i -- 1. So letD1 = bt • D and C1 = bt • C[Sl,Sl + A t ( k + 1) - 1]. Then (38) can be

rewritten as c l (B t - i • D1, (Bt-1 • C1)[s2,s2 + knt - 1]) < 6. Using the induction hypothesis we obtain

(40) d(D1 (hp~), Cl [0, kAt - 1]) < 6,

or

(41) d(Dl(hp~), C1 [1, karl) < 6,


for some p l E Zn. Let us suppose for instance that (40) holds. Then it can be rewritten as ~l((bt x D)(hpl ) , (b t x C)[Sl,Sl + kAt - 1]) < 6. Then Lemma IV.4 implies that either ~l(D((hpl ))(hp2), C[0, k - 1]) < 6 or ~l((D(hp, ))(hp2), C[1, k]) < 6

for somep2 E Zn. Let t ingp = P l +P2, these two last eventualities read as (39). []

IV.2. Special subsequenees and subblocks o f w To estimate the numbers t t (A, w) for the blocks A appearing in w, we distinguish

special subsequences o f cv and then we examine the possible appearances o f all long enough blocks in those subsequences.

Every f ragment / j := w[jnt , ( j + 1)nt -- 1],j _> 0, is o f the form Bt(h) for some h e H. So b y a t -symbol o f w we mean a fragment o f w like/j , that corresponds to

a block Bt(h).

Given a fragment A = w[q,q + s - 1] o f w we define

A ~ = w[q - 6s, q + s + 8s - 1],

where 0 < 6 < �89

Gathering the t-symbols Bt(h) from their natural positions (like for /j) we can define disjoint subsequences wt(h) of w, h E H. Precisely, let Nh =

{ j > 0 : lj = Bt( h ) } and put wt( h ) = LJjeNh 1j. In the same way we define

(41a) ~ = U / f . jer~

Other natural subblocks o f w that we need to distinguish are the blocks

Et, i(h) : Bt x ~t+l,i(h), 1 0, is equal to some Et,i(h), for some 1 o: = E,:(h)},

The blocks

f-J+t times :_~. ~.~'~.. ~ . . . Lt,i,g ~ "~, , l = l t + l , i = m + l , . , r , g E H ,


also appear naturally in wt+ 1 . Any block/~t+l,i(h) is a concatenation o f the blocks Zt,ia , where g runs over H; moreover,

1 g, h E H , m + l < i < r . fr(Lt,i., 3t+ l,i( h ) ) = n m ,

Define Mt,i,g = B t • Lt,i,g, m + 1 < i < r, g E H. The blocks Mt,i,g appear in w at the positionsjntl r + sl r-i+l, l = It+l, j E Ni(h), 0 < s < l i - l , h E H. L e t u s denote

and define the following: Illj,s = IIItj,s = w~ntl r + sl r-i+l ,jntl r + (s + 1)/r-i+l - 1],

j ~ Uh~uN~(h), 0 <_ s < l ~-l,

(41c) { Ni,g OJt+ 1 ,i,g

W ~ t+ 1 ,i,g

= {( j , s ) : j E UhEHNi (h ) , 0 < s < [ i - l , 11~,s = g t , i , g } , =

= U(/,s)~N,. 111fs.

The subsequences defined above (41a--e) enjoy the following properties (for fixed t > 0):

(42) wt(h) are pairwise disjoint when h runs over H, [.Jh~wt(h) = w, and O(wt(h),w) = 1/n m, h E H,

(43) wt+l,i(h) are pairwise disjoint when h runs over H and 1 < i < r,

r ra UhEH [-Ji=l Wt+l,i(h) = w, and D(wt+l,i(h), w) = 1/rn ,

(44) ~ wt+l,i,g are pairwise disjoint when g runs over H and m + 1 _4 and 61t+l >_4.

Now, we classify the subblocks A < w such that 1.41 _> 3no. For every such block there exists a unique t _> 0 such that

(46) A = Ez (Bt x C)E2,

where ICI > 1, C < bt+l (g~)bt+l (h') for some g~,h' e H such that (g')(h') < 03t+2, E1 (resp. E2) is a right-hand side (resp. a left-hand side) o f a t-symbol.


Given 6 > 0 and t _> 0 satisfying (45), we define three subsequences &~, &2,0J3 ofw by

~t, 1 = ~1 = Uj>OW[( j -- 6)nt, ( j + 6)nt],

~t,2 = ~2 : Uj>0 w[(j - 8)1;+i, ( j + 6)l~'+11, W,,3 = W3 : Uj>0 w[( j - - 6)lt+l, ( j + 6)lt+l].

Notice that D(&l U &2 U ~3, w) < 66, where ~bl U &2 U &3 denotes the obvious subsequence ofw. We shall examine blocks (46) satisfying the additional condition

(47) `4 n \ (Q1UQ,, UQ3)) # O.

We now distinguish six classes E l , . . . , E6 for blocks of the form (46) that are such that any A satisfying (47) belongs to at least one of these classes:

(A) . 4 E E l i f f l C l < 3 ,

(B) `4E /C 2 i f f lCl > 4 and C < fiSt+l,i(h ') where h' E H and l 8, j = 1,2, C 1 <~ fii(gt),

C2 < fii+l (gs), l 4, m + l 6, fr(Lt,i,h,, C) > 6, ~iLt,if, i C) > 6, where g', h' , f ' are pairwise distinct,

(F) `4 E ~6 i f f f i t + l ( g t) < C f o r s o m e ~ E H .

I V . 3 T h e o r e m (m, r)

Let �9 m 1 1 ,


A be as in (47) a n d I = w[q,q + ]AI - 1]. We shall estimate t~,(A,w).

L e m m a IV.6 Assume that ~I(A,1) < 8, 0 < 8 < 360, It >_ 4 a n d A E IC1 (el. (A)). Then I < UhEHAW3t(h), where Ha c H with #Ha < n. Hence t6(A,w) < 7 / (n m-l) + 68.

P r o o f Let ql = IExt (q~ < n,), and k = ICI. Using ( l f ) we have

38 > 3d(A,I) > (l(Bt x C , w [ q + q l , q + q l + k n t - 1]).

So, applying Lemma IV.5, we get

(48) (l(B t x C,w[snt ,(s + k ) n t - 1]) < 38,

where [sn t -- q - ql[ < nt. This implies

(49) }snt - q] < 2nt.

Next, (48) gives ~l(C, wt+l[S,S + k - 1]) < 38, and Corollary IV.1 (4k8 < 1 holds) implies that either C(hp) = wt+l Is, s + k - 1] or C(hp) = wt+l [s + 1, s + k] for some

p E Z~. Putt ingpl = C[1] + hp i f I CI = 3 and C[0] + hp i f I Cl < 2, we get that either wt+1 Is + 1] = p l ( i f k = 3) or wt+l [s] = P l ( i fk < 2).

Assume that k = 3, for instance. The equality Wt+l Is + 1] = C[I] + hp means that the fragment w[(s + 1)nt,(s + 2)nt - 1] < Up~z wt(C[1] + hp). Taking into consideration two neighbouring t-symbols from the right and left sides, plus the

blocks El and E2, we deduce that I < Uh~H~ w3(h) where # H a < n. In the case k < 2 we obtain the same conclusion. Using (42) and (47) we deduce

that 7n 7 t~(a,~) < n-~- + 66 = n-WZi-_l + 6&

[]

L e m m a IV.7 Assume that~l(A,I) < 62/6, 0 < 6 < 360, a n d A 6 K~ 2 (see 03)). (26)

Then I < Oh~HWt+l,i(h). Hence

1 + 4 6 t62/6(A,w) <_ ~ + 66.

r

P r o o f The same arguments as those appearing in Lemma IV.6 lead to

~l(C, w t + l [ s , s + k - 1]) < 62/2.


The conditions (47) and 61t+l _> 4 imply that C contains a subblock C such that [C[/[C[ > 1 andC < 3t+l,i(h') for some h' E H. Hencecl(C,w[sl ,s l +[C[ - 1]) < 62,

where s l , . . . ,sl + [C[ - 1 are the positions of C in C. Writing C and w[s,s + k - 1] instead of C and o:[sl,sl + [C[ - 1] respectively,

suppose that wt+ 1 Is, s + k - 1 ] contains a subblock Dl such that ID1 [ > 4, ID1 [/k > 6,

D 1 < UhEHWt+l,i(h) and i # i'. Denoting by C1 a subblock of C appearing at the same positions as D1 in Wt+l Is, s + k - 1] and using (If) we get

62 > cl(C, oJt+l[S,S + k - 1]) _> ~-~3(C1,D1) > 6d(C1,D1),

which gives d(Cl,D1) < 6. However, the property (25) says that d(Cl ,DI ) > 1. Hence either ID11/k < 6 or ID11 _< 3. In both cases this means that

B, • o:t+l[S,s + k - 1] ,( 0 W:+l, i(h) hEH

because k < 13il and 3/lrt+l < 6 (el. (45)). Enclosing the (wings) blocks El ,E2 we obtain I < 2~ Uh~HWt+l,i(h) on the basis of(49) and the inequality 2/nt < 6.

We conclude as in the above Lemma, using (47) and (43), to obtain

1 +46 t62/6(A , w) < ~ + 66. []

r

L e m m a IV.8 Let d(A,1) < 63/36r, 0 < 6 < 360, and assume A E E3 (see (C)). Then either

1 +46 1< 0 ~'t+2'(28)//~3k")' ~HA _<n, thent63/36r(A,w ) _< n--~sy-1 + 66 ,

hEHa

or

(51) I < [..J ,(2~) t'b~ i ' = i o r i " i + 1 , thent63/36r(A,w)<_ 1 + 4 6 ~t+l,i'~"l' = ~ + 66, hEH r

where i is defined in (C).

P r o o f As before we have ~t(C, wt+l [s,s + k - 1]) < 63/12r. Let D1 and D2 be subblocks ofwt+1 [s,s + k - 1] occupying the same positions in it as C1 and C2 do in C, respectively. Then we have

lea IGIIAI a ICl - ICl ->


Hence

(52) d(C1,D1) < 62/6 and d(C2,D2) < 62/6.

(26) It follows from the proof of Lemma IV.7 that 01 <~ [jh~HWt+l,i(h) and D2 <s. (26) I.Jh~H %+1 :+1 (h) (these last inclusions are also valid for i = m and i = r). From the

above inclusions we deduce that O102 for some gl

I f in addition D1 </3)+2~),i(gl ) (or D2 < 3)+2~),i+1 (gl)), then (51) holds. I f not then

(53) (26) (26) 0102 <s 1,i(gl ),3~+ 1,i+1 (gl)-

Further, (lc, e) and (52) imply d(bt x CI C2, bt x DID2) < 62/3. Applying Lemma IV.4 we deduce

d((C1C2)(hp), (DID2)[0, IClC2l - 1]) < 62/3,

or

(55) d((ClC2)(hp),(D1D2)[1,[ClC2[]) < 62/3, fo r somep E Zn.

Suppose that (54) holds. Then we can write Cl = E(go), C2 = E'(go), D1 = E(gl), D2 = E'(gl ), where E is a subblock of the right side of 3i and E' comes from the left side of 3i+l. Thus (54) has a form cl((EE')(go + hp), (EE')(gl)) < 62/3. Hence we must have gl - go = hp. Putting HA = {go + hp : p E Xn }, we deduce from (53) that (50) is satisfied. In the same way we deduce (50) from (55). If(50) holds, then using (42) and (47) we have

1 + 4 6 t:/36~(A, ~) <_ ~ + 66.

Else for (51) we use (43) instead of (42) and deduce

1 +46 t63/36r(A,w ) < ~ + 66. []

r

L e m m a IV.9 Letd(A,1) < 6/2, 0 < 6 < 360, andA E K;4 (see (D)). Then

r

I < [,3 ,(1+26) for some g' E H. wt+ 1 ,i,g' i=ra+ 1

Hence 3 +46

t6/2 (A, 60) <_ n-------~ + 66.


P r o o f The assumptions of the Lemma imply

(57) s tot+l[s,s + k - 1]) < 6/2.

First assume that C < Zt,io,g,. Then (57) implies directly that wt+l[s,s + k - 1] < W/+l,;0,g, for some m + 1 < i0 < r. Using the same arguments as in the preceding

r ,(26) Lemmas we get I < Ui=m+l -t+l,i,g"

L ~ L ~ The conditions (47) and the inequality Now assume that C < t,io,g, t,io,h'. 6lt+1 > 4 imply that C contains a subblock C1 such that [CI[/[C[ > �89 and C1 Lt,i,g,,, where g~ = g' or h'. Repeating the above arguments we obtain I1 =

I I r w (26) . This implies (56). Then (47) and (44) imply that (Bt • C1) < k.3i=m+l t+l,i~'

3 +46 t6/2(A, w) <_ n ~ + 66. D

L e m m a IV.IO Letcl(A,I) < 62/18, 0 < 6 < 360, A E K;5 (see(E)). Then

(58) I < U .(2~)f~,~ #HA < n. "~t+2 k"/~ hEHA

Hence 1 +46

te2/18(A,w) < nm_ 1 +66.

P r o o f As before we obtain

(59) d ( C , got+ 1 [s, s + k - 1]) < 62/6.

Let us assume that C contains exactly three kinds of subblocks Lt,i,g', Lt,i,h', Lt,if,, each appearing in its natural positions. Let el, g2, g3 be the families of all subblocks

Lt,i,g',Lt,i,h',Lt,if, of C appearing at their natural positions. Each of el, C2, C3 is a union of disjoint subblocks of C.

We pick subblocks D1,D2,D3 from wt+l[s,s + k - 1] occupying in it the same

positions as the blocks C1, C2, C3 from the families Cl, C2, g3 do in C, respectively. They define the families 791,792,793. For each Cj. E Cj, we let Dj(Cj.) denote the

block of 79j appearing in the same positions as Cj., fo r j = 1,2, 3. Then we define

I il(Cj, vj) : = c jil( cs,mA G ) ).

Then (59) implies d(Cj, 79y) < 6/2, 1 < j < 3. It is not hard to deduce that

I < wt+2 v.J for some h E H. Let g be another element of H such that ] < wt+ 2 ~6J.


Then using Lemma IV.4 and repeating the arguments of the proof of Lemma IV.8 we obtain g - h = hp for some p ~ 7_.,. This implies (58). The same arguments

apply to the case where C contains more than three blocks of the form Lt,i,g'. Then using (42) and (47) we deduce that

1 +46 t62/18(A,w ) < nm------- ~ + 66. []

L e m m a IV.11 Letd(A, I ) < 62/18, 0 < 6 < 160, andA E K~6 (see (F)). Then

(60) I <~ U '(26+2)t'/~h #HA < n. "~t+2 k"], hEHA

Hence 5 +46

t6v18(A,w) < nm_l +66.

P r o o f Using the same arguments as in the (numerous) preceding Lemmas we

prove that C contains a subblock C1 such that ICll/ICI >_ �89 and C1 contains at least one block/3t+1 (g'), and C1 < bt+l (gt) (or C1 < bt+l (h')). Then

(61) a(C1,D1) < 6/3,

where D1 is defined in the same way as the one in the proof of Lemmas IV.8 and

IV. 10. Then (61) implies that

(62) D1 < bt+l (g) for some g E H.

Then we use Lemma IV.4 to deduce that g - ~ = hp for some p E Zn. This, with (62), implies (60). Then (47) and (42) imply

t~2/18(A,w ) _< 5 +46 nrn_-------- T + 66. []

T h e o r e m (m,r) The system (X • H , T , , , # | mn) is such that r(T~,) = r. m(T , , ) = m, and the essential range o f its multiplicity function is {1,. . . ,m}. It is measure-theoretically isomorphic to a strictly ergodic continuous Morse automorphism.

P r o o f Let 0 < 62 < 16o and let 61 = 63/36r. Then select to such that 62nto >_ 4, 62lt0+1 _> 4. Let A < w be a block such that [A I _> 3nto. Then A has a form (46)

234 J. K W l A T K O W S K I A N D Y. L A C R O I X

with t > to. With Lemmas IV.6-IV.11 we deduce that, using the fact that for 0 < 6' < 6, t6(A,w) > tv (A ,w) ,

{ 7 1+462 3+462 1+462 5+462} t61(A'w)<--max n ~--l' r , nm , nm_~ , nm_l + 6 6 2 : = a .

Hence F*(T~o,) _< a. We can select n and 62 in order to ensure that a < 1/(r - 1). Then using Section 1.4 we deduce that r(T~o,) > r. []

R e m a r k IV.1 A "Chacon type" modification of the candidate to the (m, r) pair (see Sections II and IV) can lead to a weakly mixing system realizing the pair (m, r) (2 < m < r < oo). Namely, we define the base system (X, T, #) as follows (it will no longer be an adic adding machine):

(1) pick Ao, A1,... as in Section II; (2) define the generating partition of the continuous Lebesgue probability space

(X, #) inductively to be a refining sequence (~t)t>_o of T-towers

= ( D ' o , . . . , D , , - 1 ) ,

where no = A0, #(D ~ = l /no (0

nt+l = /~t+lnt + 1 , and

TD~ = D~+I, 0 < i < n t - - 2,

< i < no), and if ~t is defined, then we let

[ At+l --2 t + l _ t / u ':';.,+,/u':'+l

nt+l-nt+i+l -- D i , 0 <_ i < nt, \ j=o /

and #(D t+l) = 1/nt+l, 0 o exactly as in Section II if m = r or in

Section IV if m < r;

(4) with the new sequence (nt)t>o from (2) above, define the blocks (Bt)t_>o by Bo = bo and

Bt+l = Bt(bt+l [0])'-" Bt(bt+l ([~t+l - - 2])OBt(bt+l [~t+l - - 1 ] ) ;

(5) let the M-cocycle ~o : X --, H (H = G if m = r) be defined by qOl~ =

Bt[i + 1] - Bt[i], 0 O is a rigid time for T, and following the argumentation from [KwJLe, See. 4], (X x H, T~, # | m/4) is seen to be weakly mixing with the desired spectral multiplicity equal to m. As is seen along the lines of [Kw] or [Le], the system has a strictly ergodic shift representation (fl,o, S, #,,) where w 6 / f f is defined by

r nt-1]=Bt , t>_O.

The computation of its rank uses this symbolic representation and may be done in a very similar way to what was done in Sections III and IV.


REFERENCES

[Ch] R. V. Chacon, A geometric construction of measure preserving transformations, Proc. Fifth Berkeley Symposium of Mathematical Statistics and Probability, II, part 2, Univ. of California Press, 1965, pp. 335--360.

[dlR] T. de la Rue, Rang des syst~mes dynamiques Gaussiens, preprint, Rouen, 1996. [d J] A. del Junco, A transformation with simple spectrum which is not rank one, Canad. J. Math.

29 (1977), 655--663. [Fe] S. Ferenczi, Syst~mes localement de rang un, Ann. Inst. H. Poincar6 Probab. Stat. 20 (1984),

35-51. [FeKw] S. Ferenczi and J. Kwiatkowski, Rank and spectral multiplicity, Studia Math. I02 (1992),

121-144. [FeKwMa] S. Ferenczi, J. Kwiatkowski and C. Mauduit, A density theorem for (multiplicity, rank)

pairs, J. Analyse Math. 65 (1995), 45-75. [GoKwLeLi] G. R. Goodson, J. Kwiatkowski, M. Lemaficzyk and P. Liardet, On the multiplicity

function of ergodic group extensions of rotations, Studia Math. 102 (1992), 157-174. [GoLe] G.R. Goodson and M. Lemaficzyk, On the rank of a class ofbij'ective substitutions, Studia

Math. 96 (1990), 219--230. [IwLa] A. Iwanik and Y. Lacroix, Some constructions of strictly ergodic non-regular Toeplitz flows,

Studia Math. 110 (1994), 191-203. [Ke] M. Keane, Strongly mixingg-measures, Invent. Math. 16 (1972), 309--353. [Kw] J. Kwiatkowski, Isomorphism of regular Morse dynamicalsystems, Studia Math. 72 (1982),

59-89. [KwJLe] J. Kwiatkowski Junior and M. Lemaficzyk, On the multiplicity function ofergodic group

extensions. 11, Studia Math. 116 (1995), 207-215. [KwSi] J. Kwiatkowski and A. Sikorski, Spectralproperties of G-symbolic Morse shifts, Bull. Soc.

Math. France 115 (1987), 19--33. [Le] M. Lernaficzyk, Toeplitz Z2-extensions, Ann. Inst. H. Poincar6 Probab. Stat. 24 (1988), 1--43. [Ma] J. C. Martin, The structure of generalized Morse minimal sets on n symbols, Tram. Amer.

Math. Soc. 232 (1977), 343--355. [M 1 ] M. Mentzen, Some examples of automorphisms with rank r and simple spectrum, Bull. Polish

Acad. Sci. Math. 35 (1987), 417-424. [M2] M. Mentzen, Thesis, Preprint no 2/89, Nicholas Copernicus University, Tormi, 1989. [Pa] W. Party, Compact abelian group extensions of discrete dynamical systems, Z. Wahrsch Verw.

Gebiete 13 (1969), 95-113. [R1] E. A. Robinson, Ergodic measure preserving transformations with arbitrary finite spectral

multiplicities, Invent. Math. 72 (1983), 299-314. [R2] E.A. Robinson, Mixing and spectral multiplicity, Ergodic Theory Dynam. Systems 5 (1985),

617--624.

d. Kwiatkowski INSTITUTE OF MATHEMATICS

NICHOLAS COPERNICUS UNIVERSITY UL. CHOPINA 12/18

87-100 TORUlq, POLAND E-MAIL: [email protected]

Y Lacroix DEPARTEMENT DE MATHI~MATIQUES

U.B.O., FAC. DES SCIENCES 6 AVENUE V, LE GORGEU, B. P. 809

29285 BREST CEDEX, FRANCE E-MAIL: [email protected]

(Received July 23, 1996 and in revised form September 10, 1996)

Documents

Multiplicity, rank pairs