Non-singular isomorphism and weak isomorphism on the class of dynamical systems with discrete spectra

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1981)

J.

(Torun)

Non-singular isomorphism and weak isomorphism on the class of dynamical systems with discrete spectra

Introduction. In [5] given a dynamical system X with discrete spectrum

(in general not ergodic) a complete system 9(X) of invariants of X has been constructed, and given 9 a standard dynamical system S(0) has been defined in such way that S(9(X)) is isomorphic to X. In the present paper necessary and sufficient conditions for fh and 92 are found under which the standard dynamical systems S(0d and S(02) are non-singularly isomorphic (see Definition 1 and Theorem 1). Similarly conditions on 9± and 92 are given in order that S(0i) and S(02) be isomorphic by partition [2].

In Section 3 it is proved that any two weakly isomorphic dynamical systems with discrete spectra are isomorphic modO. Moreover, the concept of a weak non-singular isomorphism is introduced as a non-singular counter­

part of the weak isomorphism. It turns out that in general two weakly non-singularly isomorphic dynamical systems with discrete spectra are not non-singularly isomorphic. Finally one shows, under some assumptions on 9t and 92, when the standard dynamical systems S(91) and S(92) are weakly non-singularly isomorphic. 1

1. Notation, definitions and theorems. Let X = (Q, 2F, p, T) be a dy­

namical system with a Lebesgue space such that p(Q) = 1 and let U be the unitary operator on l3(Q,fi) defined by (Uf)(co) = f (Tea) for any f e l 3 ( Q , p ) and almost every (a.e.)

Q. Recall that X is called a dynamical system with discrete spectrum if there exists a basis of l3(Q,p) consisting of eigenfunctions of U ; X is said to be an ergodic dynamical system if for every T-invariant subset A of Q we have p(A) = 0 or p(A) = 1.

Denote by A the set of all eigenvalues of U. It is known that if X is an ergodic dynamical system, then A is a countable subgroup of the circle K. If X is an ergodic dynamical system with discrete spectrum, then A is a complete system of invariants of X [3]. Furthermore, if A is any countable subgroup of K, then A is the set of all eigenvalues of a dynamical system S(yl) with discrete spectrum, namely S(zl) = (À, M , p, T), where A is the dual group of А,0й is the cr-field of borelian subsets of A, p is

9 — Roczniki PTM — Prace Matematyczne XXII

the normalized Haar measure on M, and f ( j ) = «•)(, where /е Л and â eA is defined by а (Л) = A, Ae A .

Assume A is a dynamical system with discrete spectrum. In [5]

a complete system 0(A) of invariants of X has been constructed. For the use below we recall shortly the construction of these invariants.

Let £ be a partition of Q on ergodic components [7]. If (M , Me)

= ( Q, 'îF, n) |£ and if {

}

is a canonical system of measures of £ then for a.e. C e M , X c = (С, .¥c , цс , T\, — .¥\ C, is an ergodic dynamical system with discrete spectrum and its group of all eigenvalues Ac is contained in A. Furthermore, if А еЛ then for every integer и,А"еЛ [4]. Now, let

£ be a partition of M defined in the following way: Ct, C 2e M determine the same element of £ iff ACl = ACl. Then for every А еЛ , the set

M(A) = { C e M ; А еЛ с}

is a measurable subset of M with /^(M(A)) > 0 and the family of sets {М (А )}ЯбЛ forms a basis of £. In particular £ is a measurable partition.

Consider the quotient space

( N , , /if) = (M , , /rç) I £

and denote by {^ y }/6jv a canonical system of measures of £. In what follows, we shall construct a mapping F from N to the set <4 of all sub­

groups of A (the set У is non-empty since the cyclic groups (A") belong to ^). Using the mapping F one can transfer the measurable structure of N to the set У . For this purpose we use the following notations: S — the family of all subsets of Л, S(A) — the family of subsets of A which contain A, 2Î — the a-field of subsets of S generated by the family of sets |5(А)}Яб/1.

On 21 one can define a measure v as follows:

v (S (AO n S (A2) n ... n S (A„) n 5 (A'j) n S (A'2) n ... n S (AJ)

= ^ (M (AJ n M (A2) n ... n M (A„) n M (Aj) n M (A2) n ... n M (Xmj), where S(A) = S\S(A), M(A) = M\M(A) and A1? A2, ..., A„, X\, A2, ..., X'meA.

Let N (A) = £(M(A)), where £: M -* N is the homomorphism induced by the partition £ (N (A) is a measurable set in N because M(A) is a £-set in M).

Since (М (А )}ЯбЛ is a basis of £, then the family {TV (А)}ЯеЛ is a basis of the Lebesgue space (iV, J^*, ^*) whence almost every element & of N is of the form â? = f) i\P(/)(A), where e(A) = 0 or 1 and iV°(A) = N(A), N l (A)

ЯеЛ

== N(A). Now, we define a mapping F : V -*• S by formula: F{&) = П SC(/)(A).

ЯеЛ

It is easy to verify that F is an isomorphism of the spaces ( N , ^ * , м * ) and

(S, 2Ï, v). Moreover, for a.e. £AeiV, F (&) is a group contained in A (namely

F(£A) = Ac , where C is any element of £A). Thus F ( N ) а У

S, and since

v (F(N)) = 1, ^ is a measurable subset of S and v(Cf) = 1. Putting 21 = ,

v = v f S , we obtain a Lebesgue space , 21, v) and an isomorphism F from (N, , H*) onto (C^,3I,v). Observe that 31 is a cr-field generated by the family of sets {^ (А )}ЛбЛ, where ^(A) = {G e ^ ; AeG }.

Finally, let mn, n = 1,2,..., be a sequence of real measurable functions on ^ defined by m„(G) = the type of the measure Д;,_ 1(0). It is clear that

O O

mn(G) ^ 0, m„(G) ^ mn + 1(G), £ m„(G) ^ 1 for n = 1,2,... and a.e. G e . n= 1

In this way we have obtained a triple ' G(X,C) = (Л ,(^ ,3 1 ,у ),{т „ (С )}; G e S .n = 1,2,...).

Since the set ^ and the cr-field 31 are uniquely determined by the set A, we can write shortly

0 ( X , O = (Л ,у ,{т „ (С )}).

The role of objects explains the following theorems proved in [5].

Theorem

A. I f X — (Q, 37, p, T) is a dyngmical system with discrete spectrum and £, я tire two different partitions of Q on ergodic components, then 9 { X , Q = в ( Х , п) .

Thus, we can write 0(2Г) = (Л, v, (m„(G)}).

Theorem

B. Let X { = (Qt, p{, 7j), i = 1,2, be two dynamical systems with discrete spectra. The systems Х г and X 2 are isomorphic iff 9 ( Х г)

= в ( Х 2).

T^heorem

C. Suppose Л is a countable subset of the circle К such that A e Л implies A” e Л for all integers n and denote by У the set of all subgroups of A, and by 31 the о-field of subsets of <4 generated by the family of sets {^ (А )}ЯбЛ, ^(A) = (G e 1 ^; A eG }. Then for any normalized measure v on 31 with v(^(A)) > 0 , А еЛ and for any sequence |m„} of measurable functions defined on У such that m„(G) ^ 0, m„(G) ^ m„ + 1(G) and

O O

£ m„(G) ^ 1 for a.e. G e У and n = 1,2,... there exists a dynamical system П = 1

X with discrete spectrum for which 9(X) = [A, v, |m„(G)}).

Rem ark 1. In [5] (Theorem 5) is used the fact that (Т/, 31, v) is a Lebesgue space. This may be proved at once. In fact, let (S, 31) be the measurable space constructed previously and let v be a normalized measure on 3Ï defined by

v(S(A1)n S (A 2)n ...n S (A „ )n S (A ,1)nS (A'2)n ...n S (A J )

=

( « (A,) n 'S (A2)-n ... n « (Я.) n 9 (Ai) n 9 (A'2) n ... n 9 (Ai,)), where, At , A2, ..., A„, A), A'2, ..., Л'те Л.

It is known [8], that (S ,3 I,v) is a Lebesgue space. In order to show

that (^ , 31, v) is a Lebesgue space it remains to prove that ^ is a measurable

subset of S.

Let A = {Aj, Д2, ...} and let Z be the set of all integers. For any positive integer n and for any subset P = {it , i2, ..., ik} <= { 1 , 2 , n] = Z„ we define

_ | 0 ,

if there exist cu ...,cke Z ; A-J • A^ •-... • Х\к кф A,

\ П S (A • J • X\2 2 .... A •*) n П S (A,), otherwise.

( c v ...,ck)e Z k l ^ K i ,

1фР

Let R(n) — П Я(Р). It is clear, the sets R{P) and R(n) are measurable p^z„

00

in S. Moreover, also ^ is a measurable set because У = f] R(n); thus,

n= 1

we have shown ( y , $ l , v ) is a Lebesgue space.

Now, we describe a dynamical system S(0) with a given invariant в constructed in the proof of Theorem 5 [5]. This description will be used in the sequel.

00

Let в = (A, v, (m„(G)}, С е У , п =* 1,2,...) and let a(G) = 1 - £ W„(G),

n — 1

G e ^ , I

= < 0 ,a (G ))u (J {pf}, where {p^,pt,- •} are distinct abstract i <t

points, t = к if > 0, mk(G) = 0 and t = oo otherwise. Define a measure pG on I G in the following way: pG is equal to the Lebesgue measure on the interval <0, a(G)> and pG({pf}) = mf (G), i < t. Let Q = {(G ,x); G e y , x e I G} , Q i = {(G, pf); G e ^ }, i = 1,2,..., and let £20

= {(G ,x ); a(G) > 0, x e <0, a(G)>}. Making use of the natural identification of with У i = { G e y ; m,(G) > 0}, i = 1,2,... one can define on Qt a (T-field of

. Denote by a cr-field of subsets of Q generated by the cr-fields 3Ff, i = 1,2,..., and by those sets of Q0, which have the form A x B , where АеШ\§, У — { G e y , a (G) > 0} and В is an interval such that A x B c= Q0. Then, one may check that for any set E e ^ , E r \ l G is a measurable subset of I G (here I G denotes the set ( IG, G)), (p(G) — pG(E n

is a measurable function of variable G. This permits to define a measure Д on # by

Д(£) = f <p(G)v(dG)

‘S

and as a result we have the measure space (0 , # , Д ) .

Now, we shall construct the dynamical system S(0). Let G be the dual group of G, G e y , and let

Q = {(m, x);

= ( G , X ) e Q ; x e G ) .

In order to define a <r-field 3F of subsets of Q, take Al5 A2, ..., A„, k\, X'2, ...

..., A|„ e A and put

( 1 )

A = ^ (Ax) о У (Аз) n ... n ^ (Аи) n У (X\) г\...слУ (Am).

Suppose v{A) > 0 and denote by A the family of all subsets of Q consisting of the sets A x В , where В is an interval such that A x В c= Q0 and the sets {(G ,p f); G e A }, i = 1,... Furthermore, denote by G (A) the group generated by Al5 Л2, ..., A„ and observe that G (A) is a quotient group of G because G (A) cz G for any G e A. If C is any borelian subset of G{A) and H e  , then we define H ( C ) = {(ct>, hG1 (c)); c o e H , c e C ) , where hG is the natural homomorphism from G onto G (A). All sets H(C), where A runs over all sets of the form (1) and H e  , C is a borelian subset of G (A), are clearly subsets of Q and generate a <7-field in Q denoted by . On the cr-field one can define a measure \i by formula p, (H (С)) = Д (H) mA ( Q , where mA is the normalized Haar measure on G {A). Thus we obtain a measure space ( Q , ^ , p ) . The needed transformation T on / we define as follows:

T(co,x) = (fflJ 'ie ),

where со = (G , x ) e Q , ^eG and j G is an element of G such that j G(g) = g, ge G. It is easy to verify that T is a measurable measure-preserving transformation. From definition S(0) = (Q, p, T). S{0) is the system with discrete spectrum and its invariant 0(S(0)) is the given triple 0. We shall call S(0) the standard system for 0 (or for X , if 0 = 0(A)).

2. Non-singular isomorphisms. Let X { = ( üif i? 7J), i = 1,2,..., be dynamical systems.

De f i n i t i o n 1.

The dynamical systems X x and X 2 are non-singularly

isomorphic if there exists a measurable invertible non-singular transformation S : Q1-+£22 with = 1 such that the diagram

&21 ---► Qx

s s

T2 +

Çi2 * Ü22

is commutative.

Rem ark 2. If X x is an ergodic dynamical system and X 2 is non-singular isomorphic to X l5 then X 2 is an ergodic system and S is a measure-preserving transformation. In fact, if В a Q2 is a T2-invariant set with 0 < p2(B) < 1, then 0 < p l (S~1B) < 1 and S~1B is Ti-invariant set. Hence X 2 is an ergodic system. In order to show that S is a measure-preserving transformation, we define a measure fl2 on Q2 by fi2(B) = p l (S~1B). Then Д2 is a T2-invariant, ergodic measure and Д2 is equivalent (Д2 ~ g2) to p2. It follows from [1] that Д2 = p2 i.e. S is a measure-preserving transformation.

Let A be a dynamical system with discrete spectrum and let 0(A)

= (A,v, {m„(G)j, G s ^ , n = 1,2,...) be its invariant. We define two mea­

surable functions fh and l on ^ as follows:

00

0, i f £ m „ ( G ) = l , n= 1

1, if £ m„(G) < 1,

n~ 1

if m„(G) > 0 for n ^ 1, if m„+1(G) = 0, mn(G) > 0.

If 5(0) is the standard dynamical system for X, then 1(G) is the number of all atoms of the measure pG and m(G) = 0 iff pG is purely atomic. Now, we are in position to prove the following

Theorem 1. Let X v = (Qt, 3* {, pb 7J), i — 1,2, be dynamical systems with

discrete spectra and 6(Xi) = (A(, vt-, {mln (G)}) be its invariants. Then X x is non-singular isomorphic to X 2 iff

1° A, = A 2 = A,

2° Vj is equivalent to v2 and m1(G) = m2(G)» h (G) = l2 (G) mod vi?

i = 1,2,...

Proof. Necessity. Let 5 : Q1^>Q2 be non-singular isomorphism'and let Ci be a partition of on ergodic components. It is easy to see that C2 = S(Ci) is a partition of Q2 on ergodic components. Denote by M l and M 2 the spaces Q1/Çi , 0 2/C2 and suppose that {рс^сем^, {Рс}сем2 are canonical systems of measures of Ci and Ç2 respectively. First we show that if C = 5(C), C e M j, C e M 2, then the dynamical systems (C, f i e r i , 7i) and (C, JF2

, pi, T2) are isomorphic mod 0. For this purpose we define a measure p on and pc on f C e M x, by p(A) = p2(SA) and p c ( A n C )

= r f (S A nSC), A e ^ t and observe that {рс}сем1 is a canonical system of measures of Ci with respect to pç .

The assumption 5 is a non-singular transformation implies p ^ p x that is there exists f e L l (Q1, p 1) such that f (со) > 0 mod px and

p(A) = J f ( c o ) p v(dco) for A&tF x.

A

If g(C) = J f (со) pc(dco), then g(C) > 0 mod p. and it is easy to verify

pc ( A n C ) = - j p r - I f(co)ri(dco), A

^ x.

g(C) Anc

Thus we obtain pc ~ p'c m o d ^ . It follows pc = ph because pc ,Pc are m(G) =

and

1(G) = oo,

n,

ergodic measures [1], what means that S : C ->S (C ) is a measure-preserving transformation. Thus

(2) Ac = Ac, whenever C = S(C).

By definitions of the sets M X(Â) and M 2(A) and by (2) we conclude S ( M X(A)) = M 2(Â) for A e A x or A e A 2 and hence /r?1(M 1(A ))> 0 iff

^ 2(M 2(A)) > 0 what implies A x — A 2 = A.

Further, by definitions of the partitions Çx and £2 and by (2) we have S (^ ) = £2. Then S define an invertible non-singular mapping S: N X^ N 2, iVj = M x/Çx, N 2 = M 2/Ç2. Take the isomorphisms F x: N 1-> ('^ ,v 1) and F 2: N 2 - > ( ^ ,

2) and observe that if H = F 2SFX1, then (2) implies H (G)

= G mod Vj, whence Vi % v2. Now, let {P^}oeNx, (Д^}эбЛг2 be canonical systems of measures of Çx and £2 respectively. It is easy to check Д^ % Д|2, whenever ^ 2 = S{&x). Taking G e ^ and putting 3>x = F f^ G ), S>2 = F2 *(G) we have £b2 = and therefore p lQl ^ Д^2 that is mx(G) = m2(G), lx (G) = /2(G)mod vf, i = 1,2.

Sufficiency. The asumption Vi » v2 implies 5 = F 2 1 H Fi is a non-singular transformation of N x onto N 2, where H(G) = G for G e ^ . From the equalites fhx(G) = m2(G) and /i (G) = /2(G)modv,, i = 1,2, it follows that the measures p\ l and Дс2 have the same non-singular types, whenever Q>2 = S{3)x}. In this situation there exists a non-singular invertible transformation S: M x -► M 2 (see Lemma 1, below) so that the diagram

(4)

M x — — ♦ M 2

É1 %2

N i ---- ---> N 2 is commutative.

Now, we define a measure Д on M 2 by p(B) = ^ (S~i B), ^B

^ ^ 2 and a measure Д on &2 by

l*(A) = i n l ( A n C ) p ( d C ) , A e & 2,

m

2

and observe that Д is a T2-invariant measure on Q2 and Д ~ ju2. So we

obtain a dynamical system У2 = (Q2, J^2, Д, T2) for which (2 is a partition

on ergodic components and ( M x, /х?1) -» (M 2, Д?2) is a measure-preserving

transformation. It follows by (4) that Лс = /ls(C) for a.e. C e M x and therefore

the systems (C, ^ xc, Hc> Tx) and ( C , ^ 2c, P c ,T 2) are isomorphic, where

C = £(C). Applying Rokhlin’s theorem [7] we conclude there exists an

isomorphism mod 0, S of the dynamical system X x to the system T2. Clearly,

S is a non-singular isomorphism from X x to X 2 as was be shown.

Rem ark 3. From the proof of Rokhlin’s theorem [7] it follows a map­

ping S can be constructed in such way that the diagram

П S ГЛ

*£1 * S^2

5l 52

M j — ----► M 2

is commutative. In the case when and X 2 are dynamical systems with discrete spectra and S is a non-singular homomorphism, then a transform­

ation S can be constructed directly. For this purpose we use the standard representations SiOf) and S(62) constructed in Section 1 for X t and X 2 respectively.

Suppose S: Ü 1- > Ü 2 is a non-singular homomorphism i.e. pCl(5~1(B))

> 0 iff ъ 2(В) > 0 and ju?2(SQ^ = 1, such that if со = (G, x ) e Q 1, then S(a>) = (H , y ) e Q 2 with H <= G. We define

S(œ, x) = (5 со, Л(х)),

where (c o ,x )e 0 1 and h: G -> Й is the natural homomorphism.

Analyzing the a-fields and &2 of 5(0^ and S(62) respectively one can verify that S is a measurable non-singular transformation from Ql onto Q2 and that the diagram

Qx Tl ,

s s

i t 2 i, I22 ~ ^ £22

is commutative. Thus S is a non-singular homomorphism of the dynamical system S(0i) to S(02).

In the sequel the following lemma shall be useful:

Lemma

1. Let ( A , 3F, p) and

^ , v ) be Lebesgue space, Ç and rj be measurable partitions of X and Y respectively, M = A/Ç, N — Y /rj and let {jUc}c6M. {vc}ceN be canonical systems of measures of Ç and r\. I f S: M -* N is a non-singular transformation satisfying the conditions pc and Vç have the same non-singular types for a.e. C e M , whenever C = S(C), then there exists a non-singular transformation S: X -> Y such that the diagram

X ---- ---► Y

(5) «

M --- ► N

is commutative.

P roof. Let {m „(C)}, C

M , (m „(C)}, C e N , be the types of Ç and rj respectively [8]. Then there exist a pairwise disjoint measurable subsets P l , P 2,... of X such that

(6) ^ card {Pi n C) = 1 or 0, /хс (Р {^ С ) = mf(C) for i ^ 1 and a.e. C

M .

oo

If we denote X 0 = X\ (J P t, then pLc is a non-atomic measure on i = 1

X 0 n C, whenever juc (X 0 n C) > 0. Similarly, there exist measurable subsets

° f T having the same properties as P l f P 2,...

Put = {C e M ; mf(C) > 0), IV; = {C e N ; т г(С) > 0), i = 1,2,...

and note that 5 (М £) = N f, because the measures цс and Vç, C = S(C), have the same non-singular types. Further, conditions (6) permit to define a mapping Vt: P i - * M i which is non-singular transformation because

ц(Р) = J H c ( P ^ C ) ^ { d C ) = J mi(C)^(dC),

Mi Mt

where P a P ( and М,- = {C e M f; цс {Р n C ) > 0}. Thus, we define S: P t g,-, i = 1,2,..., as follows S = Vi~ i - S - V i, where Q i ^ N ( is a non-singular mapping defined in the same way as Therefore, we obtain a non-

00 oo

singular transformation S from (J P { onto IJ Qt and it may be easily i = 1 i = 1

verified that diagram (5) is commutative.

In order to prove the lemma it remains to construct S from X 0 onto 00

Y0 = Y\ IJ Qi- For the use later we shall construct S on X 0 in such way i = 1

that the measures v and v = juS-1 on Y0 are countable equivalent (see Definition 2, below).

Suppose f i( X0) = m > 0, v(T0) = m > 0 and denote

M 0 = {C e M ; fic ( X 0 n C ) > 0}, N 0 = {C e N ; V c(C n T 0) > 0}.

Since Цс is non-atomic measure on X 0 n C , whenever C e M 0, we may O O

assume C n X 0 = Ic = <0,a(C)>, where a(C) = 1— ]T m„(C), and цс is n = 1

the Lebesgue measure on Iç [8]. Similarly, C n T0 = /? = <0, j?(C)>, where 00

P (C) = 1 — Y

™n (C), and vc is the Lebesgue measure on I* for C e N 0.

n = 1

Then a(C) = j j ç { X 0 n C), P{€) = Vc(70 n C), whence a(C) > 0 and_ ]8(C) > 0

for C e M 0, C e N 0. From the assumption that Цс and v^, C = S(C), have

the same non-singular types it follows S ( M 0) = N 0 and from the assumption

that S is a non-singular transformation it follows there exists a positive function q>eL1( M , ju:) such that

v,(SA) = f <p(C)^(dC),

A

for A с M and J (p{C)n^{dC) = 1.

M

Take 0 < £ < 1/3. Then we can find 0 < Ô < e m such that pi: {A) < 3 , A <= M , implies j (p(C)nc(dC) < e m. Let

A

r{C) = P(S(C)) ■ (p{C)ja{C) for C

M 0 and let

M n,k = {C e M 0; к/2" < r(C) ^ (k + l)/2"}, n = 1,2,..., к = 0, 1,2,...

Now, we find К so that k/(k + 1) > 1—£ for к ^ K. Then for sufficiently large n

(?) I < г.

k < K

Putting M K = U M n<k we have

k < K

• Z vn(SMnk) = X J <p(C)/t: (t/C) = J <p(C)/*; (dC) < e m,

к < K k < K M n k M,.

because < Ô.

Take к ^ К and denote

*„,k = {(C ,x ); C e M „it; 0 ^ x ^ (l - e ) a ( C ) },

(9)

П.к = (C, v); C( СЛ/f

ГН1

and define a mapping S from X nk onto Ynk as follows:

5(C, x) = ( S (C ), x k/2n (p(Cj).

Thus, the mapping S is defined on the set = (J k and we have

k>A.

= yl5 where Yt = [J Ynk. It is not difficult to check that S is an k^K

invertible measurable transformation and diagram (5) is commuting. Further, if Î c X„tk, then

v(SX) = j vc (SX nC)v„(dC) = J vS{C)(SXnSC)<p(C)nAdC)

v0 M0

= 1 4 r - J æ

l k- ( XnC) <i >( Q Я- (d C) = X f , (x ) .

м0 I (p{L) 2"

The above equality means that the measures v and v = p - S ~ l on Yj are countable equivalent.

Now, using (7) and (9) we can establish > (1 — 2 • e) p( X0). Similarly from (8) and from the inequality

a (C )(l —e) к a ( C )(l—e) k + l

<P(C) 2" <p{C) 2” k + l

<p{Q 2" (p(Q

= ( l - e ) 2iS(S(0) > ( 1 - 2

) P ( S ( Q ) ,

v ( Q

for C e M n k we obtain v ^ ) > (1 — 3 • e) • v(70).

Next’, we can repeat the arguments above for the sets X 0\Xl and Y0\Yt replacing the numbers m, m by mi = = v(Y0\Yi) and the functions a and f$ by â and defined as follows:

s ( 0 =

H Q =

a(C), C € M E, e-a(C), C $ M E,

P ( C ) ,

« ( S - ^ a - e ) к P { ) cpiS- 'C) ‘ 2 " ’

C e S (M £),

CeSfAf,,,*), к ^ K.

Hence we obtain sets X 2 <= ХоУХ^, Y2 <= and a non-singular mapping S from X 2 onto Y2 such that diagram (5) is commutative and p ( X 0\(Xl u X 2))

< (2 • e)2 • m, v(Y0\(Yi u Y2)) < (3 -e)2m.

In the above manner we can find the sets X 1, X 2,... and Yl t Y2,...

00 00

with U X t = X 0, U Yi = Y0 and a non-singular transformations S from

i=l i = 1

X t onto Yt satisfying the conditions required in the lemma and such that the measures v and v = pS~1 on Y0 are countable equivalent.

Definition 2. The normalized measures ц and v defined on a measure

space (X , $F) are said to be countable equivalent if there exists a positive function f e L x (ЛГ, p) such that/ admits countable many of values, J f ( x) p( dx )

x

= 1 and

v (A) = j f (x)p (dx) for any A s SF.

A

Now, we give a necessary and sufficient conditions in order that two standard dynamical systems with discrete spectra be isomorphic by partition.

The concept of an isomorphism by partition was introduced by J. R. Choksi [2].

Definition

3. The dynamical systems Xx and X 2 are isomorphic by

partition if there exist two partitions { A j and {Bt}, i — 1,2,... of C2t and

Q? respectively into disjoint measurable sets of positive measure such that (i) the At are invariant under Ti and the Bt under T2,

(ii) there exists an invertible measurable transformation P t of At onto Bi for which

/ TJ

B

(Bf)

H

(Pi C) = ---

— • px (C), Hi (A)

where C c= A{ and T2 P t = P t Tx.

Rem ark 4. If X x and X 2 are isomorphic by partition, then the mapping S : Qx - * Q 2 defined by S (со) = P,(co) if œ e A t, is a non-singular isomorphism of X x onto X 2 such that the measures p2 and p2 = H i ' S 1 are countable equivalent. Thus X x and X 2 are non-singularly isomorphic.

Defintion

4. Let 0 = (A, v, (m„(G)}) be the invariant of a dynamical system X with discrete spectrum described in Section 1. We say that n is a measurable partition of в if for a.e. Ge @ n(G) = { Z X(G), Z 2( G ) , ...}

is a countable partition of the set of natural numbers and the sets В{ = {(G,pj), j e Z i ( G ) , Ge&} are measurable subsets of Q, i ^ 1.

We shall consider below the sets Z f(G) as ordered sets (by the relation of inequality).

Theorem

2. The dynamical systems S(0i) = (Л,-, vf, (mJ,(G)}), i = 1,2, with discrete spectra are isomorphic by partition iff

1° A x = A 2 = A,

2° Vi is equivalent to v2 with (p(G) = dvx/dv2, and mx(G) = m2(G) mod vf, i = 1,2,

3° there exist measurable partitions nx of 6X and n2 of 62 satisfying the conditions card {Z} (G)} = card { Z f (G )}, ml(G) = Ciml.(G)/(p(G), C, > 0 for i = 1,2,... and a.e. Ge&, where nj, hj are j-th elements of Z\ (G) and

Z f ( G ) respectively.

P roof. The necessity of conditions Г and 2° follows from Theorem 1.

In order to prove condition 3° we assume S(0t) is isomorphic by partition to S(02) and let S : S(0X) -> S(02) be the mapping as in Remark 4. Now, we denote by N the set of natural numbers and put

Z/(G) = { n e N ; p ' ° e A , } , Z f ( G ) = { n e N , i = 1,2,...

Repeating some arguments from the proof of Theorem 1, we obtain S (Ia) = I

for a.e. G e ^ . Further, from the above and from the conditions S(^4f) = Bif p2(SÂ) = Ci • px (T )for any A a A (, i = 1 ,2 ,...,it followsp^G6 At iff S(p^G) e В,- what implies card {Z\ (G)} = card { Z f (G)} and mf.(G) = C1m^.(G)/<^(G) if nj,nj are the y'-th elements of Z f (G) and Z f ( G ) respectively. Thus the necessity of condition 3° is proved.

Now, we assume that conditions Г , 2°, and 3° hold. First we shall

construct a non-singular transformation 5 from Qx onto Q2 such that the measures

are countable equivalent and

a.e. G e ^ (see Section 1). Having such mapping S we can extend S to a non-singular isomorphism S from S(#i) onto S(92) according to Remark 3.

It follows from Lemma 1 that there exists a non-singular mapping S0 from Q

onto Щ satisfying the above conditions. It remains to define S

00

on the set y Ü}. Let j= 1

Ai = { ( G , p j G) , j e Z i ( G ) , G

^ },

B i

= {(G, Pj G); j e Z ? (G), Ge&}, i = 1,2,...

From assumption 3° we have A t , A 2, ...,

,

..., are measurable subsets

00 00 00 00 00 00

of У Üj and У Qj respectively and У At = У Ü), U

= У Qj.

j = 1 ^{J =} 1

i

Observe that (Л,) > 0 iff p2(Bi) > 0. In fact,

Pi (Bi) = J /4 (#г n I I ) v2 (dG) = J E w» (G) v2 (<*G) Q

nezf(G)

V(G)

Z <

rn‘ (G)v2(rfG)

= C ,J ( Z m» ( G))

neZhG)

cp(G)

(P(G)vl (dG) =

Let i ^ 1 be such that (Af) > 0 and define the measures pu on Ai and p2i on Bi as follows: pu (-) = Д1 (-)/Pi (AJ, p2i(•) = Д2 О/Д2 (Bt). Dénote a(G) = X m*(G), b(G) = £ m^(G) and remark that b(G) = Ci Q(G)/

n e Zi ( G) n e z f ( G)

/<p(G) what implies a(G) > 0 iff b(G) > 0. Let = {G e ^ , a(G) > 0}- If = <*;2|В,, then it is easy to verify that {pb/a(G)}G^ 0, {pG/b(G)}G&0 f° rm a canonical systems of measures of and Çf respectively.

From assumption 3° it follows that the measures {Ac/a(G)}, and

have the same types. Simultaneously the measures and v2 on <^0 defined by

»,(® ) = - У y r M G K f d G ) ; v ^ f ^ ^ S T Î H G I v ^ G ) , g c ÿ

М Л ) 4 Pi (Bd

4

are identical. Now, applying Rokhlin’s theorem [8] to Çf and we obtain that there exists a transformation Sf: 4 { whi ch preserves the measures Ди and Ji2i and ^ ( Д п/о) = for a.e. G

^ . If Л,- <= Л;, then

Д 2 ( И () = p 2 l(SÂt)-p 2 [B,) = ^ А ^ - ^{р а в} ,) = c r ptw -

Pi w

The mappings S0, Sx, S2, ... determine a non-singular mapping 5 from Qx onto Ü2 such that the measures Д2 and fx2 = are countable equivalent on Q2.

extending

to Qx (as in Remark

we obtain the desired non-singular isomorphism 5 of the dynamical systems 5(0!) and 5(02). The theorem is proved.

Example

1. There exist a dynamical systems with discrete spectrum which are non-singularly isomorphic but not isomorphic by partition. Let px, p 2,... be the set of all prime numbers and let Z p. be the group of all

G O

Pi-roots, i ^ l , with unity and let A = ]T Z p. be the direct sum of the

i = 1

groups Z p , i > 1. If G <= A is any subgroup of Л, then G = £ Z p., where

ieN

N is a subset of N. Thus, one can identify the set ^ of all subgroups of A with the unit interval / by the formula x (G) = £ af/2*, where a* = 1 if i e N and а,- = 0 otherwise. It is easy to check that the cr-field 31 generated by the family of sets \fS (А)}ЛбЛ is identical with the cr-field of borelian subsets of I. Let v be the Lebesgue measure on / and let m„(G) = 0, mi(G) = 1/2, mn(G) = 0, m1 (G) = | (l+ x (G )), for n = 2 ,3 ,..., and Ge%.

Put 0! = (A, v, (m„(G)}), 02 = (Л, v, (m„(G)}). Using Theorems 1 and 2 it is easy to show that 5(0!) is non-singularly isomorphic to 5(02) but not isomorphic by partition.

3. Weak isomorphism. Let h: Qx -*■ Q2 be a homomorphism of the dyna­

mical systems X x = (Q1, ^ x, pl5 7i) onto X 2 = {Q2, ^ 2, p 2, T 2) and let Ci, C2 be partitions on ergodic components of X x and X 2 respectively.

Then Ci = h~1{C2) is a Ti-invariant partition of X x and we have Ci < Ci- Thus, the homomorphism h induces a homomorphism h: M x - + M 2, where (M i5 J^c.,p; .) = { Q i , ^ i , р Л Ь, i = 1,2. Let {рЬ}сем1 Ы) се М2 be canonical systems ol measures of Ci and C2 respectively.

Now, we prove

Theorem

3. I f the dynamical systems X {, i = 1,2, with discrete spectra are weakly isomorphic, then X x is isomorphic to X 2.

Proof. Let h i ’. X x ^ X 2, h2: X 2 - + X x be homomorphisms and let 9 ( X x) = (Л 1, vj, {ml (G)}), 0 (X 2) = (Л2, v2, (m^(G)}). The weak isomorphism of X x and X 2 implies X x is spectrally isomorphic to X 2, hence A x = A 2 = A and У x = У 2 = У . One can show that hxc = hx\C is a homomorphism of the dynamical system ( C , p c , T xc) onto (C, pi, T2e), C = hx(C), for a.e.

C

M

. Thus we obtain A~hlC c= Ac and similarly A

,2

<= A

for a.e. C e M x, C e M 2 what implies

Ях 1( М 2(Хх) г л . . . п М 2(Хк)) с= M 1 (AJ n ... n M 1 (Ak),

^ ^ h2 1 (M 1 (Я1) n ... n M 1 (Afc)) c M 2 (Я1) n ... n M 2 (Ak),

for any Xx, Я2, ..., Xke A. Hence we have

(11) /j£l(M ’ (A1)n . . . n lW 1(At)) = By above we obtain

(12) ^ l (Bi (Xx) n B 1(X2) n . . . n B l (Xk)) = pC2(B2(Xx) n . . . n B 2(Xkj),

where B2(A) = M 2(A) or M 2\M2(A) and В1 (A) = M 1 (A) or М Д М 1 (A).

(10), (11) and (12) implies

hx 1 (B2 (Ax) n B2 (A2) n ... n B2 (Ak)) = Bl (AJ n B1 (A2) n ... n B1 (Ak), h2 l (Bl (Aj) n В 1 (A2) n ... n B1 (Ak)) = B2(A1) n B 2(A2) n . . . n B 2(Ak).

If F t : N 1- » '(^ ,v 1) and F 2: N 2 - * ( ^ , v 2) are the mappings constructed in Section 1 and if G = F x( & x) = F 2(Sn2), G e 1 ^, S>

N x, @ 2

N 2, then using (13) we have h2{fy2) = & x and hx{2cx) — fy2. This means that vx = v2 = v and that the measures F ^ -i(0) and F ^ -i(C) are of the same types. Thus, m* (G) = m2(G) for a.e. G e # and n = 1,2,..., what gives X x is isomorphic to X 2.

Definition 5.

We say that a mapping h: Qx -> Q2 is non-singular homomorphism of the dynamical system X x to X 2 if h is a measurable transformation, px (h~l (B)) > 0 iff p2(B) > 0 for B

3F2, p2(hQx) = 1 and h - Ti = T2 h.

Definition

6. The dynamical systems X x and X 2 are called weakly non-singularly isomorphic if there exist non-singular homomorphisms hx: X x

—> X 2 and h2: X 2 - * X x.

Rem ark 5. If the dynamical systems X x and X 2 are non-singularly isomorphic, then they are weakly non-singularly isomorphic. The inverse statement is false.

Example

2. Let A 0 c= A x c= ... be an infinite sequence of countable subgroups of the circle К with ЛДЛ{_ 1 # 0 , i = 1,2,... Let X x

= № . 3F 7j) be an ergodic dynamical system with discrete spectrum with A t as the point spectrum, i ^ 0. We may assume that the sets Q0, Qx, Q2, ..., are pairwise disjoint. We define Yx = £20 u (J Q2i, Y2

i= 1

= Q0

(J Q2i~i and let ! F X, 3 ' 2 be the c-fields of subsets of Yx and Y2

‘ =i

generated by the families { ^ 0, SF2, ...} and (J%, SFx, 3Fъ, ...} respect­

ively. Further, we define measures p x and p2 on Yx, Y2 in such way that px(Xi) > 0, i = 0, 1 ,2 ,4 ,..., ji2\(Xi) > 0 , i = 0, 1,3,... Finally, the auto­

morphisms T0, T X, T 2,... determine automorphisms Sx on Yx and S2 on Y2 and we have two dynamical systems X x = (Yj, # l5 Цх, Sx) and

*2 = (Y2, ^ 2 , p 2 , S 2).

Now, there exists a homomorphism ht from X t onto X t_ x, i ^ 1,

because X f is the dynamical system isomorphic to 5 (Л ;). We define two non-singular homomorphisms hx: Yl - * Y 2 and h2: Y2 - +Yx as follows:

h i y )

h2 (z)

y e Q 0, h2i(y), yetin,

z,

Q(

z e

Q:

Thus the dynamical systems X 1 and X 2 are weakly non-singularly isomorphic.

Observe that X x is not non-singularly isomorphic to X 2, because the purely atomic measures vx with atoms (Л 0, Л 2, Л 4, ...} and v2 with atoms (Л0, A x, Л 3, ...} are not equivalent.

Let X be a dynamical with discrete spectrum and let 9(X) = (A,v,{m„ (G)}) be its invariant. Define a function к (A) on Л in the following way:

f oo, if the measure и on M(A) has the continuous part, (14) к (A) = < .

[п, if ju on M(A) is purely atomic measure with n atoms.

In [4] is proved that к (A) is the multiplicity function of the unitary operator U T on l3{Q,p) so that the pair (Л,к) is a complete system of spectral invariants of X.

• Th e o r e m

4. Let X f = ( й (, 3F t, pt, 7]), i = 1,2, be two dynamical systems with discrete spectra. I f X 1 is weakly non-singularly isomorphic to X 2, then X l and X 2 are spectrally isomorphic.

P roof. Let hx: Qx - +Q2 and h2: Q2 - > Q X be non-singular homo­

morphisms. The conditions T2 ■ h1 = hx ■ Tx and Ti • h2 = h2- T2 imply h ï 1 Ci ^ Ci and h2 xC\ ^ Ci, i-e- we can define the induced non-singular homomorphisms ht : M 1 -> M 2 and h2: M 2 - + M l . Now, repeating some arguments from the proof of Theorem 3, we can establish that for a.e.

C e M j, hie — ht \C is a homomorphism of the dynamical system ( C , ^ l c ,Pc, Tic) to { C , ^ 2c , p l , T2C), C = hi (C). Similarly, h2c = h2/C is a homomorphism for a.e. C e M 2, whenever C = h2(C). Thus we have (15) h ï l ( M 2(A)) <= M 1 (A), A

A 2, h2 l ( M x(A)) a M 2(A), A e A lt where A t and Л 2 are the point spectra of X x and X 2 respectively. If A

A 2, then pç2( M 2(A)) > 0 and pCl (M 1 (Я)) ^ (M 2(Я))) > 0, and so A

A x.

Thus, we obtain A 2 a A x and also A x <=. A 2 so that A x = Л 2 = A.

Further, (15) implies M 1 (A) has the continuous part iff M 2(A) has the continuous part. Similarly, if /i?1 on M 1 (Я) is purely atomic measure, then p^2 is also purely atomic with the same numbers of atoms. Using formula (14) we obtain kx(A) = k2(A) for any A

A, i.e. the dynamical systems X x and X 2 are spectrally isomorphic. In this way Theorem 4 is proved.

Let S(9) be the standard dynamical system with a given triple

9 = (Л, v, (m„(G)}). Suppose v is a purely atomic measure on '/J with atoms {G l5 G2,

Definition 7.

The group G, is called maximal in Т/ if m(Gt) =

/ (Gt) < oo and G, is no contained in any group Gj, j Ф i.

Let be the set of all maximal groups in У and let У 2 be the set of all maximal groups in If we have defined the sets a < f , where P is an ordinal number, then define Ур as the set of all maximal groups in T/\ U • The sets TP are defined for all ordinal numbers a with

ï

<p

a < a0, a0 ^ X0. Denote T = U The set T is called the set of maximal groups of T. Put T = {G t , G2,

Given two standard dynamical systems S(0j) and S(02), where vl5 v2 are purely atomic measures with atoms {G 1? G2, { H lf H 2, •••} respectively and let T , T 2 be the sets of maximal groups of and TJ2. Using the

oo 00

notations of Section 1, we have = (J If., Ù2 — (J Iflr Denote £2n i = 1 i — 1

= U ^c> ^ 2i = U -f

» ^12 == ^i\^n>

= @

\@2i ’ <*nd let Q2(G)

Ge'A'j He'A'2

= { ( Н , у ) е й 22, у е 1 2, Н э G}, ^ ( Я ) = {(G ,x )e Ô 12, х е / £ ,Я c G}.

D^efinition 8.

We say that the group G e l j can be covered by the set § 2 if there exists a non-singular homomorphism from a subspace of (Q2(G),fi2) to (I f, pa)-: i.e. m1(G) = 1 implies m2(H) — 1 for some H e Q 2(G), mx (G) = 0 and m2(H) = 0 for any H e Q 2(G) implies lx (G) < £ /2(Я).

ИеП2(( i)

9. The group G e i s called covering for the set { H 1, H 2, ...}

if one can construct a non-singular homomorphism from ( I f , p f ) to ( U lb* Дг)-

HcG

Now, we may prove:

Theorem 5.

The standard dynamical systems SiOf) and S(62) satisfying condition that vls v2 are purely atomic measures, are weakly non-singularly isomorphic iff

Г Л х = Л2 = Л;

2° the sets of maximal groups # 1 and C?. are identical and lt (G) = /2 (G) for G e T , f = T 2;

3° each group G e l j can he covered by the set $ 2, each group H e l 2 can be cornered by # x;

4° all groups G e f j , Я е # 2 яге covering for the sets { H l , H 2,...} and { Gl t G2,...}.

P roof. Necessity. Let hji h2: Q2 - + Q 1 be non-singular homo- morphisms and let hl : Û 1- + Ü 2, h2: Q2 - + Q l be the induced non-singular

10 — Roczniki РТМ — Prace Matematyczne XXII

homomorphisms. Theorem 4 implies Л х — Л2 — Л. By the definitions of the sets and C?2 d follows that the measures Д, are purely atomic on A v i = 1,2. It is easy to verify the equalities h ^ I f ) = /

for G e ^ } and

h2(Ic) — 1}, for 2- Further, by the transfinite induction it follows that /it (A ) = I

and h2{Ia) = I f for any G e-I] or G e l 2- Hence we obtain the equalities = ff2 = C? and (G) - 12{G), G e .l (note that mx (G)

= m2(G) = 0, whenever G e # ). Thus for G e ^ we have /?i {Iff) c; 0 22 and }

2( I

)

Û 12, G e # 2, what implies conditions 3° and 4°,

Sufficiency. We shall construct a non-singular homomorphism h2:

Q2 - > Q X, such that

(16) h2( H,y) = (G ,x)

implies G a H .

Applying Theorem 1 to the spaces 0 21 and 0 21 we obtain a non­

singular isomorphism h2: Q 2 l - + Q l l such that if h2(H, y) = (G, x), then G — H, because condition 2° holds. It remains to construct h2 on Q22.

Assume $ x = {G l5 G2,...}. Let = (Git, Gh , ...} be the set of all G e $ 1 for which there exists H = H (G)

Q2(G) such that a2(H) > 0. Now, we may successively define a non-singular homomorphism h2 from <0, (l/2J)a 2(H)>, H = H (G f ), to I f n , because H => G, .. In this way h2 is defined on a part of Q22 onto (J I f . Further condition 3° implies that the measure Jix on

.

Q =

12\ у i f is purely atomic. Let Q = У У (G j,p/). It is easy to

Ce '?j, />1 />1

remark that for any / ^ 1 the set Q2 (Gj) contains infinity numbers of atoms.

Denote Ùs = { ( G f p j 1), i + j = s}, s = 1,2,!.. Taking s = 1,2,... we may successively define a non-singular homomorphism h2 from arbitrary atom of Q2(G'i) to (G-,p/‘), i + j = s. Thus, h2 is defined on a part of Q2 onto Qx. Condition 4° permits to define h2 on Q2 in such way that (16) holds.

Now, we may extend h2 to a non-singular homomorphism h2 of S(02) onto S{6X) as in Remark 3. Similarly, one can construct hx: SiOf) -> S(62) so that theorem is proved.

References

[1 ] J. R : B lu m and D. L. H a n s o n , On invariant p ro b a b ility m easures, Pacific J. Math. 10 (1960), p. 1125-1129.

[2 ] J. R. C h o k s i, N o n -e rg o d ic transform ations w ith disc rete spectra, Illinois J. of Math. 9 (1965), p. 307-320.

[3 ] P. H a lm os and J. v o n N eu m a n n , O p era to rs m ethods in classical m echanics II , Ann.

o f Math. 48 (1942), p. 332-350.

[4 ] J. K w ia t k o w s k i, U n ita ry o p era to rs w ith d isc re te sp e ctra induced b y m easure preserving tran sform ations, Reports on Math. Physics 4 (1973), p. 203-210.

[5 ] J. K w ia t k o w s k i, C lassification o f non-ergodic dynam ical sy stem s with d isc rete sp e ctra , this fascicle p. 263-274.

[6 ] J. C. O x t o b y , E rgodic sets, Bull. Amer. Math. Soc. 58 (1952), p. 116-136.

[7J V. A. R ok hi in, An expansion o f a dynam ical sy stem on tran sitive com ponents, Mat. Sb.

25 (67) (1949) (in Russian).

[8 ] —, On fu n dam en tal notion o f the th eo ry o f m easure, ibidem 25 (67) (1949), p. 107-150 (in Russian).

INSTITUTE OF MATHEMATICS

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