C Classification of non-ergodic dynamical systems with discrete spectra

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) . ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXII (1981)

J.

Classification of non-ergodic dynamical systems with discrete spectra

Abstract. In this paper a complete system of invariants of non-ergodic dynamical systems with discrete spectra is constructed. Any dynamical system X = (Q , .F , ц , Т ) with discrete spectrum determines a triple 0 (X ) = ( Л, (^ , 31л , v), {m „(G )}„= where Л is the point spectrum of X , У is the set of all subgroups of А , 31л is a а -field of subsets o f generated by the family of sets {^ (A )} (А е Л ) (^ (A ) = { G e ^ ; A e G }), v is a normalized measure on 31л . Finally, {m „(G )}„ = 1 2,... is a sequence o f measurable functions defined on '§ such that m„(G) ^ 0,

X

m„(G) ^ mn + 1(G) (n = 1 ,2 ,...) and £ m„(G) < 1 for a.e. G e ' S . The triple (fiX ) is a complete system of invariants of X .

1. Introduction. In the present paper a complete system of invariants of the dynamical non-ergodic systems with discrete spectra is constructed.

The isomorphism problem for the ergodic dynamical systems with discrete spectra, was solved by P. Halmos and von Neuman [2]; V. A. Rokhlin proved that a type of a given dynamical system is uniquely determined by types of its ergodic components [5]. However, the looking for an iso­

morphism between quotient-spaces (with respect to some partition on ergodic components) which preserves the types of the corresponding components, can be difficult. For that reason, the construction of objects, which form a complete system of invariants of non-ergodic dynamical systems, appears to be needed and useful. That is what is done in this paper.

If X — (Q, & , i i , T ) is a dynamical system with disrete spectrum and

C is a partition on ergodic components, then we can define a partition c, of the space M ; = Q\C, as follows: two elements of 0|£ determine the same element of Ç iff they have identical types. To each element of M : |£

corresponds a group (a type) which is contained in the point spectrum

A of A . Using this correspondence, one can transfer the measurable structure

of Mr I £ to the set '</ of all subgroups contained in A. In this way we

obtain a <7-field of subsets of which is identical with the cr-field

generated by the family of sets {-^(A)};6/1, where 'У (A) = {Ge'#'; 2 e G }. The

types of measures of the elements of £ define a sequence of real non-negative

functions {

„(G )}„=1>2j..., defined on 3. The triple 9(X) = [A, (3, A, v), 1 w„(G)}n=1>2>...) does not depend on the partition ( on ergodic components and turns out to be a complete system of invariants of X . Moreover, for a given triple (A, (3 , , v), {m„(G)}„=li2>. ) satisfying certain conditions, there exists a dynamical system X with discrete spectrum such that 6(X)

= (Л ,р # ,«Г „ у ), {ffin(G )}n = lt2,...)-

2. Notations and definitions. Let X = (Q, ,W, p, T) be a dynamical system, where Q is a Lebesgue space with p(Q) = 1 and let 41 be the unitary operator on l3{Q ,p ) defined by (Uf)(co) = f (Tea) for any f e l 3 ( Q , p ) and almost every (a.e.) coeQ.

Definition

1. A dynamical system X is a system with discrete spectrum if there exists a basis of Û ( Q , p ) consisting of eigenfunctions of 41.

We denote by A the set of all eigenvalues of 41. Let E be the spectral measure of °U and let d1 be the maximal spectral type of 41 [1], i.e. for any Borelian subset A of the circle group К (K = (z; |z| = 1 }),E (A ) is a projection of the space L?(Q,p) and a1 is a normalized Borelian measure on K. It is well known that there exists a function f0el3(Q, p) such that Qfo = dr1 and qf < Qf о for any f

L2(Q, p), where gf (A) = (E ( A ) f , j ) , A c z K . By above E(A) ф 0 iff <тх (zl) > 0, or equivalently E(A) = I iff o x (A) — 1, where I is the identity projection on l3 (Q ,p ). From ([1 ], p. 58, Lemma 3) it follows that Я is an eigenvalue of Ш iff E({X}) Ф 0, whence Я is eigenvalue of 41 iff crt ({Я }) > 0 . Therefore, by above and by ([3], Theorem 1) we have the following:

Theorem

A. The following conditions are equivalent:

(a) X is a dynamical system with discrete spectrum, (b) E(A) = I,

(c) the maximal spectral type o x of JU is purely atomic, and A is the set of all atoms of о x.

In the sequel we use the following notations: £ — a partition of Q on ergodic components with respect to T [5], (M ,F ç,/iç) = ( Q , ^ , p ) |£ — the quotient-space with respect to ( [6], {рс}сем — a canonical system of measures of £, Ac , ( C e M ) — the point spectrum of the dynamical system ( C , ' ^ c ,p c , Tc), where Tc = T\C, c = J4 C. 4lc — the unitary operator on L2(C , p c) induced by Tc , Ec — the spectral measure of 41 c .

For f e l f ( Q , p ) let f c = / | C be the component of / over the element C, (C e M ). From properties of the canonical system of measures { рс}сем, we obtain that for almost every (a.e.) C e M , f c el3{C, juc). For a.e. C

M the set Ac is a countable subgroup of the cricle group К ((C , » Me > Tc) is an ergodic dynamical system).

3. The partition of M defined by the types of £. Assume that X

= (ü, 4F, p, T) is a dynamical system with discrete spectrum.

Non-ergodic dynamical systems 265

Th e o r e m

1. For a.e. C e M , (C , t F c ,Pc,

is a dynamical system with discrete spectrum and Ac c= A.

Prpof. V. A. Rokhlin [4] proved that for a.e. C e M

(1) 1 Е {А )Л с = Ec (A) f c

for any f e l } ( Q , p ) and any Borelian subset A of K . Let Mç be the set of all C e M satisfying (1). Take a dense sequence j / 1, / 2,...} of functions of l } ( Q , p ) and denote by M 2 the set of all C e M such that the sequence {fc , f c , •••} is dense in l } ( C , p c)- In Lemma 1 we shall show, that M 2e . ^ ; and /iç(M 2) = 1. Further, by Theorem A, E{A) = 1 what implies E(K\A) = 0 i.e. E(K\A) / ' = 0 for i = 1,2,... Now, if we put

M c(0 = { C e M ; [£ ( К \ Л ) Я с = 0 in L2(C ,p c)},

and M ç* = M l c\ M l n f] Mr (0, then clearly e and pz (AT*) = 1. It is i = 1

easy to see that if C e M £ , then EC(K\A) = 0 i.e. EC(A) = I. Therefore Q

(A) = 1, where Qc is the maximal spectral type of °UC. Thus, we have shown Qc is a purely atomic measure and Ac a A. The theorem is proved.

Le m m a 1.

I f

is a dense sequence'of functions in

then for a.e. C e M , {fc , fc , •••} IS a dense sequence in l } ( C , p c)-

P roof. Let {Di}ieI (Г ^ X) be a basis of measurable subsets of Q such that Dkn D j e { D i] ieI, for any k , j e l . It is obvious that the sets (D { n C }lW form a basis of measurable subsets of C.

Consider the sequence {gl , g 2,...} of all functions of the form C\X

+ C2X

2 + ••• + CkXDk, where X

are the characteristic functions of Djy and Cj are complex rational numbers (je I). Then {g1, g2, ...} is a dense set of functions in l3 (Q ,p ) and {gl, Qc,...} is a dense set in Ü ( C , p c) for a.e. C e M . This implies that for any positive integers n and i there exists j Un such that

2dg < - f j .

Let A (i,n ) = { C e M ; J \gt-fc‘’ \z » l/(n-2*)} (i,n = 1,2,...).

Then we have

- f -тг > f y - F ' f d n = j ( l lg‘c - f h 2d ^ ) d ^

n Q M C

= J ( ) + J ( )

A(i,n) M\A{i,n)

> j ( f \Sc ~ f c in12 dpc) d^ ^ Pç (A(i, nj).

A(i,n) С П •

2

Hence fy(A{i,n)) ^ 1/(

- 2f) and putting A(n) = (J A (i,n), A = f] A(n)

i= 1 n = l

we obtain p^(A(n)) ^ 1/n and Pç(A) = 0.

Suppose СфА. Then there exists n0, such that СфА(п0) whence C $ A ( i , n 0) for i = 1 ,2 ,3 ,... Let К (/ С,в) be any open disk in l3{C,pc) with the centre f c and the radius e (e > 0). Since { gh,9c, •••} is a dense set in l3 {C ,p c), then there exists a number i0 such that g'ç e К (f c , e/2) and

\/ l/(n0 • 2l°) < e/2. But С ф A ( i 0,n0), i.e.

v 7S \g‘ê - j i ‘0"0\2dnc < V l/ t n »^ '0) < 6/2.

c

Therefore f c ° n° ^ K{g'c , e/2). Consequently f c l°n° e К ( f c ,

) because g'c & К {fc , e/2). This shows that {/с1, /c2, ...} is dense in L2(C, /tc) if C eM \d.

Since fiç{A) = 0 the lemma follows.

We denote M ç = { C e M ; {C,£Fc , n c , Tc) have a discrete spectrum and Ac c A}. By Theorem 1, M^

J2^ and p^{M^ = 1.

Rem ark 1. Because ( is a partition on ergodic components, then for every C eM ç the group A c is a countable subgroup of the circle K.

D

2. Let ^ be a partition of M ç defined in following way:

Cl5 C2

M^ determine the same element of £ iff ЛС1 = ЛС2. The partition ç, we shall call the partition induced by the types.

T

2. The partition Ç induced by the types is measurable.

Proof. For ÀeA let M (A) = { C e M c; A eA c}. In what follows we shall show that M(A) is a measurable set (Lemma 2) and that the family of sets {М (А )}ЛбЛ form a basis of Ç. Let ^ denote the partition of induced by the family {М (А )}ХеЛ. Any element <3 of can be represented in the form

(2) 2 = П A f(A )n П M'(A),

j ЯбЛ\Л j

where A x is a subset of A. If CeQ), then A e A x iff AeAc , i.e. A t = Ac . In particular A l is a group. Note also that 3) is the set of all elements C

M^ for which Ac = A 1. It follows 2 is an element of £. Since each element of Ç has the form (2), then Ç = The theorem is proved.

L

2. For each AeA the set M (A) = ( C

M?; AeAc} is measurable in the space (M ç, , p^ and p^ (M (A)) > 0.

Proof. Let Я А с: l } { Q , p ) be the eigenspace of A and let {f\ , f £ , •••}

be a basis of H x. Further, let

M } = ( C

M ?; J \fc\2 d pc > 0}, i = 1,2,...

c

Since Ф,(С) = J \Jc\2dpc is a measurable function of variable C, then M f

Non-ergodic dynamical systems 2 6 7

is a measurable set. From the equality

1 ( № c t à - * - - f K \ 2dnc)dK = = 0

M r C Q

s

it follows that °UC■ — X ■/£ for a.e. C e M ç i.e. is an eigenfunction of X in l3(C,fic). Therefore, if C e M f , then XeAc and C e M (X ) whence

00

M f a M(X) and U M f a M(X).

/= l

In order to show inverse inclusion we consider spaces Hr (C

M ç, ХеЛ), where He is the space spanned by {fie, F \c , F o r a.e. C e M ç we have Нс1 1 He2 if Ax Ф X2 and since the spaces Hç {ХеЛ), generate Ь2(С ,ц с) (Lemma 1), then He is the eigenspace of X in L2(C,/xc)- It is clear that if C eM (X ), then X eA c and Нс Ф 0, hence there exists i such that /iG ф 0

00

and C e M f. As a consequence we have the equality M(X) = (J Mf, what

gives that M{X) is measurable. 1 = 1

Furthermore by the equality

1 = \\fik\2dn = J (Jl/icl2^ c ) ^ = 1 ii\fic\2dnc)d^,

q c m a c

> 0 for i = 1,2,... In particular цс(М(Х)) > 0. The lemma follows.

4. Construction of invariants. Below we use the following notations:

= (M C, J ^ ) | £ ,

— a canonical system of measures of = {^ cl C e M ç}, ^ — the family of all subgroups of К contained in A. Now, consider a map F : ^ -*■ N c defined by

F(G ) = П M ( X )n П M'{X) for G e ^ .

A e G А б Л \G

Observe that the map F is one-to-one.

Let $1* = F ~1 ( ^ * ) and let (T) = /^* (F (A)) for A e 21*. Moreover, let 2lç be a cr-field of subsets of ^ such that AeSH^ iff АслУ^еЩ . A measure on 2IÇ is defined by v^{A) = v ^ ( A n ^ ) , for A e 2 Iç

Rem ark 2. Note that the space (^ ,2 Iç,vç) is a Lebesgue space, vç is a complete measure on 2lç, ^

2Iç, vç (^ ç) = 1 and 21* = 2lç | . Moreover, the map F : (^ , s é vç) -»■ (Nç, 3F%, ц%) is defined a.e. on ^ with respect to

and F is an isomorphism mod 0 between spaces ^ and JVÇ.

For a.e. G e ^ we have the measure jùF(G) defined on the element F(G) of £. Let (m„(G)} (n = 1,2,...,) be the type of p.F(G) [6]. It is known that the functions m„(G) of the variable G are measurable, m„(G) ^ 0,

00

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

L

3. For every Л е Л the set &(Л) = {G e ^ ; X eG } is measurable in (^ ,2 lç,yç) and vç(^(A)) > 0. Furthermore, the family of the sets {&(Л)}ЛеА generates the а-field

P roof. The partition £ determines a homomorphism of the space M ç onto iVj [6], which we denote by £ also. So we have the following mappings:

(M j, JF{ , Л£) (N s, Щ) f - (<ÿ, 9I{ , »5).

The sets {М(А)}лел are £-sets and by definitions of the mappings F and £ holds F _1(^(M (A))) = ^ n ^ ( A ) . Hence # çn # (A )e 2 lf, i.e. ^(A )e2Iç. Since the mappings £ and F preserve the measures and р^{М(Л)) > 0, then vç(^(A)) > 0. Further the partition of the space (^ ,5 tç,vç) induced by the sets {^ (А )}ЛбЛ is the partition on the points of this space, therefore is generated by the family {&(Л)}кеЛ. The lemma is proved.

Now, to the dynamical system X under consideration and to its partition

c on ergodic components, one can attach th£. following triple:

д(Х,С) = ( А , ( $ , Ъ л,ч,),{т„(С )}п=i >2J , where 91л is a о -field generated by the family {^(Я)}яел*

Rem ark 3. The set У(Л) is non-empty since the cyclic group {A"}„ = o,± i,..., belongs to &(Л) [3].

T

3. I f Ç and n are two different partitions of the dynamical system X on ergodic components, then 9 {X ,Q = 9 {X,n).

P roof. In what follows the following notations are used:

( M ', i F ; , ^ ) = (Q , ^ , p )\ n ,

{FC'}c'6M' — a canonical system of measures of n, Ac■ — the point spectrum o î { C ' , ^ \ C , p a , T) M n = { C 'e M '; ( C , 3F'\ C , p^>, T) is a dynamical system with discrete spectrum and Ac> a A}, Ç — the partition of M n induced by the types, (N n, = {M n, &'n, /4)1 {/4'k'ev* - a canonical system of measures of (Sn = { A d C ' e M n}, F': y n^>Nn — the mapping defined analogously as F for Ç 21* = ( F ' y 1^ * ) , v*(-) = ^ * (F '{)).

By [5] there exists a set Q0 <= Q, p(Q0) = 1 such that Ç \Q0 = n\Q0.

We may suppose that Q0 is a Fin variant set. For each CeÇ such that C r\Q0 Ф 0 there exists exactly one С е л such that C n Q 0 = C n Q 0. Let j be the identity map on the set Q0 and let j * { C ) = C iff C n U 0 = C n Q 0.

It is obvious that j * : M -*■ M ' is an isomorphism mod 0 and its restriction to Mç is an isomorphism onto M„ also. We have the following commutative diagram :

Q — Q

- —► M n

Non-ergodic dynamical systems 269

where £ and n are the mappings corresponding to the partitions C and n respectively.

It may be easily verified that Ac = A f {C) for a.e. C e M This means that ;'*(£) = £ 'mod 0 and therefore j* determines a mapping j* * : N ^ - * N n which is an isomorphism and such that the diagram

M; » M n

(3) C ?

is commutative and Д, is isomorphic to for [6]. Now, the measure space can be constructed in just the same way as the space (^, 21л , vç). As a result we obtain the following commutative diagram:

Q ___ i___ > Q

M n

F F

where H = (F 'f f lF . It is easy to see that H (G ) = G for a.e. G e <§ (with respect to vç). Since H preserves the measures vç and v„, then vç = vn.

Let {m£(G)}„ = 1>2>... be the type of p!F{G). It follows from the commutativity of diagram (4) that j* * (F (G )) = F '(H (G )) = F'(G) for a.e. Ge&. Hence according to (3) the measures JiF(G) and pF(G) are isomorphic, i.e. mn(G)

= m'„(G) for a.e. G e ^ and n — 1,2,... This completes the proof.

The theorem we have just proved justifies the following notation в (Х ) = ( А , ( % ® л , v), {mn(G)}m=1>2>J .

5. Isomorphism theorems.

Th e o r e m

4. Let X t = (Q{, 7]) (i — 1, 2) be two dynamical systems

with discrete spectra. The systems X x and X 2 are isomorphic mod 0 iff OiXJ = в ( Х 2).

(4)

Mr

P roof. Necessity. Let f: (Qlt T J -> (Q2, ^ 2, ^

->T2) be an isomorphism and let ^ be a partition of X x on ergodic components. Then C2 = f (Ci) is a partition of X 2 on ergodic components and therefore / determines the mapping / *: M Cl -> M ?2 such that the diagram

/

--- ► Çl2

Cl Î2

► M ?2

is commutative. Now, repeating the arguments from the proof of Theorem 3, we get 0 (X J = 6 (X 2).

Sufficiency. Let us suppose 9 (X J = 6 (X 2). In particular A 1 = Л 2 = A and (0, 21л , vx) = ( У , Шл, v2). Let Ci and Ç2 are partitions on ergodic com­

ponents of X i and X 2 respectively. Using £i and Ç2 we may construct all objects needed for the definition of the invariants and B(X2).

We have the following mappings:

Q2

Cl C2

(5)

M

M

C2

Cl

where / = F 2 Fffi. For G e ^ we put = F 1 (G) and = F 2(G). By the assumption m\ (G) = m%(G) for a.e. G e@ and n = 1,2,..., i.e. the spaces ( C ï x x) and (C

3 2, jïy2) are isomorphic. By [6], there exists an isomorphism / : M ç -*■ M ?2 such that / ( ^ ) = £2 and the diagram

M , i ---♦

C2

Cl C2

AT

(6)

Non-ergodic dynamical systems 271.

is commutative. Let A 1Cl and Л 2с2 be the point spectra of the dynamical systems (C lt & i\ C lt ph^ Tt) and (C2, F 2 \ C2, Цс2, T2) respectively and let / (C i) = C2. Then by the commutativity of diagrams (5) and (6) A 1Cl

= ( F ^ t i H C i ) , A 2

2 = (F 2 - 4 2)(C 2) and

A lCl = ( F T ' Z M C J = (F 2 - 1F 2F r 4 i ) ( C 1)

= ( F î ' f S M C i ) = ( F ; l s2f ) ( c t)

= (F j Ч гН / С ,) = (F J 1 f 2)(C 2) = Л 2

2.

It follows that the dynamical systems (C lf ^ ’1jC 1, pclf Tt) and (C2, JF2|C2, ^ 2,T 2) being discrete spectra are isomorphic for a.e.

By Rokhlin’s tneorem [5] we conclude that the dynamical systems X t and X 2 are isomorphic as was to be shown.

T

5. Let A be a countable subset of the circle К such that AeA implies A"eA for all integers n. Let be the set of all subgroups of A, SUA be the а-field of subsets of У generated by the family of sets У (A) = {G e& ; A eG} (A eA ) and let v be a normalized measure on ^lA such that v(^(A)) > 0. Moreover, let {m„(G)}Ce^ (n = 1,2,...) be a sequence of measurable functions defined a.e. on such that mn(G) ^ 0, mn(G) ^ m„ + 1(G)

00

and £ m„(G) ^ 1. Then there exists a dynamical system X = {Q, 3F, p, T) n — 1

with discrete spectrum such that

в (Х ) = { Л , & , 2ГЛ, v), {m „(G )}„,1>2,J .

P ro of. First assume mt (G) = 1 and m„(G) = 0 for n = 2 ,3 ,... and a.e. G e ^ . If S is a subgroup of K , then by S we denote its dual group.

Let Q be the set of all couples (G, x), where G e ^ and xeG . By A* we denote the smallest group containing A. Then Л* is a separable abelian compact topological group with the borelian <r-field $ and the normalized Haar measure p0. Now, let H : x А*, <т(21л x J 1), v x p0) Q be a mapping defined by H (G , (p) = (G , hG (<p)), where hG is natural homomorphism Л*

onto G, (G <= A c= A*). Furthermore the mapping H permits to define a (T-field of subsets of Q and a measure p in following way:

A e ^ o H - 1(A)e(j(SAA x ^ ) , p(A) = (v x p0) ( H ~ l A).

The measure space ( Q , ^ , p ) is isomorphic to the space ^ x A * \ q , where q

is the partition of ^ x A* on the sets of the form {G } x C, where C is any

abstract class of Л* with respect to kerhG. We shall show (Lemma 4)

that q is a measurable partition of ^ х Л * , whence (O, ^ , p) is a Lebesgue

space.

In what follows we shall construct a transformation T: Q^>Q. Let j e A* and j GeG be characters defined by j(X) = X for XeA* and j G(X) = X for XeG. In particular hG(j) = j G for each G e 'S . Put

T0(G, (p) = (G, (p j ) for (G, <p)eGx Л*, T ( G , x ) = ( G , x - j G) for (G ,x )e Q .

The mapping T0 is an automorphism of ^ х Л * . Note also that the diagram

У х A*

Q

To У x A*

Q

is commutative, whence T is an automorphism of (Q ,iF ,n ). The Lebesgue space and the automorphism T form a dynamical system X — T ). In a sequel we show that I is a dynamical system with discrete spectrum and that its invariant в(Х ) is the triple (А, ( У , Шл, v), {m„(G)}n= i >2,...) given in the theorem.

Let now C be the partition of Q given by the sets of the form { G } x G . It is easy to see that Ç is a partition of X on ergodic components and T\{G} xG is an ergodic automorphism with discrete spectrum with G as the point spectrum. We have

1 Е (К \ А )П с = Ec (K\A)fc

for a.e. CeÇ and / eL2 (й, ц). Since EC{K\A) = 0, then E{K\A) = 0, i.e.

the spectral measure E is concentrated on A. In this way AT is a dynamical system with discrete spectrum and the point spectrum of (Q, 3F, ц, T) is contained in A.

The condition v(&(X)) > 0 implies X is an eigenvalue of T for each XeA. By definition of £ it follows (Q, ^ , ц)\£ = Further, the partition £ of Q |£ induced by the types is the partition on points of У . According, the sequence of functions {mx (G) = 1 and m„(G) = 0, n = 2, 3,...}, is the type of £. By above

6(X) =

In general case, let I G (Ge&), be a measurable space with {w „(G )}n=1)2i...

as the type. Since {wn(G )}n = li2>... are a measurable functions of G, then

there exists a measurable space ( Y , R , p ) and a partition /1 of У such that

Y\f$ is isomorphic to (^,31

,

) and the elements of /7 are isomorphic

to I G respectively [6]. Now, we can identify the points of Y with the

couples(G, yG), where G e ^ and yGe/G. Let Q = {((G, yG), x); (G, yG)e Y, * eG} .

Non-ergodic dynamical systems 273

We define a mapping H : Г х Л * ° ^ 0 by H ((G, yG), (p) = ((G, yG), hG((p)) (<реЛ*), and a a-field 0* of subsets of Q and a measure \i on 0* as before.

Put

T0((G, yG), q>) = ((G, yG), (p j), ({G, yG), <p)e Y x A * , T ( ( G , y G) , x ) = ((G,yG) , x j G), ((G,yG) , x ) e Û .

In a similar way we can show that X = (Q p, T) is a dynamical system with discrete spectrum and 6(X) = [Л, {У, $1Л, v), { m„(G )}II=i >2,..). The theorem is proved.

Lemma

4. The partition rj of У x A* defined in the proof of Theorem 5 is measurable.

P roof. We shall construct a sequence of measurable partitions of У х A*, say r\n such that rj„ ]rj. Let A = {X1,X2,...} and let e„ be the partition of (У,Ш

,

) generated by the sets У (Xt), У (X2) , ..., У (X„) (n = 1,2,...). Obviously, et ^ e2 ^ ... and e„te, where £ is the partition on points of У.

Let C

£„. Then C has the form С = У (Х{1) п У (Xi2) n ... п У (Xik) n n f ] y' ( Xj ) .

j=

We denote by Gc the group generated by Xix, Xi2, ..., Xik and let hc be the natural homomorphism A* onto Gc . Now, one can define a partition t]n of У х A* as follows: the sets of the form CxS>, whered Ceen and Q) is any abstract class of Л* with respect to ker hc , are the elements of t]n.

It is easy to see, that r\n is a measurable partition of ^ х Л * and rj„ j rç.

Therefore, rj is a measurable partition of У x A * . The proof of lemma is completed.

Example.

Let S be an abelian separable compact topological group, p the normalized Haar measure on <r-field 0b of borelian subsets of 5.

Fix aeS and let 7^,(5) = a-S. It is known that X a = {S,0b,p, Ta) is a dynamical system with discrete spectrum with A = {q> (a); q>eS] as the point spectrum. Let Sa be a subgroup of S generated by a i.e.

Sa = {^n}n=o,± i,±2,...• The partition Ç of S on the abstract class of S with respect to Sa is a partition of X a on ergodic components. Each element of

C is a dynamical system isomorphic to (Sa,0ta, p a, Ta), where 3âa is a-field of borelian subsets of Sa, and pa is the normalized Haar measure on dSa.

Therefore, S|£ is the factor group of S with respect to Sa and the partition

£, induced by the types is trivial. Let as before У be the set of all subgroups of A, be the ст-field generated by the family of the sets {У(Х)}лел- Denote by v the ^-measure on ftA concentrated on А (Л is an element of У). Let m„(A) = 0 for n = 1 ,2 ,3 ,..., if S|Se is infinite and m„(A) = 1/k, n = 1,2, . . ., k and m„{A) — 0, n = k + 1,... if S|Se is finite and card {S|Sa} = k. It is easy to verify that

0 ( X J = ( Л , ( » , Я Л.»),{И 1 .(Л )},.1.2....).

Rem ark 4. The invariant 9( Xa) one can identify with the pair (A,k), where к = card {S|Se}, if S\Sa is finite, and к = K0 if S|Sfl is infinite.

Simultaneously, the pair (Л , к) is complete system of spectral invariants of X a, because the number к is the multiplicity function of the operator ШTa on L2(S,n). Therefore the spectral isomorphism of dynamical systems of above form is equivalent to isomorphism mod 0 of this systems.

References

[1 ] N. D u n fo r d and J. T. S c h w a r tz , L in ea r operators, vol. 2, Moscow 1965 (in Russian).

[2 ] P. R. H a lm o s , J. v o n N eu m a n n , O pera to r m ethods in classical mechanics II, Ann. of Math. 43 (1942), p. 332-350.

[3 ] J. K w ia t k o w s k i, U n ita ry operators w ith d isc rete spectru m induced b y m easure preserving tran sform ations, Reports on mathematical physics 4 (1973), p. 203-210.

[4 ] V. A. R ok hi in, S elected problem s o f the m etric th eo ry o f dynam ical system s, U M N 4 (1949), p. 57-128 (in Russian).

[5 ] —, An expansion o f a d yn am ical sy stem on tran sitive com ponents, Mat. Sb. 25 (67) (1949) (in Russian).

[6 ] —, On fu n d a m en ta l 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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