^n) — the Lebesgue space Strong topology on the automorphisms group of

R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O Séria I : P ItA C E M A T EM A T Y C Z N E X X I I I (1983)

Kr y s t y n a Pa r c z y k (T o ru n )

Strong topology on the automorphisms group of the Lebesgue space

1. Introduction. In [1 ] Halmos introduced the notion of a strong topology on the automorphisms group G of a Lebesgue space ([0, 1], 38, m).

This topology was investigated by Bohlin in [2]. Topological properties of the group G and the sets of periodic, aperiodic and ergodic automorphisms have been investigated ([1 ], [2 ], [3]).

The strong topology is giveii by a metric d which is invariant under right and left translations. The group G is a complete topological group [1] and it is contractible [3]. Moreover, the strong topology is equivalent to the topology induced by the uniform convergence on 38, [2 ].

The sets 38n, n > 2 , of periodic automorphisms of period n are closed and nowhere dense in the strong topology and Clos ( hJ ^n) — $ U where

2 2

ЭД is a set of aperiodic automorphisms, [2 ].

' A set of ergodic automorphisms is nowhere dense.

In this paper it is shown that not much information can be deduced from the fact that a set is nowhere dense in G. For a metric d there exists an isonletry from G onto a nowhere dense subset of G. There exists a homeo- morphism from finite or countable Cartesian product of G onto nowhere dense subset of G.

Moreover, a direct proof of the well-known fact that the set of ergodic automorphisms is nowhere dense in the strong topology is given.

2. Notations and definitions. Let [0 ,1 ] denote the unit interval, 38 a cr-algebra of the Lebesgue measurable sets of [0,1] and m the Lebesgue measure on 38. We shall identify sets which are different only on a set of measure zero.

In 38 we consider the metric given by the formula

q

(A, B) = m(A + B)

where + denotes the symmetric difference of sets. In this metric, 38 is a

complete, separable and not locally compact space.

We shall call the triple ([0,1], £3, m) the Lebesgue space.

The transformation T : [0, 1 ]->■[<),1] is called an automorphism of ([0,1], âS, m) if T is one-to-one, onto and measure preserving. We shall identify automorphisms which are different on a set of measure zero.

Let G be the group of automorphisms of the Lebesgue space ([0,1], £8, m). We define the strong topology by the metric

d(S, T) = m {x e [0 ,1 ]; Sx Ф T

}, S ,T <

G.

We shall write T = lim Tn if Tn is convergent to T in the strong topology.

Now, let <f, 51, п ф 2 denote the sets of ergodic, aperiodic and per­

iodic automorphisms:

cf = {T e G : for every E e & Т Е = E only for E — 0 and E = [0 ,1 ]}, 51 = {T e G : T nx ф x for positive integer n and every x},

éPn= {T e G: T nx = x, Tn~lx Ф x, ..., Tx Ф x}.

We shall write I for the identity automorphism, lx — x.

• 3. The strong topology. For the strong topology we prove the fol­

lowing

Th e o r e m 1.

The closed dish {T : d(T, S)

a} is the closure o f the open dish {T : d(T, 8) < a}.

P ro o f. Since d is an invariant metric, it suffices to prove the theorem in the case S — I .

We have e

Clos{T: d(T, I) < a} c {T : d(T, I) < a} .

Let T e G and d{T, I) = a. For any 0 < <5 < a, take a subset A, with mA — ô, of the T-invariant set В = {x: Tx Ф x}. Define an automorphism Tô by the formula

x T0(x) = Tx

Ux

for x e A ,

for x e AcпТ ~ гА 0, for x e A cnT ~ 1A ,

where U is an invertible measure-preserving transformation from Acr\T~lA onto A Gn T A . Such U exists since m (Acn T ~ 1A) = m(A°r\TA).

We have d(Tô, I) < a since

d(Tô, I) = m {x e A °n T ~ lAc: Tx Ф х }ф т {х e Acn T ~ lA : Ex ф x}

< m(A° n Т~гА с n В) + m (A c r\ Т~гА) = m(Acn (T ~ 1A cnB\jT~1A))

— m (B \ A ) = a — ô.

Now

d(Tô, T) — mA-f'm{x e A°r\T lA\Tx Ф Ux} < mA +m (T L4.nAc) < 2<5 what implies lim Tô = T. This proves the theorem.

<5->0 S

Co r o l l a r y

1. The sets n = 2, 3 , . . . , % and S are nowhere dense in G.

P ro o f. For T e T e % T e S we have d(T , I) = 1. Therefore,

£?n, %, «? are contained in the unit sphere {T : d(T, I) = 1 } which is nowhere dense. So 0>n, n = 2 , 3 , . . . , $ and S are nowhere dense too.

n —1

Th e o r e m

2. There exists a homeomorphism from f j G, n > 2, onto a no- where dense subset o f t?n. For n — 2 this homeomorphism is an isometry.

71— 1

P ro o f. We define Un_i: [ ] G->0>n, n ф 2, by the formula

. . . , T n_i) {x) — ^

n

i = 0, . . . , n — 2 ,

for x e F?-}

One can verify that 8 = Un_1{T1, Tn_x) e 0>n, and

<*) 1) = (mod 1 ), i = 0,

\|_ n n J/ l n n J

n —l

The set Vn_x ( [ J G) is nowhere dense in 0>n. In fact, let 8 e

n —l 1

^ - 1 ( П e) and 0 < б < 1 jn. Let 80 be an automorphism given by the formulas

B,

г=0

= 8

г=0

^als*[o,a] ~

V M w i — Qy . y n

where Ui is a periodic automorphism of 8г [0, б], г — 0, ..., п — 1, of period п.у Thus, 8 Ô e 8 Ô does not satisfy (*) and d(8â, 8) = nô. So 8a

n —l n —l

f Un_ i ([JG ) and lim 8Ô = 8 which proves that Un_x ( f[ G1) is nowhere dense

1 3-.0 8 • 1

Ш

71— 1 77- — 1

Notice that TJn_ l is a homeomorphism from /7 G onto Î7n_1(] 7 G)>

i i

n = 2 ,3 , ... Indeed, we have

d(Un-i{T i) •••? Tn_1), Un_ 1(81, . .., $„_!))

[r ^]; Ti (* (æ - ^{i ) )} # s* (* ^{(x ~} n))}'+

+m {• e 1];Tf1'-T”-‘ (и (e- n r))

n—2

V f

= у m\x e i=0 I

# ^ ... 8 п1г \n\x Hæ -V))l

Since

I Г i )m c: \ --- m ix G

[ b t J ^ \ T (n (x - ^ ) H( ⁴ ^))l

= ^ {— 2/; У e [0, 1]; Ту Ф = — tf(T, 8), i = 0 , . . . , » - l , we have

ft—1

d (T {, $i)

d(U n-

Un-iiS-L, ..., 8

i=0

7 1 — 1 71— 1

= i У О Д , Щ + - ÆfTr1 . . . T - :„ S r 1. . . S - l j X - У â(Tt, 8 {).

n n n 4—J

i=0 i=0

77 — 1 71—1

Therefore f7w_! is a homeomorphism from /J О onto ( n e > ,

i i

w = 2, 3, ...

For Uj : G->3?2 we have

d{Vx{T), TJX{8)) = = d(T,£)

so it means that Z7j. is an isometry. This completes the proof.

R e m a r k 1 . Similarly, we can construct a homeomorphism £ГТО from

00

f j G onto a nowhere dense subset of G. We put i

1 / , / 1 U 1 „ r 1 1

R em ark 2. There exists a homeomorphisms from G onto nowhere dense subset of £Pnyn = 2 , 3 , . . . We define Un:G ^ ^ n by

Ü JT ) = Vn_x( T , .. ., T ) .

R em ark 3. The set 0>n, n = 2, 3, . .., is not a locally compact space.

This statement* is a consequence of Remark 2 and the fact that the group G is not a locally compact space.

References

[1] P. R. H alm o s, Approximation theories for measure preserving transformations, Trans. Amer. Math. Soc. 55 (1944), 1-18.

[2] У. A. R oh lin , Selected problems of the metric theory of dynamical systems, Uspehi Mat. Nauk 4.2 (1949), 57-127 (in Russian).

[3] M. K e a n e , Contractibility of the automorphisms group of a non-atomie measure space, preprint.

IN S T IT U T E O F M A T H EM A T IC ^ N ICH O LA S C O P E R N IC U S U N I V E R S IT Y 87-100 T oron , P oland