ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATHEMATYCZNE XXI (1979)
B. Ka m in sk i (Torun)
Some properties of coalescent automorphisms of a Lebesgue space
The concept of coalescence has been introduced by Auslander [3] in topological dynamics and by Hahn and Parry [4] in ergodic theory. In [5], Newton has investigated the spectrum of coalescent automorphisms.
The present paper consists of three parts. The first part contains defini
tions and lemmas. The second part concerns the decomposition of a Lebesgue space on ergodic components with respect to coalescent automorphisms and coalescence of automorphisms with discrete spectrum. In the third part we prove that any Bernoulli shift is not coalescent. This theorem gives us a posi
tive answer to the question stated in [5]. From this theorem we conclude that unitarily coalescent automorphisms have zero entropies.
The author is indebted to Professor E. Sqsiada for suggesting the idea of the proof of Theorem 2.
1. Definitions and lemmas. First we recall the definition of coalescent and unitary coalescent automorphisms [5].
An automorphisms T of a Lebesgue space (X,.Jd,p) is coalescent if each endomorphism Ф of (X,:J#,p) such that Ф о Т = ТоФ is an automorphism.
We denote by Ut the unitary operator defined on L2 ( X, p ) by the formula UTf = f o T .
An automorphism T of ( X , M, p ) is unitary coalescent if each linear isometry U of L2( X, p ) such that Ut oU = UoUt is an unitary operator.
Let, for any measurable partition £ of X , L2(£) denote the subspace of L2( X, p ) consisting of functions constant on elements of
It is easy to prove the following
Lem m a 1. T is coalescent if and only if for each measurable T-invariant partition Ç of X such that the automorphisms T and are isomorphic we have Ç = e.
Lem m a 2. The class of unitary coalescent automorphisms is closed under factors.
Proof. Let T be unitarily coalescent and let T* be a factor of T.
It is well known that there exists a measurable T-in variant partition £ of X such that T* and T* are isomorphic. Let U0 be a linear isometry of L2(£) such that U0oTt = U r o U 0 and let E be the identity on
L2( ÿ = L! ( i , / i ) e i . 2 ({).
The operator U = U0 @ E is a linear isometry on L 2( X , p ) and UTo U
= U о UT. Since T is unitary coalescent, U is an unitary operator. Thus we obtain
L2 Ю 0 L2 = L2 {X, p) = UL2 ( X , p) = 170 L2 (£> © L2 ( ^ .
Hence U0 L2 (Ç) = L2 (£), i.e. f/0 is an unitary operator on L2 (£). Therefore, T, and T* are unitary coalescent and the lemma is proved.
Co r o l l a r y. If T is unitary coalescent, then the partition sT of X on ergodic components with respect to T is countable.
Proof. It is enough to show that the measure pEj. is discrete. Suppose, on the contrary, that p, r is not discrete. Then there exists an endomorphism Ф of X/ eT which is not an automorphism. Since TEJ. is the identity, we have Т1тоФ = Ф оТЕ.г , i.e. TEJ. is not coalescent and hence it is not unitarily coalescent. It follows by Lemma 2 that T is not unitary coalescent. The contradiction establishes the corollary.
2. Coalescence and spectrum. Let ( Х, Ш, р ) be a Lebesgue space and let T be an automorphism of ( X, j $ , p ) . Let us suppose that the decom
position £t = { X i , i e l } of X on ergodic components with respect to T is countable. Let us put Tk = T\x , k e l .
Th e o r e m 1. T is coalescent if and only if, for each i e l , 7J is coalescent.
Proof. It is easy to see that coalescence of T implies coalescence of 7], i e l . Now, let us suppose that 7] is coalescent for each i e l . Let Ф be an endomorphism of ( X , t $ , p ) such that Ф о Т = ТоФ. It is enough to prove that Ф is a one-to-one transformation. For a fixed k e l we define the following sets of indices:
П = { i e I ; f i ( Xd > Ж , ) } , II = { . € / ; , ! « = » ( X t)}, П = { i s / ; fi(Xi) < лг(ЛГк)}.
Let us put
n = г ^ ^
X I
= u
x t, X I= u
x h X I= и
X,.iell iel\ ielsk
First we shall prove that Ф-1 ( Xk) e , £ k. Since Т о Ф = ФоТ, and T X k = X k, the-set Ф-1 ( Xk) n Xi is 71-invariant for each I e l l . It follows from ergodicity of 7] and the definition of Щ that ц( Ф~1 ( Xk) n X,) = 0, i el f . Hence р(Ф-1 (Xk) n XI) — 0, i.e. Ф-1 ( Хк) фЩ. It remains to show that Ф~1( Хк)ф
Ф Я , i-e. /л(Ф~1(Хк) rv X Ф) — 0. Let us suppose, on the contrary, that ц(Ф~1 ( Xk) n X k) > 0. Since 7] is ergodic for i e l k, there is a non-empty set 7* c= Ik such that
ф 'М а д = и ie4 Then
ф - U x t ) с= у Xt,
l e ! k
where 71 = /£ — 7* and so
p ( * D = ц(ФкЩ) ) X t) < p ( X sk).
ieK
This contradiction establishes the desired equality. Thus Ф~1( Хк) е 1' к, k e l , i.e. Ф is a permutation on 2Ck. Therefore, using our assumption, we obtain the desired result.
Now we shall show that one cannot replace in Theorem 1 coalescence by unitary coalescence.
Ex a m ple 1. Let ( X , S , p , T) be a dynamical system having the decompo
sition £т = { X k; к ^ 1} of X on ergodic components with respect to T such that all components Tk, к ^ 1, are unitary coalescent and isomorphic. Let
<Pk: X k -> X k- ! , к ^ 2 ? be the isomorphism of 7^ <md 7^_j . We define the operator U of L2 (X , p) by the formula
(Uf)(x) = 0
( f о (pk)(x)
if x e X lt if x e X k, к ^ 2.
It is easy to check that U is a linear isometry but is not unitary and
V o U t = V t o U . Then, in view of Theorem 1, T is coalescent but not unitarily coalescent.
From Theorem 1 we get at once
Co r o lla ry 1. If T has discrete spectrum and eT is countable, T is coalescent.
Now we shall show that the countability of s t in Corollary 1 cannot be omitted.
Ex a m ple 2. Let X denote the 2-dimensional torus written multiplicatively, P the Lebesgue measure on X, and T( u, v ) = (u, Suv), where Suv — u- v For each u(modO), Su is coalescent. Let us observe that T is not coalescent since it commutes with the endomorphism Ф(и, v) = (u2, v2) which is not an automorphism.
Prace Matematyczne 21.1
Our next aim is to describe coalescent rotations on abelian, compact and separable topological groups.
Let G be an abelian compact and separable topological group, РЛ a «т-algebra of Borel sets of G, ц the normalized Haar measure of G and Ta: G -+ G given by Tax — a - x , where a is a fixed element of G different from the identity of G. Let Ga denote the closure of the cyclic subgroup generated by a. It is well known [8] that the elements of the group GjGa from the partition eTg of G on ergodic components with respect to Ta.
If /i(Ga) > 0, then £Ta is finite otherwise it is uncountable.
Corollary 2. Ta is coalescent if and only if eTa is finite.
P ro o f. The sufficency immediately follows from Theorem 1. It follows from Lemma 6 of [1] that the system (G ,ju ,T ay is isomorphic to the system {G /G a x G a, fi* x /ла, I x Ta} , where ц* and fia denote the normalized Haar measures of G/Ga and Ga, respectively, and I is the identity on G/Ga.
Let us suppose, on the contrary, that £Tg is not finite, i.e. /i(G a) = 0.
Then Ц* is continuous and so I is not coalescent. By Proposition 1 of [6], I x Ta is not coalescent and hence Ta is not coalescent. The contra
diction establishes the necessity.
3. Coalescence and entropy. Let X0 = {0, l , . . . , r — 1} and let fi0 be a probability on X 0 having the following distribution: /x0 ({*'}) = Pi, i =^0, 1 ,..., r — 1. Let { X , p) = (X 0, p f f and let T be the translation on X, i.e. (Tx) (i) — x ( i + l ) , x e X , i e ^ t . T is called the Bernoulli shift.
Theorem 2. Bernoulli shift is not coalescent.
P ro o f. We shall construct a measurable Т-in variant partition Ç of X such that T and T* are isomorphic but £ Ф e. Let Ф: X -> X be defined by the formula: (Фх)(п) = x ( n ) Q x ( n — 1), п е Ж , where © denotes the sub
traction mod r [5]. It is easy to see that Ф is measurable, r-to-one, onto and Ф о Т = Т о Ф. Let us put Ç — Ф- 1 (е), where e denotes the partition of X on points. Then Ç is measurable, T-invariant and £ Ф e. It follows from [7] that 7^ is a Bernoulli shift. It remains to show that T and 7^
are isomorphic. It follows from [2] that there exists a Lebesgue space (X, v) and a collection {Sc}c6^ of automorphism of (У, v) such that the system ( Х , ц , T) is isomorphic to the system X/ Çx Y, /i*xv, 7^x{Sc}. Since Ф is r-to-one, each CeX/H; is a finite set consisting of r elements and each (mod /и*) measure pLC of the canonical system of measures of £ is discrete and uniform. Then Y is finite and {Sc}c6£ is a collection of permu
tations of Y.
Now we use the Abramov-Rohlin formula for entropy of skew product automorphisms [2]:
h{T) = h(T() x { S c }) = A(T{) + M { S C}),
where
hr({Sc }) = sup hY({Sc }, a), aei'
h ({Sc}, a) = in f— j H (a" ) (dC) , n n x/c
< = V S f 1 ■ S f J- ...S j L , «, C e ( .
k = 1 1 s
Since Y is finite, one can easily prove that /ir({Sc}) = 0, i.e. h(T) = к(Тг).
It follows from [6] that T and 7^ are isomorphic. Then, by Lemma 1, T is not coalescent and the theorem is proved.
From Lemma 2, Theorem 2, and the Sinai theorem [9] follows Corollary. If T is unitary coalescent, then h (T ) = 0.
References
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[6] D. S. O rn s te in , Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4, No. 3 (1970), p. 227-352.
[^] Factors of Bernoulli shifts are Bernoulli shifts, ibidem 5, No. 7 (1970), p. 349-364.
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NICOLAUS COPERNICUS UNIVERSITY INSTITUTE OF MATHEMATICS, TORUN
