In this paper we charácterize weakly mixing transformation groups in terms of weighted ergodic theorems

Becker, María Elena

Becker, M. E.

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 34 (1990), 307-309 .
WEIGHTED ERGODIC THEOREMS FOR
WEAKLY MIXING TRANSFORMATION GROUPS
Abstract
MARÍA ELENA BECKER
In this paper we charácterize weakly mixing transformation groups in
terms of weighted ergodic theorems .
1 . Introduction
Let (Tt, t E R) be a one parameter group of measure preserving transfor-
mations on a probability space (S2, F, P) and let (Tt ) denote the corresponding
group of unitary operators on LZ = LZ(2, F, P) . We assume that the map t to
Ttf is continuous for each f in L2 .
The group (Ti) is ergodic if the only functions let fixed by Tt for all t are the
constante . The group is called weakly mixing if for any f in LZ
lim 1
J r J(Ttf, f) -
JE(f) 1212 dt = 0,
r-oo r 0
where E(f) = fs2 f dP .
Weak mixing implies ergodicity and then by the Birkhoff Ergodic Theorem
we have r
lim 1jTt f(x) dt = E(f) a.e .
r-oo r
From this it is easy to see that the convergence also holds in the L'-norm . For
a proof of these assertions see [1, Chapter 1] .
We denote by W the class of all bounded complex valued functions p on R
such that r
m(p) = lim 1 I p(t) dt exists .r-oo r
Let p be a function in W. For each f E LZ we consider the weighted averages
A,(f,p)(x) =
1 frP(t)Ttf(x)dt .r o
In what follows we prove that A,(f, p) converges in mean to m(p) E(f) if,
and only if, (Ti) is weakly mixing . We aleo show that this later condition is
equivalent to the almost everywhere convergence of A,(f, p), when the weight
p is a periodic function .
308
	
M.E. BECKER
2 . Statements and proofs
Theorem 1. The following conditions are equivalent :
(a) (Tt ) is weakly mixing
(b) For any p in W A,(f, p) converges in mean to m(p) E(f), for each f E
L2 .
Proof.: (a) => (b) . Let f E L2 . We may assume without loss of generality
that E(f) = 0 . For a given e > 0 let Bf = {t E R : I (Ttf, f )I > E} . Since
IT_tf,f)I = I(Ttf,f)I, we see that BE is a symmetric set respect to zero . Also
from
I rf ~(Ttf,f)j2 dt > E2 IB. n [o,r]IT r ,0
where vertical bars denote Lebésgue measure, we deduce
density, i .e .
lim ¡B, n [- r, r] I = 0 .r-oo r
On the other hand, by Fubini's theorem we have
IIAr(f , p)f = r f r frp(t)p(s)(Ti_, f, f ) dt ds
r ~ r 9
= T f p(S)(r 9 p(u + s)(Tuf, f) du) ds
EIIPII óo + II fII2IIPII2o I
[[-r, r] n B,¡ .
that BE has zero
Letting r -> oo, we obtain limsuplIAr(f,p)II2 < EIIpl1 2 . Since e is arbitrary,00r-oo
this implies IIAr(f,P)II2 --+ 0, as r -> oo .
(b) =* (a) . Suppose that (Tt) is not weakly mixing . Hence, there is a non-
constant f such that Tt f = e'A¡f , for some real number A (see [1, p . 29]) . If we
take p(t) = then Ar(f, p)(x) = f(x) does not converge to m(p) E(f) .
Under the assumption that the map t to Tif is continuous for each f E L2 ,
it is not hard to prove that if (Tt) is weakly mixing then Ta is itself weakly
mixing for any a qÉ 0 . Using this fact we can state the following result .
Theorem 2. (Tt ) is weakly mixing if, and only if, for any periodic function
p E W A,(f, p) converges almost everywhere to m(p) E(f), for each f E L2 .
Proof.- Assume that (Tt ) is weakly mixing . Let p be a bounded periodic
function and let a > 0 such that p(a +t) = p(t) for all real number t . Let f be
a bounded function . To prove the almost everywhere convergence of A,(f, p)
it sufñces to show that A,,,, (f,p)(x) tends to m(p) E(f) for almost every x, as
n -> oo, n being integer .
We have
WEIGHTED ERGODIC THEOREMS
	
309
A«n(f,p)(x) = jan p(t)Ttf(x) dt =
0
a
p(t)Tt+iaf(x) dt
7=0 0
n LTj«[a Jo p(t)Ttf(x)df
.
j=o
Since Ta is weakly mixing and therefore ergodic, it follows that
lim A«n(f,P)(x) = E(- f p(t)Ttf(x) dt) = m(p) E(f)o
for almost every x .
By virtue of the Maximal Ergodic Theorem (see [2, p . 690]) and the density
of bounded functions in L2 , a standard argument implies the convergence of
A,(f, p) for an arbitrary f E L2 .
The converse follows from the proof of (b) => (a) in Theorem 1 .
Remark . Since all almost periodic function is the limit of a uniformly
convergent sequence of trigonometric polynomials, Theorem 2 remains valid if
we set almost periodic functions instead ofperiodic functions .
References
[1] I . CORNFELD, S . FOMIN AND YA. SINAI, "Ergodic Theory," Springer-
Verlag, New York Inc ., 1982 .
[2] N . DUNFORD AND J . SCHWARTZ, "Linear operators, Par¡ I: General
Theory," Interscience, New York, 1958 .
Jaramillo 3204-1 "C"
1429 Buenos Aires
ARGENTINA
Rebut el i de Noviembre de 1989


Weighted ergodic theorems for weakly mixing transformation groups

Diposit Digital de Documents de la UAB

Publicacions Matemátiques, Vol 34 (1990), 307-309 .WEIGHTED ERGODIC THEOREMS FORWEAKLY MIXING TRANSFORMATION GROUPSAbstractMARÍA ELENA BECKERIn this paper we charácterize weakly mixing transformation groups interms of weighted ergodic theorems .1 . IntroductionLet (Tt, t E R) be a one parameter group of measure preserving transfor-mations on a probability space (S2, F, P) and let (Tt ) denote the correspondinggroup of unitary operators on LZ = LZ(2, F, P) . We assume that the map t toTtf is continuous for each f in L2 .The group (Ti) is ergodic if the only functions let fixed by Tt for all t are theconstante . The group is called weakly mixing if for any f in LZlim 1J r J(Ttf, f) -JE(f) 1212 dt = 0,r-oo r 0where E(f) = fs2 f dP .Weak mixing implies ergodicity and then by the Birkhoff Ergodic Theoremwe have rlim 1jTt f(x) dt = E(f) a.e .r-oo rFrom this it is easy to see that the convergence also holds in the L'-norm . Fora proof of these assertions see [1, Chapter 1] .We denote by W the class of all bounded complex valued functions p on Rsuch that rm(p) = lim 1 I p(t) dt exists .r-oo rLet p be a function in W. For each f E LZ we consider the weighted averagesA,(f,p)(x) =1 frP(t)Ttf(x)dt .r oIn what follows we prove that A,(f, p) converges in mean to m(p) E(f) if,and only if, (Ti) is weakly mixing . We aleo show that this later condition isequivalent to the almost everywhere convergence of A,(f, p), when the weightp is a periodic function .308	M.E. BECKER2 . Statements and proofsTheorem 1. The following conditions are equivalent :(a) (Tt ) is weakly mixing(b) For any p in W A,(f, p) converges in mean to m(p) E(f), for each f EL2 .Proof.: (a) => (b) . Let f E L2 . We may assume without loss of generalitythat E(f) = 0 . For a given e > 0 let Bf = {t E R : I (Ttf, f )I > E} . SinceIT_tf,f)I = I(Ttf,f)I, we see that BE is a symmetric set respect to zero . AlsofromI rf ~(Ttf,f)j2 dt > E2 IB. n [o,r]IT r ,0where vertical bars denote Lebésgue measure, we deducedensity, i .e .lim ¡B, n [- r, r] I = 0 .r-oo rOn the other hand, by Fubini's theorem we haveIIAr(f , p)f = r f r frp(t)p(s)(Ti_, f, f ) dt dsr ~ r 9= T f p(S)(r 9 p(u + s)(Tuf, f) du) dsEIIPII óo + II fII2IIPII2o I[[-r, r] n B,¡ .that BE has zeroLetting r -> oo, we obtain limsuplIAr(f,p)II2 < EIIpl1 2 . Since e is arbitrary,00r-oothis implies IIAr(f,P)II2 --+ 0, as r -> oo .(b) =* (a) . Suppose that (Tt) is not weakly mixing . Hence, there is a non-constant f such that Tt f = e'A¡f , for some real number A (see [1, p . 29]) . If wetake p(t) = then Ar(f, p)(x) = f(x) does not converge to m(p) E(f) .Under the assumption that the map t to Tif is continuous for each f E L2 ,it is not hard to prove that if (Tt) is weakly mixing then Ta is itself weaklymixing for any a qÉ 0 . Using this fact we can state the following result .Theorem 2. (Tt ) is weakly mixing if, and only if, for any periodic functionp E W A,(f, p) converges almost everywhere to m(p) E(f), for each f E L2 .Proof.- Assume that (Tt ) is weakly mixing . Let p be a bounded periodicfunction and let a > 0 such that p(a +t) = p(t) for all real number t . Let f bea bounded function . To prove the almost everywhere convergence of A,(f, p)it sufñces to show that A,,,, (f,p)(x) tends to m(p) E(f) for almost every x, asn -> oo, n being integer .We haveWEIGHTED ERGODIC THEOREMS	309A«n(f,p)(x) = jan p(t)Ttf(x) dt =0ap(t)Tt+iaf(x) dt7=0 0n LTj«[a Jo p(t)Ttf(x)df.j=oSince Ta is weakly mixing and therefore ergodic, it follows thatlim A«n(f,P)(x) = E(- f p(t)Ttf(x) dt) = m(p) E(f)ofor almost every x .By virtue of the Maximal Ergodic Theorem (see [2, p . 690]) and the densityof bounded functions in L2 , a standard argument implies the convergence ofA,(f, p) for an arbitrary f E L2 .The converse follows from the proof of (b) => (a) in Theorem 1 .Remark . Since all almost periodic function is the limit of a uniformlyconvergent sequence of trigonometric polynomials, Theorem 2 remains valid ifwe set almost periodic functions instead ofperiodic functions .References[1] I . CORNFELD, S . FOMIN AND YA. SINAI, "Ergodic Theory," Springer-Verlag, New York Inc ., 1982 .[2] N . DUNFORD AND J . SCHWARTZ, "Linear operators, Par¡ I: GeneralTheory," Interscience, New York, 1958 .Jaramillo 3204-1 "C"1429 Buenos AiresARGENTINARebut el i de Noviembre de 1989

Weighted ergodic theorems for weakly mixing transformation groups

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