Books in black and white
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 12 13 14 15 16 17 < 18 > 19 20 21 22 23 24 .. 99 >> Next Proof, (due to A. Garsia) Clearly Fs e L]{(m). For 0 < n < N we have FK > f, so UFn > Ujn by positivity, and hence UFN -r f >fŌĆ×+i. Therefore
UFx{x) + f{x) > max fŌĆ×{x)
1 ┬Ż*i┬ŻjY
ŌĆö max fn(x) when F,v(x) > 0 0&n┬ŻN
Tin┬╗ / i s - I- I',\ 1)11 Ō¢Ā ŌĆó [ą╗-: /-"ą╗ l-V) > 0}, i?o film > j ( I1 silm ŌĆö J ^ Ul'silm
ŌĆó r
ŌĆö j ^ I- ą╗ i/m ŌĆö J ( L l's dm since Fv = 0 on A'Vl.
j^ 1\ dm ŌĆö j UTVdm since F-V > 0 and hence UFS > 0.
> 0 since |jL'|| < 1. Ō¢Ī
Corollarj 1.16.1. Let T.X ŌĆö> X he nwasurc-prcxijruiiuj. If ge Lj.(ni) and
[ 1 "'J
Ux = <ą╗ e ,Y: sup V <j[T\x)) > a ( n> i n i= ąŠ
then
I \$ dill > ąŻčł(ąÆ, ą│ A)
JIlar\A
if T~1 1 = .1 din! m{A) < x.
Proof. VVe lirsl prove lhis result under tire assumptions m(X) < cc and -1 = .Y. Let f ŌĆö >j ŌĆö y.. then ŌĆö |J v - ąŠ ( v' I'\(v) > 0J so that jBi /dm > 0 by Theorem 1.16 and therefore čā dm > y.iinbJ. In the general case we apply the above to T\A to get \AŌĆ×,iai)dm > %m(A n BJ. Ō¢Ī
Proof of Tiii:oki;M 1.14. VVe first assume m(X)<oc. By considering real and imaginary parts it su(Vices to consider /Ōé For such an / let
jS*U) = IimsiipŌĆ× ., U/'U V;,= o/(T4)and./#(x) = limmfŌĆ×.,oc,(l/)i)X:'ri/(TŌĆśx). We have /* T - f* and /ŌĆ× T = /* because if ą░ŌĆ×(čģ) = (1/ą╗) ąŻ^ąŠąöąóčģ) then ((n + 1), ii)di: n(.v) ŌĆö uŌĆ×(Tx) ŌĆö f(x)/n. We have to show that /* = /* a.e. and that they belong to L'(m).
For real numbers a, /> let Ey /, = [.v e X\ fjx) < /? and a < /*(x)]. Since (xUJx) < / *(.v)j = (J{/*&{!> < čā and a, /) both rational} we shall show m(Ea ji) ŌĆö 0 if l> < a because then we shall have/* = /* a.e. Clearly T"1 Eajt = EaJ and if we put Ba = {x e X |supŌĆ×-., (l/ą╗) ąŻ>~^/(ąóŌĆśčģ) > a} then ┬Ża/J ąŠ Bx = ┬Ża p. From Corollary 1.16.1 we get
j/.ŌĆ× = J,ŌĆ× n/JŌĆ× felm - am(Eu: n = ŌĆ£'┬½(┬Ż┬½>ŌĆó)ŌĆó
Therefore _ / <//┬╗ > ą░čł(
If we rcplace f, a, /> by -/, ŌĆö fi, ŌĆöa, re nectivcly. then since (ŌĆö/)* = -/* and (-/\ = -/* wc get
| ^ f dm < ąĀą┐čäąĢąźčł0).
Therefore am(/;ii(,i < lini(Exll), so if /f < a then m(Ey fl) - 0. This gives/* = /* a.e. as we explained above. Therefore (l/ąĖ) ąŠ /(ąōčģ) ŌĆöŌ¢║ /* a.e.
M.miu;
39
ąōąŠ 41 low /'* čü L'\iu) we use liie part of Faion's lemma lhal asserts lim c/ŌĆ× e if [čā,,] is a (poimwise) comergenl sequence of non-negative integrable functions with lim inf |(/ŌĆ×dm < x. Let gj.x) = |( 1 /ą╗) ąŻ)'=(5/(7',(ą╗'))|- Then | j. lldm so wc can apply the Fa ton lemma to assert limŌĆ×_ ^ ą│/ŌĆ×(ą╗) = ŌĆ×(1 n) ┬Ż"~l,flT'x)\ = |./ *| belongs lo LŌĆśM It remains to show that | / dm = \ f*dm if m(X) < čüąŠ. Let D'l = {x e X:(k n) < f :'(\i < Ik -t- 1)//;,ŌĆÖ where ą║ e Z, n > 1. For cach omall čü > 0 we have D'l n B[kiu)_,. = DŌĆÖl and by Corollary 1.1C-1
ąō j dm > ( ŌĆö <: |čłą©") so lhal ąō f dm > - ni(Dl).
J,y┬Ż \n J n
Then j ^ j
L I * dm < - - - /ąĮ(ą®) < - in(D'k) * L
ą╗ jj J"i<
by the above mequalii). Summing over ą║ gives \ x f* dm < (m(X)/'n) + j, / dm. Since lln.s holds for all n > 1 we have fA J * dm < ] v f dm. Applying 111 is to ŌĆö /'in.slcad of / gives | v (ŌĆö /)* din < jA- ŌĆö/ dm so that J.y/* dm > |A prim. Hut /, ŌĆö /* a.e. so j A / * dm = |A f dm. This finishes the proof when m( \ i < cc.
When m(X) = čüąŠ 'he above proof is valid once we have shown thai ///(┬Ż,.,,) < x. when (i < y. (Wc need to show this to apply Corollary 1.16.1.) Suppose firstly that a > 0. Let Ce J be any set with ąĪ čü ┬Ża/J and m(C) < oc. (Such a sei exists bccause X i* c-finite.) Then h = / ŌĆö e LŌĆś(┬╗3) so by the maximal ergodic theorem
f (I ŌĆö čÅ/čü)^ 0 f┬░r all N > 1.
(The function Hv is ussociatcd to /i by Theorem 1.16.) But C^EuP<z IjxŌĆś //Y(.v) > 0; and therefore \x\f\clm > can(C). Hence m(C) < (bx) JA \ f\dm for each ąĪ ┬Ż/ąÉ with ąĪ čü: Ea-lt and m(C) < oc. Since X is o-finite we have čé(ąĢ,ą╗1.) < oc. If a < 0 then /? < 0 so we can apply the above with ŌĆö / and ŌĆöji instead of/ and a to get m(EaiP) < cc. Ō¢Ī
┬¦1.7 Mixing
If ąō is a measure-preserving transformation of a probability space we have deduced from the ergodic theorem that T is ergodic ifT V/l, ąÆ e
1 1
lim - Yj ŌĆśA n B) = m(A)m(B).
II ~t Gf.j W |'=0
We can make changes in the method of this convergence to give the following notions.
40
1 Mca^uef-Prcscrving Tranbl'orm.iiions
DiTmiiiun 1.5. Let 7'be a measure-preserving transformation ofa probability space {X'.Sijn).
(i) T is wciik-nŌĆÖixiKt] if V.'l, ąÆ e ąø
j ąĖ -1
lim - ąŻ \m(T~'A n B) ŌĆö čł(/4)čł(ąÆ)| = 0.
n
(ii) T is struiuj-im King if V/l, ąÆ e 3#
lim m{T~"A n ąÆ) ŌĆö čé[ąÉ)čéą©). h~* f
Remarks
(1) Every strong-mixing transformation is weak-mixing and every weak-mixing transformation is crgodic. This is because if {a,,} is a sequence of real numbers then limŌĆ×_ , ┬½ŌĆ× = 0 implies
1 1
lim - ^ lf/,-1 = 0
" ,*ąōąŠ
and this second condition implies
1 1 i;m >_ iij = 0.
ii-.ą╗ n i - ()
(2) Am L.\ample of an crgodic transformation which is not weak-mixing is given by a rotation T(z) = a: on the unit circle K. This will be proved at the end of this section but one can sec it roughly as follows. If A and ąÆ are two small interval.s on ąÜ then T ŌĆśA will be disjoint from ąÆ for at least half of the values of i so that (l/n) ┬Ż"=(| \m(T~ŌĆśA n B) ŌĆö m(A)m(B)\ > \m(A)m(B) for large ii. From this one sees that, intuitively, a weak-mixing transformation has to do .some ŌĆ£stretching.ŌĆØ Previous << 1 .. 12 13 14 15 16 17 < 18 > 19 20 21 22 23 24 .. 99 >> Next 