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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 57 58 59 60 61 62 < 63 > 64 65 66 67 68 69 .. 99 >> Next ‚Äû p(B n E) u(Bn(X\E))
p(E) p(WE)
Note that p} and p2 are >n M(X, T), /z, —Ñ p2, an¬£l
p(B) = /i(¬£)/i,(B) + –ü - H<E))p2(B).
Therefore p is not an extreme point ol M(X, T).
Conversely, suppose p e M(X, T) is ergodic, and
P = –Ý–Ý i + (1 - P)P 2
where plt p2 e M(X, T) and p e [0,1]. We must show = p2. Clearly /ii ¬´ p (pi is absolutely continuous with respect to p) so that the Radon-Nikodym derivative dpjdp exists, (i.e.,
dp Ax)
i(E) = J¬£ dp{x), V¬£ e –©–•).
dp
See Theorem 0.10). We have dp^fdp > 0. Let
56.2 Invariant Measures for Continuous Transformations
153
Wc have
f , ^-d^ + –ì ^ic^ = /(1(¬£) = /t1(T'-1¬£)
=Tf f ^-J/¬´
jEr.r-iE dn h -'F'F dfl
ao that
dfh , –ì
–ì ~dfi = –ì d>‚Äò-
Jr\T-¬´¬£ –°/—Ü –õ Jr->¬£ ¬£ </,,
c//t J7' ‚ÄòL f dfi
Since d^i/dn < 1 on ¬£\T_I¬£ and <//<,iclji > 1 on T~lE,E and since —Ü(–¢~'–ï\–ï) = —Ü{–¢~1–ï) ‚Äî fi(T~ 1EnE)= —Ü(–ï) -—Ü(–¢~1–ï–≥–ª –ì) = nft\T~lE) we have —Ü(–ï\–¢~1–ï) = 0 = ;t(7'~1¬£\¬£). Therefore ^(7'_1¬£–î¬£) = 0 so ^(E) = 0 or 1. If //(¬£) = 1 then /^(.V) = \E(dfiljdfi)dfi < —Ü(–ï) = 1 contradicting —Ü}(–•) ‚Äî 1. Hence we must have ;i(E) = 0.
Similarly if F = {.v|dftjdfi > 1} we have //(¬£) = 0 so that dfi1/dfi = 1 a.e. (fx). Hence fil ‚Äî fi and so /j is an extreme point of M(X, T).
(iv) By the Lebesgue decomposition (Theorem 0.11) there are unique probability measures ar>d a unique p e [0,1] such that fj. = p/t, + (1 ‚Äî p)fi2 where /–≥, ¬´ m and //2 is singular with respect to m. But ft = –¢—Ü = pTjt, + (1 ‚Äî —Ä)–¢—Ü2 and since –¢¬´ 7 m = m and Tfi2 is singular with respect to Tm = m the uniqueness of the decomposition imply //,, —Ü2 e M(X, T). Since /i is an extreme point we must have either p = 0 or p ‚Äî 1.
If p = 0 then —Ü = –¶–≥ and so —Ü is singular relative to m. If p = 1 then // ¬´ m and we can argue with dfi/dm as in (iii) to get ft = m, a contradict.on. ‚ñ°
Remarks
(1) The second part of the proof of (iii) shows that if /^,/–≥–µ M(X,T), fil ¬´ /I and /–ª is ergodic then /i, ‚Äî —Ü.
(2) Since M(X, T) is a compact convex set we can use the Choquet representation theorem to express each member of M{ X, T) in terms of the ergodic members of M{X, T). If E(X, T) denotes the set of extreme points of M(X, T) then for each —Ü e M(X, –ì) there is a unique measure r on the Borel subsets of the compact metrisable space M(X, T) such that z(E(X, T)) = 1 and V/ 6 C(Af)
jxf(x)dll(x) = Ji;(v r) ^/(jcJr/mCx^rM.
We write ft = \–µ{—Ö.—Ç),–ø^—Ç(–ø1) and call this the ergodic decomposition of —Ü. See Phelps . Hence every —Ü e M(X, –ì) is a generalised convex combination of ergodic measures. This is related to the decomposition of a measure-preserving transformation into ergodic transformations (see ¬ß1.5).
We shall now interpret ergodicity and the mixing conditions in terms of the weak*-topology on M(X, T).
154
6 Invariant Measures for Continuous Tr.msform.itions
¬ß6.3 Interpretation of Ergodicity and Mixing
Let T:X -* X be a continuous transformation of a compact metric space. We say —Ü e \1{X, T) is ergodic or weak-mixing or strong mi.ving if the measure-preserving transformation T of the iheasure space (X, –õ(–•),/i) has the corresponding property.
Recall that is ergodic iff V/,</ e L2(j¬´)
- Z –ì/(7'–ª)–±|(–ª')^(–ª)-¬ª (fdfi fg dfi.
W ft ,--q v
Ve change this slightly for our needs.
Lemma 6.11. Let —Ü e M(X, T). Then
(i) fj. is ergodic if V/ e C(X) Vg e L'(;i)
‚Äú X Jfdft Jffdfi-
(ii) /.i is strong mixing iff V/ e C(X) Vg 6 Ll((i)
^f{Tix)g(\)dii(x) ‚Äî ^fdfi Jg d[i.
(iii) /j ts weak mixing iff there is a set J of natural numbers of density zero such that V/ e C(X) 7<y e L‚Äô(//|
^ lim ¬ßf[T‚Äòx)g{x)dn(x)-*¬ßfdn Jgdfi.
Proof
(i) Suppose the convergence condition holds and let F, G e L2{fi). Then
G 6 L4rf so i –£ J/(T‚Äòx)G(x) J/4x) ‚Äî Jyrf/i jGrf/i V/ 6 C(X).
Now approximate F in L2(fx) by continuous functions to get
‚Äú Z ¬ßF{T!x)G(x)dn{x)-> ¬ßFdn jGfa
Now suppose —Ü is ergodic. Let / e C(A'). Then / e L2(^) so if h e L2{//) we have
Z f/( –ì‚Äò*–® x) Mx) -*‚ñÝ J7 dn jhdfi.
If ge L*(/0 then by approximating g in Ll(n) by h e L2(/i) we obtain
‚Äú Z J–ê–¢‚Äò–•–ú*)Mx)dfi-
¬ß6 3 Interpretation of Ergodicity and Mixing
155
The proofs of (ii) and (i.i) are similar and use Theorem 1.23. ‚ñ°
Theorem 6.12. Let The a continuous transformation of a compact metric space. Let p e –©–•, T).
(i) —Ü h prgodic iff whenever m e M(X) and m ¬´ p then
1 " 1
- –£ f'm -> p. n ‚Äúo
(ii) p is strong mixing iff whenever m e M(.V) and in ¬´ p then Tnm ‚Äî> p.
(iii) u is weak mixing iff there exists a set J of natural numbers of density zero such that whenever m e M(X) and m ¬´ p then lim_,)n-OD Tnm -* p.
Proof
(i) We use the ergodicity condition of Lemma 6.11.
Let p be ergodic and suppose m ¬´ p, m e –©–•). Let g = dm ‚Äòdp e Lx(p). If / 6 C(X) then
(W1 I T'ni\ = - X jV¬∞ T‚Äò dm = ~ X \f(T'x)g(x)dp(x)
J \n 1 = 0 / 4 i-0 J n 1 = 0 J
->‚ñÝ jfdp^gdp = jf dp J1 dm = ^f dp.
Therefore (l/n)X"=o T m -> p.
We now show the converse. Previous << 1 .. 57 58 59 60 61 62 < 63 > 64 65 66 67 68 69 .. 99 >> Next 