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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 7 8 9 10 11 12 < 13 > 14 15 16 17 18 19 .. 99 >> Next There arc several other ways of stating the ergodicity condition and we present some of them in the –ø–µ–º two theorems.
Theorem i .5. If 7 : A' ‚Äî< X is –∏ measure-prewrvincj transformation of the probability \;>ace (X,yJ,nt) then the Jollowiiuj siattmeiU'S arc equivalent:
(il T is eri/odic.
(11) The only members –í of –õ with m[T~ '–í –î –í) = 0 arc those with iii(/\$I = 0 or in(li\ ‚Äî I.
(lii) Tor i rer\ A e Jj with m[A) > 0 m haw \ T~"A) = 1.
(iv) Tor et cry /1, –í e /–ê with m{ 1) > 0, m(B) > 0 there exists n > 0 with iiH I ! B) ^ 0.
Pkooi
(i) => (ii) Let Be.i/1 and m(T~ lB –î B) = 0. We shall construct a set –íx with T ' –í, = –í , and m(B –î–π,] = 0. For each n > Owe have —à(7 "–í –î –í) = 0 bccause –¢'–í –î –í —Å 7" '‚Äò‚Äô"–í–î–ì'–í = (Jl'-o –ì'(–ì‚Äò–í–î–ô) and
licncc m(T "–í –î –í) < –ø—Ç(–¢ '–í –î –í). Let –í–∏ = f|'= ‚Äû –ö) –¢~‚Äò–í. By the above v\—Å know m(B –î T~'B) < m(B –î T~'B) = 0 for each n > 0. Since tlic sets (_ J T ‚Äò –í decrease with n and cach has measure equal to –í we h–ª\–∞–ø(–í , –î –í) ‚Äî 0 and hencc nnB , ) = m(B). Also T~ 1–í , =
¬´ , J.¬ª‚Äû –ì .....T-''B = BV. Therefore we have
obtained ,i set –í with T~ 1–í ‚Äî–í and —à(–í^ –î –í) = 0. By ergodicity w-e must have –ø—Ü–í,) = 0 or 1 and hence –ø—Ü–í) 0 or 1.
(iij -> (ia). Let A —Å ,‚Äôi and –Ω–∑(–õ)>0. Let Ax = i T "A. We have T~ 1A i ci .3 , and since m(T ‚Äò A ;) = m(A J vvc have Z\Alj = 0.
By kii) we gel m(At) = 0 or 1. We fcannot have m(A ,) = 0 bccause T~>AczAl and ml 7" 1 ‚ñÝ!) ‚Äî m(/l) > 0. Therefore m[A ) = 1.
(ni) ro (ivj. Let m{A) > 0 and –ø'–ª–í) > 0. By (iii) we have i>!('^jn'= , T~"A) = 1 so that 0 < i)i(fi) - ni(B n Ul'_ ! T~"A) = –í n T~"a). Therefore
in(B r\ T~"A) > 0 for some n > 1.
liv) => (i). Suppose –í eana T~XB = B. If 0 < m{B) < 1 then 0 = m(B n (A B)) = m(T""B –æ (A B)} for all n > 1, which contradicts (iv). ‚ñ°
Remark. Noiice that we could replace (iii) by the statement ‚ÄúFor every A fc Mi \wih /¬´(.-I) > 0 and every natural number N we have m({j,‚Äò v T~"Ai = I" because y,; s T ~"A - T –õ(–´–ò‚Äò_‚Äû T~"A). Consequently we could replace
2S
I Measure-Preserving Transl'ormmions
(iv)b\ ".W even .-1, 4> c-.//‚Äô with iu(.!) > 0 ,m(i>) > 0 and every natural number .V there cm.sis n > N willi m(T~"A nli) > O'. One can think of (iii) and (iv) as sayinu that the orbit ''!' "A () of any non-trivia! set A sweeps out the –∏ hole sp ice A (or that each non-trivial set A has a dense orbit in a measure-theoretical sense).
The next theorem characterises ergodicity in terms of the operator UT.
Theorem 1.6. If (A\.W,in) is a probability space and T:X -¬ª X is measure-preseri ini) then the followiiuj statements are equivalent:
(i) T is eriioilic.
(ii) Whenever f is measinable and (/ ^ T)(x) = f(x) –£—Ö–µ X then f is constant a.c.
(iii) Whenever f is measurable and (/ T)(,v) ‚Äî f{x) a.c. then f is constant
a.e
(iv) Whenever f e L2(in) and ( f –æ T){x) = f(x) Va 6 X then f is constant a.e.
(v) Whenever f e L2(m) and (/ - T)(x) = f(x) a.e. then f is constant a.e.
Proof. Trivially we have (iii) => (ii), (ii) => (iv), (v) => (iv), and (iii) => (v). So it remains to show (i) => (iii) and (iv) => (i). We first show (i) => (iii). Let T be ergodic and suppose / is measurable and f ¬∞ T = / a.e. We can assume that / is real-valued for if / is complex-valued we can consider the real and imaginary parts separately. Define, for –∫ e Z and n> 0,
A'(/<,¬ª) = {.v:k/2" < J\x) < (k + l)/2"; = f~l{[k/2", (k + l)/2")).
We have
I 1 X(k,n) –õ X(k,n) c {x:(f –æ T)(x) –§ f(x)}
and hence m(T~ lX(k,n) –î X(kjufc) = 0 so that by (ii) of Theorem 1.5 m(X{k,n)) ‚Äî 0 or 1.
For each fixed n UkeZ A"(/<, –Ω) = X is a disjoint union and so there exists a unique k‚Äû wilh m(X(k‚Äû,n)) ‚Äî 1. Let Y = ff^=i X(k‚Äû,n). Then m(Y) = 1 and / is constant on Y so that / is constant a.c.
(iv) => (i). Suppose T 1E = E, E e Then yE e L2(m) and ('/E ¬∞ T)(x) -X,;( v) Vv e X so, by (iv), yv is constant a.c. Hcnce yE ‚Äî 0 a.e. or yE = 1 a.e. and m(E) = \ '/j. dm ‚Äî 0 or 1. ‚ñ°
Remarks
(1) A similar characterization in terms of Z/(m) functions (for any p > 1) is true, since in ihc last part of the proof yE is in L‚Äò\m) as well as L2(m). Also we could use real iJ‚ÄôUn) spaces.
\$1.5 Ergodicity
29
(2) Another characterization of ergodicity of T is- whenever /:X -* R is measurable anJ f(Tx) > f(x) a.e. then / is constant a.e.
This is clearly a stronger statement than (iii). To see that ergodicity implies the staled property, let T be ergodic and suppose f(Tx) > f(x) a.e. If / is not constant a.e. then there is some r e R with –í = (x e –ê^–î—Ö) > c} having
0 < m(B) < 1. But T~lB => –í so m(T~lB –î–í) = 0 and hence m{B) = 0 or
1. This contradicts 0 < m(B) < 1.
We want to analyse our examples to see which of them are ergodic. The next result will be useful for this and it also relates the idea of measure-theoretic dense orbits (Theorem l.S(iii) and (iv)) to the usual notion of dense orbits for continuous maps. We shall say more about this in Chapter 6. Recall that the —Å—Ç-algebra of Borel subsets of a topological space is the —Å—Ç-algcbra generated by the open sets. Previous << 1 .. 7 8 9 10 11 12 < 13 > 14 15 16 17 18 19 .. 99 >> Next 