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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 16 17 18 19 20 21 < 22 > 23 24 25 26 27 28 .. 99 >> Next (ii) A measure-preserving transformation always has –Ø = 1 as an eigenvalue and any non-zero constant function is a corresponding eigenfunction.
Definition 1.8. We say that a measure-preserving transformation T of a probability space (–•,–£–≤,—Ç) has continuous spectrum if 1 is the only eigenvalue of T and the only eigenfunctions are the constants.
Observe that T has continuous spectrum iff –Ø = 1 is the only eigenvalue and T is ergodic.
We shall need the following result from spectral theory to prove the next Theorem. The proof can be found in Halmos .
Theorem 1.25 (Spectral Theorem for Unitary Operators). Suppose U is a unitary operator on a complex Hilbert space –£?. Then for each f e –ñ there exists a unique finite Borel measure jij- on –ö such that
(uv,/) = JK^/W v¬´eZ-
If T is an invertible measure-preserving transformation then UT is unitary, and if T has continuous spectrum and (/, 1) = 0 then nf has no atoms {i.e. each point of –ö has zero —Ümeasure).
Theorem 1.26. If T is an invertible measure-preserving transformation of a probability space {X, &l, m) then T is weak-mixiiig iff T has continuous spectrum.
Proof. Suppose T is weak-mixing and let UTf = –Ø/, f e L2{m). If –Ø # 1 then integration give^ (/,!) = 0 and by the weak-mixing property we have
¬ß1.7 Mixing
49
and hence
Since |A| = 1 this gives (/, f) = 0 and therefore / = 0 a.e. If –Ø = 1 then f = constant a.e. by the ergodicity of T. (This part of the proof did not use the spectral theorem.)
Now suppose T has continuous spectrum. We show that if/ e L2(m) then
-"l WrfJ) -(/,i)(i, /)|2-o. n i=0
If / is constant a.e. this is true. Hence all we need to show is that (/, 1) = 0 implies
j-z1 i(^*r/,/)i2-o.
n i = 0
By the spectral theorem it suffices to show that if is a continuous (non-atomic) measure on –ö then
We have
‚ñÝ0.
1 n_1 -I
n i=o
J–Ø' dM/(X)2 = i "l (J—è–≥—Å^(–Ø) ‚Ä¢ Ja-*-d/z/–Ø) = ~ E f Ja1 ^/(–Ø) ‚Ä¢ Jt‚ñÝ ¬´fo/—Ç)
i = 0 n ‚Äî 1
1 n-1
= -E [T {ft)'d(pif x 1) (by Fubini‚Äôs Theorem) " 1=0 w
= JJ (~ E (W}<*(AV x ^/)(–Ø, t).
KxlE ' i_0 '
If (–Ø,—Ç) is not in the diagonal of –ö x –ö then
‚Äò1 - (–Ø—Ç)"'
i = 0
1 - (–Ø—Ç)
‚Ä¢ 0
as –ª -> oo. Since /.if has no atoms the diagonal has measure 0 for jiI x j.if and therefore (l/–∏) (–Ø—Ç)‚Äò -¬ª 0 a.e. {jj.f x nf). The modulus of the integrand is bounded by 1, so we can apply the bounded convergence theorem to obtain the result. ‚ñ°
We now investigate the mixing properties of the examples mentioned in ¬ß1.1.
50
1 Measure-Preserving Transformations
Examples
(1) Clearly the identity transformation I of (X,3d,m) Is ei'^od.c iff all the elements of M have measure 0 or 1 itT I is strong-mixin-..
(2) A rotation T{z) = a: of the unit circle –ö is never weak-mixing. This follows because because if f(z) = z then f(Tz) = f(az) = af(z) and we can apply the easy part of Theorem 1.26.
(3) Theorem 1.27. No rotation Tx = ax on a compact group is weak-mixing.
Proof. We know that if T is ergodic then the group G is abelian, and if —É is any character of G we have y(Tx) = y(a)y(x), which shows that T is not weak-mixing by the easy part of Theorem 1.26. ‚ñ°
(4) Theorem 1.28. For an endomorphism of a compact group strong-mixing, weak-mixing and ergodicity are all equivalent. (The condition for ergodicity was given in Theorem 1.10.)
Proof. We shall give the proof when G is abelian. It suffices to show that if the endomorphism A G -* G is ergodic then A is strong-mixing. If.7, 6s G then (UAy,S) = 0 eventually unless —É = S = 1. So always (UAy,S) ‚Äî*‚ñÝ (—É, 1)(1,–π). Fix SeG. The collection
–ñ6 = {/e L\m):{U‚ÄúAf,S) -> (/, 1)(1,<5)}
is a subspace of L2(m) which is closed. (To check –ñ–¥ is closed, suppose fk e .‚Äò/C and fk~* f e L2(m). For 3 = 1 it is clear that = L2(m), so suppose (1, (>) = 0. Then
\{U‚ÄùAf.SI\ ^ I(U*Af,6) - (UnAfk,6)\ + 5)|
< II/ - fk\\i IHI2 + \(1&–ê,–¥)\ (by the Schwarz inequality)
= \\f-M2 + \(UAfk,d)\.
If e > 0 is given choose –∫ so that l|/ ‚Äî fk||2 < e/2 and then choose N(c) so that n > N(c) implies \(V'Afk,b)\ < e/2.) Since –ñ—å contains G it is equal to L2(m). Fix JeL2(m) and consider J&f} = (ff e L2(m):(UAf,g) -> (f, l)(l,g)j. Then is a closed subspace of L2(m), contains G by the above, and so equals L2(m). Hence A is strong-mixing. ‚ñ°
(5) Theorem 1.29. For an affine transformation T = a - A on a compact metric abehun group the following are equivalent:
(i) T is strong-mixing.
(ii) T is weak-mixing.
(iii) A is ergodic.
¬ß1.7 Mixing
51
Proof. We shall give the proof in the case when G is connected. Let Bx = x_1/t(x) and recall that T is ergodic iff
(a) —É ¬∞ Ak = —É, –∫ > 0, implies —É ¬∞ A = —É, and
(b) [a,BG] = G.
If A is ergodic then BG = G since the endomorphism –í of 0 is one-to-one. ChoosebeGsothatB(b) = a. Define<f>\G-* Gby^(x) = bx.Then—Ñ–¢ = –ê—Ñ and —Ñ preserves Haar measure in. By (4) above A is strong-mixing and so if C, De3d we have m{T~nCn –ë) = —Ç(—Ñ(–¢~"–°n D) *= —Ç{—Ñ T '"C r\ —Ñ–û) = hi(/4_"0C –ü —Ñ–û) ‚Äù¬ª—Ç[—Ñ–°)—Ç(—Ñ–) = m(C)m{D). Therefore –ì is strong-mixing. Previous << 1 .. 16 17 18 19 20 21 < 22 > 23 24 25 26 27 28 .. 99 >> Next 