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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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Therefore L = H. □
The following theorem due to Halmos and von Neumann (1942) shows that the eigenvalues determine completely whether two transformations with discrete spectrum are conjugate or not.
Theorem 3.4 (Discrete Spectrum Theorem). Let Tf be an ergoaic measure-preserving transformation of a probability space (Xh т{) and suppose Т,-has discrete spectrum, i = 1, 2. The following are equivalent:
(i) Tj and T2 are spectrally isomorphic.
(ii) Ту and T2 have the same eigenvalues.
(iii) Ту and T2 are conjugate.
^.V2 Disavie Spoclrum
71
Proof
(i) => (ii) i.-> trivial.
(iii)=>(i) is always true (Theorem 2.9).
(ii) => (i). For each eigenvalue /., choose fkeL2(mi), gke L2(m2) such that VTl L = '/л, UTsj; = ki;■ and |/J = \gk\ = 1.
We define W\L2{ni2)-> Lr{mx) by W(gJ = /я and extending by linearity. We readily see that W is a bijective isometry. Moreover WUTl = VrW by checking this on the gk.
Wc now prove that (ii) =? (iii,). Let A denote the group of eigenvalues of T, which we are assuming to equal the group of eigenvalues of T2■ For each ke Л choose fk e L2^) so that |/л| = 1 and UTxfk = kfk. We know that {/я:л e Л) is a basis for L2(m,). Also choose gk e L2(m2) so that \g;\ — 1 and UT}gk — *(!;■ We know that {gk:k e /1} is a basis for L2(w2). We have for every Л, fie Л
^t,/a(i = /4lfxp
and also
uTl(A • Q = A(T) ■ ЦТ) = ■ /„).
By (iii) of Theorem 3.1 there exists a constant r(k, ц) e К such that /;(л)/„(*) = г(л, n)f^(x) a.e. We shall use Lemma 3.3 to show that we can suppose r(k, y) = 1.
Let H denote the collection of all functions X -* K. Clearly H is an abelian group under pointwise multiplication. Moreover К is a subgroup of H if we identify constant functions with their values.
By Lemma 3.3 there exists a homomorphism ф:Н -* К such that ф\к = id*. Let ft = Ф(Шх- Then |/Jj = 1, UTf* = kf\ and {/J:/. e Л} is a basis for L2(m,). Also,
ft ft = ШЪфШы, = Щ)/Д,
= <£(г(л, /ОЖДХ*. tALu = г(л, №(LJr{k, fi)f/fl = Г* ■
-/ A/i*
Thus without loss of generality wc can assume that /;/„ = /Я(1. Similarly we may as well assume gkgtl = VA, ^ e /1.
Define W:L2{m2) -* L2{mx) by W(gJ - Л and extended by linearity. The operator W is bijective, linear and preserves the inner product. Also WUT2 = UTW. If we can show that W satisfies the conditions of Theorem 2.4 then by Theorem 2.10 7, and T2 are conjugate. But
ma,gj = w{g Л(1) = A, = f.f = ^(sjm^).
Let /], fc e L2(m2) and let A: be bounded. If we fix дц and let a finite linear combination of gks converge to h in L2(m2) we obtain that W(hglt) = Then if we let a finite linear combination of g^'s converge to A in L2(m2) we get that W(hk) = W(h) W(k). It follows from this and Lemma 3.2
72
3 Mcasurc-PwciVmg Tninsformalion*. «ah Discrete Spcctrum
lhat И' niAps bounded functions 10 bounded functions because hk e L2(ni2) so U'(/i/v) e L2{n\l) V/i e L;(iii;) and so lK(/<)/e L2(/»,) for all/e L‘(/n,). □
Corollary 3.4.1. If T is an im-enible fwgodtc measure-preserving transformation with discrete spcctrum then T and T'1 arc conjugate.
Proof. They have the same eigenvalues. □
Remark. When the spaces (A',, are both Lebesgue
spaces or both complete separate metric spaces then the statements ift Tlieorem 3.4 are also equivalent to Tx being isomorphic to T2.
§3.3 Group Rotations
We now discuss a class of examples of ergodic measure-preserving transformations with discrete spcctrum.
Let T:K —> к be defined by T(z) = az where a is not a root of unity. We know that T is ergodic. Let f/.K -* С be defined by /„(z) = z" where n e Z. Then
f„(Tz) = /„(a:) = «V = а“Ш
ThuL, /„ is an eigenfunction with eigenvalue a". Since the {/„} form a basis for L2(K) we see that T is ergodic and has discrete spectrum.
These ideas carry over to ergodic rotations on any compact abelian group. Recall that G denote.-, the character group of a compact abelian group G (see ijO.7), and that we always use normalised Haar measure, m. on such a group G if no other measure is mentioned. If G is not metrisable then G is not countable.
Theorem 3.5. Let T, given b) T(g) = ag, be an ergodic rotation of a compact abelian group G. Then T has discrete spectrum. Every eigenfunction of T is a constant multiple of a character, and the eigenvalues of T are [y{a):y e C}.
Proof. Let у e G. Then
y(Tg) = y(ag) = y(a)y{g).
Therefore each character i« ar. eigenfunction and so T has discrete spectrum since the characters are an orthonormal basis of L2(m). If there is another eigenvalue besides the members of {y(a):vC} then the corresponding eigenfunction would be orthogonal to all members of G, by (iv) of Theorem 3.1. and so is zero. Hence {*(«):7 e G} is the group of all eigenvalues of T and the only eigenfunctions are constant multiples of characters, using (iii) of Theorem 3.1. С
$3.3 Group Rotations
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It turns oui that such rotations arc the canonical example., of measure-preserving nansformations with discrete spectrum.
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