# An introduction to ergodic theory - Walters P.

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Theorem 1.7. Let X be и compact metric space, 2/1 {X) the c-algebra of Borel subsets of X и ml let m be a probability measure on (X, ЩХ)) such that ni(U) > 0 for every non-empty open set U. Suppose T:X -* X is a continuous transformation which preserves the measure m and is ergodic. Then almost all points of X have a dense orbit under T i.e. {x 6 X | (T"x)'= 0 is a dense subset of X} has m-measure one.

ProGi' Let {t/„}“=1 be a base for the topology of X. Then {T"x|n > 0} is dense in X 11ТхбП“=, U*-o T~kU„. Since T~ T~kU„) с у » 0 T~kU„

and T is measure-preserving and ergodic we have га(У “=0 T kUn) = 0 or 1. Since (J*L0 T~kU„ is a non-empty open set we have »n(U*=0 T~kUJ = 1. The result follows. □

Note that this result is applicable when m is Haar measure on a compact metric group and T is an affine transformation which is ergodic.

We shall now sec when the examples of §1 are ergodic.

(1) Clearly the identity transformation on {XySS,m) is ergodic iff all members of 35 have measure 0 or 1.

(2) We have the following theorem concerning rotation? of the unit circle K.

Theorem 1.8. The rotation T(z) = a: of the unit circle К is ergodic (relative to Haar measure m) iff a is not a root of unity.

Pkuof. Suppose a is a root of unity, then ap = 1 for some p ф 0. Let f(z) = zp. Then / о т = / anJ. / is not constant a.e. Therefore T is not ergodic by Theorem 1.6(ii). Conversely, suppose a is not a root of unity and f ° T = f, f e L2(m). Let /(z) = ^ bnz" be its Founer series. Then f{az) = -* bnu"z" an<3 hence b„(ci" — 1) = 0 for each n. If n Ф 0 then bn = 0, and so / is constant a.e. Theorem 1.6(v) gives that T is ergodic. □

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1 Measure-Preserving Transformations

An equivalent way to say а & К is no; a root of unity is to say that {a"}*.^ ь dense K. (li ii r. a root oi' unity tlicn [a") Ly is a finite set and so not dense m к. 11 a is nol a root of unity we can obtain an c-dense subset of К as follows. Since the set \a")1 consists of infinitely many points there are :uo points a1'. </" w ith ilia'1,if') < <: (</ is the usual Euclidean distance on K). Then f/(l,ii'' '’) < i: so that {, is c-dense.) This formulation is used to generalise Theorem i.S to the general case.

U'l We consider a rotation I'(.x) - ax of a general compact group. The measure involved is normalised Haar measure/».

Theorem 1.9. Let G he a compact group and T(x) = ax a rotation of G. Then T is ergodic iff V1} L%l is dense in G. In particular, if T is ergodic, then G is abelian.

Piujoi-. Suppose T is ergodic. Let H denote the closure of the subgroup |(/"j _ , of G If II Ф G then by (7) of §0.7 there exists ye G w'ith у ф 1 but y(h) = 1 Vlic II. Then y(Tx) = v(fl.x) — ';(aY/(x) - y(x), and this contradicts ergodicity of T. Therefore II = G. (If G is metric we could have used Theorem 1.7 instead of the above proof.) Conversely, suppose {a”}nejl is dense in G. This implies G is abelian. Let / e L2(m) and f о T = f. By (9) of §0.7 f can be represented as a Fourier series У,- bf/i, where у,- e G. Then

Zf’ ,(«JV,(-V) = Z hiVi(x) s0 that bi ^ 0 lhcn 7i(fl) = 1 and- since 7.(a") -7= 1, 7; = 1. Therefore only the constant term of the Fourier series of J can be non-zero, i.e., / is constant a.e. Theorem 1.6(v) gives that T is ergodic. □

(4) Let G be a compact group and A .G -» G be a continuous endomorphism of G onto G. We know that A preserves Haar measure m. We first consider the special ease of endomorphisms A(z) = zp of the unit circle K. We shall show A(z) = is ergodic if |/?| > 1. Suppose / e L2(m) and f ° A = f. If f{z) has Fourier series f(z) = Y_n= a„-' then f(A:) = £*= _ r a,-’’". Therefore a,. = = up2„ = aps„ = ■ -so that if n Ф 0 w'e have a„ = 0 because the 1 ourier coefucients must satisfy Zf=-да |aj|2 < cc- Only the constant term of the Fourier series can be non-zero so / is constant a.e. Thfcretarc A is ergodic.

In the case of a general compact abelian group we have the following result due independently to Rohlin and Halmos.

Theorem 1.10. If G is a compact abelian group (equipped with normalized Haar measure) and /I: G —> С is a surjective continuous endomorphism of G then A is ergoillc iff the trivial character у = l'is the only ye G that satisfies у ° A" = у for some n > 0.

Prooi-. Suppose that whenever у A" = у for some n > 1 we have у = 1. Let / с A = / with / e L2(m). Let ffx) have the Fourier series Za«7n where yr e G

J1.5 IZrgodicily

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and V|".J2 < со- Thcn Е%й’7,(/4а') = Еа«У»(хЪso lhat Уп 0 A’ t« ° A2,. .. , arc all distinct their coefficients are equal and therefore zero. So if a„ Ф 0, y„(Ar) = y„ for some p > 0. Then y„ = 1 by assumption and so / is constant a.e. Therefore A is ergodic by Theorem 1.6(v).

Conversely let A be ergodic and yA" = y, n > 0. If n is the least such integer, / = у + у A + • • ■ + yA"~1 is invariant under A and not a.e. constant (being the sum of orthogonal functions), contradicting Theorem 1.6(v). □

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