Books in black and white
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 8 9 10 11 12 13 < 14 > 15 16 17 18 19 20 .. 99 >> Next 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. □
30
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
31
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). □ Previous << 1 .. 8 9 10 11 12 13 < 14 > 15 16 17 18 19 20 .. 99 >> Next 