# An introduction to ergodic theory - Walters P.

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We arc especially interested in the case of the и-torus K". Recall from §0.8 that a surjective endomorphism A: Kn -* Kn is given by an n x n matrix [/l] of integers and that Kn can be identified with Zn and the induced action A:Kn -»Kn corresponds to the action of the transpose matrix [X], on Zn.

Corollary 1.10.1. Let A:Kn -* Kn be a surjective continuous endomorphism of ihc n-torus. Then A is ergodic iff the matrix [-4] has no roots of unity as eigenvalues.

Proof. If A is not ergodic Theorem 1.10 gives the existence of q e Z” q Ф 0 and к > 0 with [X]J^ = q. Then [jri]* has 1 as an eigenvalue so lhat [/1]„ and hence [/1], has a /<-th root of unity as an eigenvalue.

Conversely if [/1], has a k-th root of unity as an eigenvalue then [/i]f has 1 as an eigenvalue. Therefore ([Xjf — I)(y) = 0 e R“ for some у e R", and since the matrix [Л], has integral entries we can find such aye Z". Mence [/)]*_)■ = у and A is not ergodic by Theorem 1.10. □

When G is not abelian similar necessary and sufficient conditions for ergodicity of an endomorphism can be stated in terms of the irreducible unitary representations of G. (When G is abelian these representations arc the characters of G.) This has been studied by Kaplansky (Kaplansky [1]).

(5) For atline transformations of compact metric groups necessary and sufficient conditions for ergodicity are known. The simplest case is when G is a compact, conncctcd, metric, abelien group.

Theorem 1.11. If T(x) — a ■ /Цх) is an affine transformation of the compact, connected, metric, abelian group G then the following are equivalent:

(i) T is ergodic (relative to Haar measure).

(ii) (a) whenever у о Ak = у for к > 0 then у < A = у: and

(b) the smallest closed subgroup containing a and BG (where Bx = a-- 1 • Л(а;) is G(i.e„ [a, BG] = G).

(iii) 3.v0 6 G with {T"(xQ):n > 0} dense in G.

(iv) m({x: [T"x:n > 0 is dense}}) = 1.

(Note that conditions (a) and (b) reduce to the conditions given in (3) and (4) in the special cases. The equivalence of (i) and (ii) was investigated by Hahn, Hoare and Parry.)

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1 Measure-Preserving Transformations

Prooi First note that В is an endomorphism of G (but maybe nonsurjective) and commutes with A.

(ii) => (i). Suppose (a) and (b) of (ii) hold. If / ° T = f, f e L2(m), let / = £й;'Л> V, e G, be the Fourier series of /. Then

Z й.7;(“)?;Их) = £ bf/iix). (*)

i i

If y,-, у,- o/l, y; - /1, are all distinct then = 0 or else £|;|2 < K violated. Hence, if/),- Ф 0 then у,- ° A" = y, for some n> 0, and by (а) у ° A = у But then (*) implies y,(u) — 1 and so "/Да') =* 1 Va‘ й [a, EG] and by (b) "д 1. So / is constant a.e. Therefore T is crgodic by Theorem 1.6(v).

(i) => (iv). This follows by Theorem 1.7.

(vi) => (iii) is trivial.

(iii) => (ii). It remains to show that if 3a'0 6 G with {T“x0:n > 0} dense in G then conditions (a) and (b) of (ii) hold. Suppose у о Ak = у, к > 1, у 6 G. Let

71 = У ' В. Then yi(Tkx) = у ,(a • Aa.....Ak~ ‘aJy^/J**) = y{a~ lAku)y1(x) =

y4#4- Hence yx assumes only the finite number of values y^Xo), у^ТХо)...., yl{Tk~1xQ) on the dense set {Tnx0:n > 0} and hence assumes only these values on G. Since G is connected must be constant, and so уj = 1. Hence у A = у and condition (a) holds.

If [a, £G] Ф G~\y Ф 1, у eG, with у (a) = 1 and y(Bx) = 1 (see (7) of §0.7). Then y(Tx) = y(x) and so у assumes only the value y(x0) on the dense set {Tnx0:n > 0} and therefore у is a constant. Hence у = 1, a contradiction, and we have shown that (iii) implies (b). □

When G is Kn the equivalence of (i) and (ii) becomes: T — a • A is ergodic

iff

(a) the matrix [/1] has no proper roots of unity (i.e., other than 1) as eigenvalues, and

(b) [a.BK"] = Kn.

This is easily proved by a method similar to the one used in (4) for the endomorphism case.

Conditions for ergodicity of affine transformations of compact non-abelian groups may be found in Chu [1].

(6) We now consider two-sided shift transformations.

Theorem 1.12. The two-sided (p0,... ,pk_ J shift is ergodic.

Prooi. Let .с/ denote the algebra of all finite unions of measurable rectangles. Suppose T'' E — £, £ e Я?. Let i: > 0 be given, and choose A e .;•/ with m\E Д A) < Then

|ш(Е) — ui(/1)j = \m{L n A) + m(E\A) — m(A n E) — in(A \E)j

< m(E\A) + m(A\E) < e.

Hrgodiaty

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Choose n0 so large that В = T~n"A depends upon different coordinates from /1 Then m(B n A) = m(B)m{A) = m(A)2 because in is a product measure. We have

m(EAB) = m(T~nE Д T~nA) = m(E Д A) < e,

and since £Д(/1пВ)с(£Д/1)и(£ДВ) we have m(EД{A nfi))< 2c. Hencc

|/н(£) — m(A n ZJ)| < 2e

and

|ш(Е) — »i(b)2| < \m(E) — m{A n B)j + \m{A nB) — m(£)2|

< 2c + \m(A)2 - m(E)2|

^ 2c + m(A)\m{A) — m(E)| + m(E)\m(A) — m(E)\

< 4c.

Since с is arbitrary m(E) = m(E)2 which implies that m(E) = 0 or 1. □

(7) By a similar argument, we see that the 1-sided (p0,... ,/?t_i)-shifi is ergodic.

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