# An introduction to ergodic theory - Walters P.

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It remains to prove (b). Suppose T is minimal. We need to show ф is a homeomorphism. From Theorem 6.17 we know that if M(K, T) = {»} then v(U) > 0 for all non-empty open sets U. If ф(г) = 0nv) then v([r. mJ) = 0 or 1 so that either v([z,u]) = 0 or v([w,r]) = 0. This can only happen when z = w. □

H. Furstenberg [1] constructed an example ofa minimal homeomorphism T:K2 -»Кг of the two dimensional torus which is not uniquely ergodic. The example preserves Haar measure and has the form T(r, vv) = (а2,ф(:)w) where («”}-„ is dense in К and ф.К -» К is a well chosen continuous map.

Recall that if ц e M(X, T) is ergodic then there is a Y e ЩХ) such that li{Y) = 1 and

- ДГх)- {fdfi Vx e Y, V/ С(Г) n 1 = 0 J

(Lemma 6.13). When T is uniquely ergodic we get much stronger behaviour of these ergodic averages.

Theorem 6.19. Let T:X -» X be a continuous transformation of a compact metrisable space X. The following arc equivalent:

(i) For every f eC(X) (1/n) £"= 0* f(T‘x) converges uniformly to a constant.

(ii) For every f e C(X) (1 /«) Yj = of(T‘x) converges pointwise to a constant.

(iii) There exists ц e M(X. T) such that for all f e C(X) and all x e X,

^ Z‘f(T‘x)^ ffdM. n 1 = 0 J

(iv) T is uniquely ergodic

$6.5 Unique Irgodicily

161

Proof

(i) => (ii) holr* trivially.

(ii) => (iii). btnne k:C(X) -» С by

k(f) = lim - V /T'(.v).

n-x 11 1 = 0

Observe that к is a linear operator and is continuous since

1 n-l

- I n 1 = 0

Also ..(1) = 1, and / > 0 implies k{f ) > 0. Thus by the Riesz Representation Theorem there exists a Borel probability measure f.i such that k(f) = j/dp But k{fT) = k(f) and so J/T d[i = If d/л. Hence це M{X,T) by Theorem 6.8.

* (iii) => (iv). Suppose that v e M(X, T). We have

1 1,-1

- £ /T'(x)->/* VxeX П i = 0

where f* = J/d/л. Integrating with respect to v, and using the bounded convergence theorem we get that

J/dv = J>dv = /* = J/^ V/ 6 C(A').

Hence v = ц by Theorem 6.2. Therefore T is uniquely eigodic.

(iv) =?• (i). If (l/n)Yj=o fT'(x) converges uniformly to a" constant then this constant must be J/ dp, where {ц} — M(X, T). Suppose (i) is false. Then 3g e C(X), 3e > 0 such that VN 3n > N and Зкл e X with

> e.

If

L‘n

n

i-= 0

/I .r

i = 0

then |Jgr — \gdji\ > e. Choose a convergence subsequence of {/(„}. If -* then Hr, fe M(X,T) by Theorem 6.9. Also — jcj > к so

^ # fi. This contradicts the unique ergodicity of T. □

Results about unique ergodicity known before 1952 are given in Oxtoby

[1]. More recent results of Jewett [1] and Krieger [2] imply that any ergodic invertible measure-preserving transformation of a Lebesgue spacc is isomorphic in the sense of Chapter 2 to a minimal uniquely ergodic homeomorphism of a zero dimensional compact metrisable space. In particular there are minimal uniquely ergodic homeomorphisms with any prescr.bcd

162

6 Invariant Measures for Continuous. Tiansformalions

non-negative real number for their eniropy. Hahn and Katznelson [1] had previously found minimal uniquely ergodic transformations with arbitrarily large measure-theoretic entropy.

§6.6 Examples

We now investigate M(X, T) for the examples listed in §5.1.

(1) The space of invariant measures for the identity map of A' is the space M(X) of all probabilities on (Х,ЩХ)).

(2) Theorem 6.20. If T(x) = ax is a rotation on the compact metrisable group G then T is uniquely ergodic iff T is minimal. The Haar measure is the only invariant measure.

Proof, If T is un quely ergodic then T is minimal b> Theorem 6.17, since Haar measure is non-zero on non-empty орел sets. If T is minimal then is dense lr G. Suppose ц e M(G, T). Then

J/(«"х)ф(х) = ]7(х)ф(х) V/eC(X) V/i Z.

If b e G there is a sequence a'" converging to b and by the dominated convergence theorem

j*/(i>x|<fy(x) = lim j7(fl"Jx)dpi(x) = j*/(x)dpi(x) V/ e C<X).

This shows /i is invariant for every rotation of G and is therefore Haar measure. □

(3) If A:G-*G is a surjective endomorphism of a compact metrisable group G then M(G,A) contains many measures. Two of the members of M(G,A) are always Haar measure and where e is the identity element of G. When A :Kn -» /<" is an automorphism of a torus Theorems 5.11 and 6.16 give us many atomic measures contained in M(G,A). Clearly A:G -» G can only be uniquely ergodic when G = {e}.

(4) When T = a ■ A :G -* G is an affine transformation of a compact metrisable group the set M(G, T) is sometimes small (as in (I'j) -ind sometimes large (as in (3)). When T is abelian we have that Г is uniquely ergodic ilT T is minimal The ‘only if’ part follows by Theorem 6.17 and the fact that T preserves Haar measure. The “if” part can be proved by checking statement

(i) of Theorem 6.19 holds. This was done by Hahn and Parry [1].

(5) The one-sided and two-sided shift maps have many invariant measures. For every probability vector (p0,... ,pk-!) on the state space Y the corresponding product measure belongs to M(X, T). Other members of M(X, T) are provided by Markov measures: if P — (ру) is а к x к stochastic matrix

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