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 .. 60 61 62 63 64 65 < 66 > 67 68 69 70 71 72 .. 99 >> Next It remains to prove (b). Suppose T is minimal. We need to show čä 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 čä(ą│) = 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. Ō¢Ī
H. Furstenberg  constructed an example ofa minimal homeomorphism T:K2 -┬╗ąÜą│ of the two dimensional torus which is not uniquely ergodic. The example preserves Haar measure and has the form T(r, vv) = (ą░2,čä(:)w) where (┬½ŌĆØ}-ŌĆ× is dense in ąÜ and čä.ąÜ -┬╗ ąÜ is a well chosen continuous map.
Recall that if čå e M(X, T) is ergodic then there is a Y e ą®ąź) such that li{Y) = 1 and
- ąöąōčģ)- {fdfi Vx e Y, V/ Ōé ąĪ(ąō) 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) ┬Ż"= 0* f(TŌĆśx) converges uniformly to a constant.
(ii) For every f e C(X) (1 /┬½) Yj = of(TŌĆśx) converges pointwise to a constant.
(iii) There exists čå e M(X. T) such that for all f e C(X) and all x e X,
^ ZŌĆśf(TŌĆśx)^ 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) -┬╗ ąĪ by
k(f) = lim - V /T'(.v).
n-x 11 1 = 0
Observe that ą║ 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/ą╗. Hence čåąĄ M{X,T) by Theorem 6.8.
* (iii) => (iv). Suppose that v e M(X, T). We have
1 1,-1
- ┬Ż /T'(x)->/* VxeX ą¤ i = 0
where f* = J/d/ą╗. 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 = čå by Theorem 6.2. Therefore T is uniquely eigodic.
(iv) =?ŌĆó (i). If (l/n)Yj=o fT'(x) converges uniformly to a" constant then this constant must be J/ dp, where {čå} ŌĆö M(X, T). Suppose (i) is false. Then 3g e C(X), 3e > 0 such that VN 3n > N and ąŚą║ą╗ e X with
> e.
If
LŌĆśn
n
i-= 0
/I .r
i = 0
then |Jgr ŌĆö \gdji\ > e. Choose a convergence subsequence of {/(ŌĆ×}. If -* then Hr, fe M(X,T) by Theorem 6.9. Also ŌĆö jcj > ą║ so
^ # fi. This contradicts the unique ergodicity of T. Ō¢Ī
Results about unique ergodicity known before 1952 are given in Oxtoby
. More recent results of Jewett  and Krieger  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  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 (ąź,ą®ąź)).
(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 ąŠčĆąĄą╗ sets. If T is minimal then is dense lr G. Suppose čå e M(G, T). Then
J/(┬½"čģ)čä(čģ) = ]7(čģ)čä(čģ) V/eC(X) V/i Ōé 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. Ō¢Ī
(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 Ō¢Ā 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 ąō is uniquely ergodic ilT T is minimal The ŌĆśonly ifŌĆÖ part follows by Theorem 6.17 and the fact that T preserves Haar measure. The ŌĆ£ifŌĆØ part can be proved by checking statement
(i) of Theorem 6.19 holds. This was done by Hahn and Parry .
(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 ŌĆö (čĆčā) is ą░ ą║ x ą║ stochastic matrix Previous << 1 .. 60 61 62 63 64 65 < 66 > 67 68 69 70 71 72 .. 99 >> Next 