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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
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The following important result is an easy consequence ol the compactness of the unit ball of C(X)* in the weak*-topology, but we give a direct proof.
Theorem 6.5. If X is a compact metrisable spacc then M{X) is compact in the weak*-topology
Proof. We shall v.rite //(/) instead of jf dp. Let [//„}/ be a sequence in Af(A") and we shall show it has a convergent subsequence.
Choose /,, /2,... dense in ClX). Consider the sequence of compta# numbeis This is bounded by ||/,|j, and so has a convergent sub-
sequence, say {/<*„*'(/j)}- Consider the sequence of numbers [p{nu(.f2)}: this is bounded and so has a convergent subsequence {/ijf’f /,)}. Notice that {^!t2>(/i)} also converges. We proceed in this manner, and for each i > 1, construct a subsequence {juJ,0} of (/(„} such that {/ij/*} £ "] £ ■ ■• £
W) £ {/<„}, and so that converges for / =/„./,.....f. Consider
the diagonal {pi"*}. The sequence {^"’(/i} converges for all i; thus [pl"\f)} converges for all J e C’(A') (by an easy approximation argument'. Let J(f) = !im„_a,/jJ,",l/). Clearly J:C(X)-*C is linear and bounded, as ]./(./)] < l|/j|. Also J(l) = 1, and if / > 0 then J( f) > 0. By Theorem 6.3, there exists a Borcl probability measure p on X such that J{f) — Jv/dp for all fe CLY), i.e.,
Hence M(X) is a compact convex metrisable space and this will allow us to use the fixed point theorems valid for maps of such spaces.
§6.2. Invariant Measures for Continuous T ransformations
Let T:X -*X be a continuous transformation of the compact metrisable space X. We shall show in this section that there is always some p e M(X) for which T is a measure-preserving transformation of (X,.sf(X),p.).
We first notice that T~ lttf(X) cr J\$(X) (i.e. T is measuraole) because {£e .ЩХ)\Т~1E e ЩХ)} is a ст-algebra and contains the open sets. Therefore we have a map T:M(X)-*M(X) given by (Tp)(B) = р('Г~1B). We sometimes write p° T~1 instead of Tp. We shall need the following.
Lemma 6.6
J7</(fM) = J/oTdp V/e C(X).
Proof. It suffices to deal with real-valued / e C(X) By definition of T we have §y_Bd(Tp) = \/_b ° TdpVBe ЩХ). Therefore jhd(fp) = J/i ° 7 dp if h is a simple function. The same formula holds when h is a non-negative
Jtf.2 Invariant Measures for Continuous Transformations
I SI
measurable function, by choosing an increasing sequence of simple functions converging pointwise to h. Therefore the formula holds for any continuous / X -* R by considering the positive and negative pa it of /. □
Theorem 6.7. The map T:M(X) —> M(X) is continuous and affine.
Proof. If / e C(X) then J/dT p = \ f - Tdp.. Therefore if p„ -»p in \i(X) then jY dfp„ = ff T dpn -> \fc T dp = \f dT p and so f p, — f p. This proves T is continuous.
If m, pe M(X) and p e [0, l] then T(pm + (1 — p)p)(B) = pm(T~iB) + (1 — p)p(T~lB) = [pTm + (1 — p)Tp)(B)VB e MX). This shows T is affine. □
We are interested in those members of \I(X) that arc invariant measures for T.
Let M(X, T) = {p e M(X)\Tp = p). This set consists of all p e M(X) making T a measure-preserving transformation of (.Y,J}(X),p).
Theorem 6.8, If T:X -* X is continuous and p e M(X) then p e M(X, T) iff J f'Tdp = \$fdp V/eC(X).
Proof. This is immediate from Lemma 6.6 and Theorem 6.2.

Since T:A/(_Y)-> M(X) is a continuous alline map of a convex compact subset of C(X)* we could use the Markov-Kakutani theorem (see Dunford and Schwartz [1], p. 456) to show T has a fixed point. However we will show directly that M(X, T) is non-empty. The following gives us a method of constructing members of M(X, T).
Theorem 6.4. Let T:X —> X be continuous. If {(?„}"= i is a sequence in M[X) and we form the new sequence by p„ = (l/n)Yj = o T'on then any
limit point p of {//„} is a member of M(X, T). (Such limit points exist by the compactness of M(X).)
Proof. Let pnj -> p in M(X). Let / e C(X). Then
J7 ° T dp — §fdp = lim j/° T dpnj - ^ f dptlj
= lim
j-ao
- Г x (/°Ti + 1-/c T')dcnj njJ i = 0
Ц(/° т>-л<К
Slimill.o.
= lim
j—<Xl
j- со rlj
Therefore p e M(X, T).

152
6 Invarian* Measures for Continuous Transformations
Corollary 6.9.1 (Krylov and Bogolioubov). If T.X—>X is a continuous transformation of a compiict metric space X then.M(X, T) is non-empty.
Proof. Wc can make any choice for a„ in Theorem 6.9; in particular choose jeX and put a„ = Sy for each n. □
We have the following properties of M(X, T).
Theorem 6.10. If T is a continuous transformation of the compact metric spacc X then
(i) M(X. T) is a compact subset of M(X).
(ii) M(X, T) is convex.
(iii; fi is an extreme point of M(X, T) iff T is an ergodic measure-preserving transformation oj (X,M(X\,p).
(iv) If p, m e M(X, T) are both ergodic and in Ф p then they are mutually singular.
Proof
(i) Suppose is a sequence of members of M[X, T) and p„-> /< in M(X). Then \fdfp = У °Tdp = 1тл,ш [f ° T dpn = lim^^ j/ dp„ = J'f dp, so // e M(X, T).
(ii) This is clear.
(iii) Suppose p e M(X, T) and p is not ergodic. There exists a Borel set £ such that T~yE = E and 0 < p(E) < 1. Define measures Pi and p: by
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