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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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chj = Z ЧиЧи < Z cl“ch = ch for a111>
r I
and this contradicts the definition of qj. Hence qu = c/j for all i.
(ii) => (i). To show T is ergodic it suffices (by Theorem 1.17) to show that for any two blocks A = o[i0,... ,ir~\a+rB = b[j0,... ,js]b+s we have
§1.7 Mixing
43
lim.v-<b(1/N) Y£= о "'(T "/1 n B) = m(A)m(B). For n > b + s — a we have m(T~”A nB) = PjoPjojl • • • Pjj._ , pf/0"~b~s)pi0i, Pir. lir and since we know that (ii) implies qy ~ pj this gives
j N- 1
lim 77 I m(T~nA n B) = PjoPjojl ■ • • Pjt_ j'PioPioil ■ ■ ■ ptr_
JV-ao /V ,|=0
= ni(/4)/7i(B).
(ii) => (v). We know (ii) implies qtJ = pj so that the only left eigenvectors of Q for the eigenvalue 1 are multiples of p. ByLemma 1.18 there are also the only left eigenvectors of P for the eigenvalue 1.
(v) => (ii). Suppose 1 is a simple eig«ivalue of P. Since Q = QP each row of Q is a left eigenvector and so they are identical. □
The condition (v) gives a practical way to test ergodicity. We shall use Theorem 1.17 at the end of this section to find necessary and sufficient conditions for a Markov shift to have the mixing properties (Theorem 1.31).
We shall use the following result, about sequences of real numbers, to obtain other formulations of weak-mixing.
20.
Theorem 1. .If {a,,} is a bounded sequence of real numbers then the following are equivalent:
(i) lim - £ |q(| = 0.
n-+ oo W i —0
(ii) There exists a subset J of Z+ of density zero (i.e.,
f cardinality (J n {0, 1,..., n — 1})\
I » )
0),
such that lim„ a„ = 0 provided пф J.
(iii) lim - £ |a;|2 = 0.
п~* со И i = o
Proof. If M cz Z+ let aM(n) denote the cardinality of {0, 1,..., n — 1} n M.
(i) => (ii). Let Jk = Jne Z+ :|a,,| ;> 1 /к) (к > 0). Then J, с J2 c • • •. Each Jk has density zero since
< n-i i ,
- I h-l * — «,»■
И i = 0 п к
Therefore there exist integers 0 = /0 < h < li < ''' such {hat for n > lk,
1 ч 1
k+ 1
44
1 Measure-Preserving Transformations
Set J = UB,o[A+i n We now show that J has density лего.
Since J, с J2 с • • •, if lk < n < lk+! we have
J n [0,ii) = [J n [0, /J] u[in [/*,«)] c [Jk n [0,/*)] u [J*+i n [0, n)], and therefore
^ a/n) < ^ [ос,Д.) + aJk. ,(n)] < i [aJfc(n) + . ,(n)] < ^ +
Hence (l/n)a.j{n) -* 0 as n -* oo, and so J has density zero. If n > lk and n$ J then n ф Jk + 1 and therefore |«„j < l/(k + 1). Hcnce
lim |fl„| = 0.
J ф n — ao
(ii) => (i). Suppose |a„| < К Vn. Let e > 0. There exists Ne such that n > Nc, n$ J imply |a„| < e, and such that N > Nc implies (a.j(n)/n) < e. Then n> Nc implies
;ЕН-;Г E Ы+ E H
” i = 0 "LieJrifO.l.....#i-l) itJn{0.1.....л- 1) _
< — а,(н) + e < (K + l)e. и
(i) => (hi). By the above it suffices to note that lim^,,-.^ |a„| = 0 iff
limj^n^a, |a„|2 = 0. □
Theorem 1.21. If T is a measure-preserving transformation of a probability space (X,JS,m) the following are equivalent:
(i) T is weak-mixing.
(ii) For every pair of elements A. В of 3d there is a subset J(A,B) of Z + of density zero such that
lim m(T~nA n B) = m(A)m(B).
J{A,B)tn—ao
(iii) For every pair of elements A, В of 3$ we have
1 л_ 1
lim - £ IniCT ‘A n B) — m(A)m(B)\2 = 0.
n -• oo ^ i = 0
Proof. Apply Theorem 1.20 with an = m(T~nA n B) — m(A)m(B). □
We now show that for most useful measure spaces we can strengthen statement (ii) to obtain a set of density zero that works for all pairs of sets A and B.
Definition 1.6. The probability space (X,3S,m) has a countable basis if there is a sequence {Bk}k= v of members of such that for each e > 0 and each Be & there exists some Bk with m(B Д Bk) < e.
{■1.7 Mixing
45
This condition is equivalent to the condition that L2(m) has a countable dense subset (see §0.5).
If X is a metric space with a countable topological base and 36 is the (T-algcbra of Borel subsets of X then (X,J,m) has a countable basis for any probability measure m on (X, 3d). This follows from Theorem 6.1. This is also true if ЗА is the completion, under m, of the ст-algebra of Borel subsets of X.
Theorem 1.22. Let (X,33,m) be a probability space with a countable basis and let T: X —> X be a measure-preserving transformation. Then T is weak-mixing if there is a subset J of Z+ of density zero such that for all A, Be S3 lim},,,-.* m(T~nA n B) = m{A)m{B).
Proof. It suffices to prove that the stated condition holds if T is weak-mixing. Let [Bk}x be a countable basis for (X.38,m). Put
” |m(7'_nB, n Bj> - m(B>(B,.)|
a" 2- 2‘rj
i, j= 1
Since T is weak-n..xing we have (l/n)Z"=o a,^0 so by Theorem 1.20 there is a subset J of Z+ of density zero such that HmJ#n_o0a„ = 0. Therefore limJfn^ajm(T~"Bi n B) = m(Bi)m(B]) for all i,j, and the result follows by a simple approximation argument. □
Remark. We can use Theorem 1.21 to give an intuitive description of weak-mixing. It means that for each set A e 36 the sequence T~nA becomes independent of any other set В e 36 provided we neglect a few instants of time.
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