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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 73 74 75 76 77 78 < 79 > 80 81 82 83 84 85 .. 99 >> Next the two-sided shift. Then T has a unique measure with maximal entropy and this unique measure is the (l/ą║, 1 /ą║,..., \/k)-product measure.
Proof. We know /i(T) = logk. Suppose htl(T) = log Ac. Let ┬Ż = {/10,... ,Ak_,} be the natural generator (i e. Aj = , x0 = j}). Then log A: = /;ŌĆ×(T) <
(1 /n)HjSJ1Zo T~'E) < (l/n)logk'ŌĆś = log ą║ by Theorem 4.10 and Corollary 4.2.1.
Therefore HJ\/"=o T~lc) ŌĆö log kn so, by Corollary 4 2.1 each member of \J "Jj* 7 has measure (l/ą║"). Hence čå is the (l/k,, l/k)-product measure. Ō¢Ī
This was generalised to the case of topological Markov chains by Parry. Recall from ┬¦6.6 that if T: XA -* XA is a two-sided topological Markov chain and A is an irreducible matrix then there is a canonically defined Markov measure given by a probability vector (p0,... ,pk_ t) and stochastic matrix () as follows. If ą» is the largest positive eigenvalue of A and (n0,...., uk. is a strictly positive left eigenvector and (r0,. .. ,^-j) is a strictly posithe right eigenvector with ]ąō?=ąŠ čŗ.-ąĖ,- = 1 then pt = ąĖ{čī{ and pu = a^vp.v,. We call this measure the Parry measure for T:XA-> XA.
Theorem 8.10. If T:XA -* XA is a two-sided topological Markov chain, where A is an irreducible matrix, then the Parry measure is the unique measure with maximal entropy for T.
Proof. We know h{T) = log ą» by Theorem 7.13.
Let čå denote the Parry measure. We first show p e Mmax(XA, T) by showing hJiT) = ą║^ą». By the formula for the entropy of a Markov measure
┬¦8.3 Measures vith Maximal Entropv
195
(Theorem 4.27) we have k-l
4T)=~ Z Wil,^IogŌä¢ i.j= 0 / li \ AVi .
fc- l
= - Z
uie,tjLj
i, j = 0
pOgUy + loglŌĆÖj - log;.'- log L,f]
1 k-l
= 0 - Z uivi log L'j + l┬░g^ + Z ą░ą┤ l┬░g ^ since u.j6 {0,1} j=0 i=0
= log/..
We now show ^ is the only measure with maximal entropy. We know // is ergodic since the matri" (cii;) and hence (p;j) is irreducible. By Theorem 8.2 and 8.7(v) we know that if {/*} ąż Mmix(XA, T) there is another ergodic member m of Afma!C(A'M, T). By Theorem 6.10(iv) m and /t are mutually singular so IE e ą®ąźą╗) w,th čå(ąĢ) = ąĪ and m(E) = 1.
Let <!; = {ąø0,...,/4ą║_denote the natural generator i.e.
Aj= c-.Y (|A'0 =./}.
Since j/(Vr= n T~li) * %{XA) we can choose EŌĆ× e t/(V?=-ią┐-n T~'┬Ż) with (hi + /<)(┬ŻŌĆ× A E) -┬╗ 0. Hence čå(ąĢą┐) 0 and m{EŌĆ×) -* 1.
Let i]n denote the partition ijŌĆ× = {┬ŻŌĆ×, A ',┬ŻŌĆ×}. Then
.eg. - ;ur> s ^ B.(V r-{) - V ŌĆ× r-ŌĆśi)
i
[ ŌĆö m(┬ŻŌĆ×) log(┬Żn) - (1 - /┬½(┬ŻŌĆ×)) log(l - )┬╗(┬ŻŌĆ×))
~2n- 1
+ m(EŌĆ×)log0n(EŌĆ×) + (1 - /┬½(┬ŻŌĆ×))log0ŌĆ×(X\En)]
where 0ŌĆ×(B) denotes the number of elements of V/7= -m- n T~'┬Ż that intersect ąÆ ┬Ż j*(V"=-(ŌĆ×-i) T~'q). (Here we have used Corollary 4 2 to estimate the entropy of the partitions of the sets ┬ŻŌĆ×, X\EŌĆ× induccd by
Therefore
'-log/. <
1
2n ŌĆö 1
V
;= -in- ąĖ
/E.ŌĆ× 6ąöąĢ.) , ŌĆ× ,rŌĆ×1 , 0XX\En) w(┬Żn)logŌĆö+ (1 - ni(┬ŻŌĆ×))log-
ŌĆÖm(En) v "" ┬░l-m(┬ŻŌĆ×)
However if ąĪ ┬Ż \/?=-(čÅ- n T~l┬Ż, say
C {{Xn} - OO I (x-(n- 1) Xn - l) 0" ŌĆö(nŌĆö 1)ŌĆÖ ŌĆó ŌĆó ŌĆó ┬╗ąŻąĖ- l)j
(*)
196
8 Relationship Between Topological Entropy and Measure-Theoretic Entropy
then
So if
then
čĆ=-(čÅ-1)
- ŌĆösince ŌĆ£w,.. ŌĆ£ L
a = min' UjUj, b = max u,r,-
05i. j┬Żk- 1 05 i. i2┬╗*~ 1
T herefore
^<H(C)<-VC e V ąó-ą│ ^ 0n(B) < pi(B) < b -S┬«, VBe.c/1 V ^'ąø
* ąø \i=>-(n-l) /
Using this with ąÆ = EŌĆ× and ąÆ = X\En in equality (*) gives
l0sis2^b[",(┬Ż-|l08(ŌĆ£fi&r)
+ (1 - m{En))\og
and hence
ME,
0- fi(En))l2n-1\
a(l - m(En)) J
0 < m(En) log
ą®
En)J
\am(En
+ (1 - w(┬Ż:ŌĆ×))log(l - pi(EŌĆ×)) - (1 - m(EŌĆ×))log(a(\ - m(EŌĆ×))).
When n tends to oo the limiti of the three right-hand terms are ŌĆö ąŠčü, 0, 0 respectively. This contradiction shows that ą£čéą╗čģ(ąźąÉ, T) = {pi}. Ō¢Ī
Remark. If T:X -* X has a unique measure, pi, with maximal entropy then one would expect pi to be important because it w as characterised in a natural way from the variatior principle. We shall generalise the varsational principle in Chapter 9 and this allows us to characterise other measures in a similar way.
┬¦8.4 Entropy of Affine Transformations
In this section we study the relationship between topological entropy and the Haar measure entropy of an atfine transformation. We also calculate these entropies for an affine transformation of an ą┐-torus. The first result shows Haar measure is a member of Mmax(G, T) when T:G -* G is an affine transformation of a compact metric group.
;v4 E'ntropy of Affine Transformations
147
T heorem 8.11. Let G he a compact metrisable (/čéąĖčĆ and 7 a ŌĆó I an affine transformation of G. Li t m denote (normalised) Haar measure on G. I lien hm{T) = hm(A) = ąø(ą¦) = ą½ąó). It d denotes a leh-inrariant metr,c on G then h{TI = limf_0 limsupn^ , ŌĆö(1 n) log;┬╗!((")'' .,1 -1 ~'B(e: <:)). where e denotes the identity element of G and Hie', r.) is the open hall centre čü and radius r. wlih respect to the metric d. (This limit clearly exists or is r .)
Proof. By Thco.crn 8.6 wc have hJT) < /i(T). Suppose d is a lefi invariant metric on G. Put DŌĆ×(x,s, T) ŌĆö P|"-n T~kB{Tkx: >:). By induction wc shall show that T~kff(T*x: c) = x Ō¢Ā (A kB(e:i:)). h is true for ą║ = 0 b\ the invariance of the metric d. Assuming it holds for ą║ we prove it for ą║ + 1: Previous << 1 .. 73 74 75 76 77 78 < 79 > 80 81 82 83 84 85 .. 99 >> Next 