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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 71 72 73 74 75 76 < 77 > 78 79 80 81 82 83 .. 99 >> Next By Lemma 8.5 choose a partition ^ = {At,... ,Ak} of (ąź,ą®ąź)) so that diam(4j)<┬Ż and čå(čüąÉ,)=0 for 1 </<k. Then HŌĆ×j\/?=o T~ŌĆś┬Ż)ŌĆölog^fc, X) since no member of \/?=o T~ŌĆś┬Ż can contain more than one member of En and so sŌĆ×(e,X) members of \/;=o each have <TŌĆ×-measure \/sŌĆ×(e, X) and ' the others have c^-measure zero. Fix natural numbers </, n with 1 < q < ąĖ and, as in Remark (2), define a(j), for 0 <7 < q ŌĆö 1, by a(j) = [(/1 ŌĆöj)/</].
190
8 Rel-iiionship Between Topological Entropy and Mca*ure-Thuaretic Entropy
Fix 0 < j < q ŌĆö 1. From Remark 2(ii) wc have
V?=0 ąó~ą¦ = V?=V 7",f, + 'ŌĆÖ V?=o T-'Z V Vie6 T~l4 and S has cardinality at most 2q Therefore
\ogsn(e,X) = hJ^ T-'tj
mj)-l / q- 1 \
< I ąØą▓čÅ(7ŌĆÖ-ŌĆś'*+ą╗ V T-'cW V ą»ą¤ŌĆ×(ąó-^)
r=0 \ 1=0 / *eS
by Theorem 4.3fv iii)
ą╗J)-1 /ąĖ-i \
< Z He+T-ri'A V TŌĆÖŌĆØ,Ōé )+ 2qlog(/i) by Corollary 4.2.
r=0 \l = 0 /
Sum this inequal ty over j from 0 to q ŌĆö 1 and use Remark 2(iii) lo get qlogsJfcX)*^ ąó-ŌĆśąÉ + 2/1ąŠg(k).
p= ąŠ \; = o /
If we divide by n and use Remark (1) we get
- logiŌĆ×(┬Ż, X) < HVn fV T~lz \ + 2-^ log(A). (*)
ą¤ \i ŌĆö 0 J n
By Remark 3 we know the members of \J\Zl T~have boundaries of ^-measure zero, so limjJ(n unj(B) = čå(ąæ) for each member ąÆ of \'(=l T~ŌĆś┬Ż, (Remark 3 ┬¦6.1) and therefore lim^ ąØ^{\JtZl T~l┬Ż) = ą»ŌĆ×(\/7=ąŠ ąó~ŌĆś┬Ż). Therefore replacing n by n} in (*) and letting j go to čüąŠ we have qs(c, X. T) < HjSfIZąŠ T~ŌĆś┬Ż). We can divide by q and let q go to oo to get s(e,X, T) < hŌĆ×{T,Z) < /ifT),. Ō¢Ī
Corollary 8.6.1. Let T:X -* X be a continuous map of a compact metric spa^e. Then
(i) h(T) = sup{^(T)|^ ┬Ż ┬Ż(X,T)}.
(ii) h(T) = h(T\nm).
(iii) h(T) = h(T\ ft rx).
*-o
(iv) If, for i = 1,2. T,: X, -┬╗ X,- is a continuous map of a compact metric space and if there is a bijection čä:ąź1 ŌĆö> X2 which is bimeasurahle (i.e. čä and čä~1 are measurable) and čäąó1 = ąó2čä then h(Tx) = h(T2). (This generalises the fact that topological entropy is an invariant of topological conjugacy.)
Proof
(i) Let e > 0 be given. Choose čå e M(X, T) such that ',-(r)>tl/e = čüąŠ.
┬Ż8.3 Measures with Maximal Entropy
191
fax.T)m dr(m) is the ergodic decomposition of/i then, by Theorem 8.4, hJT) = jE(x.r)^m(T)dr(m) and so hJT) > hJ^Tj ŌĆö e. for some m e E(X, T). Hence
i ,t)^SHT]~2e if,'(r)<3C
U ' jl/e-ąĢ tfUJT) = oc
so that sup{/jŌĆ×|T)]m e E(X, ąō)} = h(T).
(ii) By Theorem 6.15 we have u(Q(T)) = 1 V/; e V/( A. T) so th^t
sup(/7ŌĆ×(r)|/i e M(X, T)} = sup'/iJ((7')|/< e ąø/(0(7"), T|n,n)}.
(iii) If čåąĄą£(ąź.ąó) then čå(ąóą│ąź) = čå(ąó'ą┐ąó''ąź) = fi{X) = 1. Therefore ą£ą¤ą┐*= i TnX) = 1 VfieM(IJ) so that we can identify ,\/(A, 7 ) and M(| lo TnX,T). The result follows from the variational principle.
(iv) We have i.i&M(Xl,T1) iff čå čä~1 e A/(A\, T2). Also hll(Tl) = ą║čå.čä-ąÉąóą│) so the result follows from the variational principle. Ō¢Ī
┬¦8.3 Measures with Maximal Entropy
The variational principle gives a natural way to pick out seme members of M(X,T).
Definition 8.2. Let T:X -* X be a continuous transformation on a compact metric space X. A member p. of M(A, T) is callcd a measure of maximal entropy for T if /iM(T) = h(T)
Let MmaJ X, T) denote the collection of all measures with maximal entropy for T. After the next theorem, which gives the properties of A, T), we shall give an example where Mmax(A, T) is empty.
Theorem 8.7. Let T: X-* X be a continuous transformation of a compact metrisable space. Then
(i) Mmax(A, T) is convex
(ii) If h(T) < oc the extreme points of WJA', T) are precisely the ergodic members of Mmax(X, T).
(iii) If h{T) < čüąŠ and Mmax(X, T) čä 0 then Mraax(A\ T) contains an ergodic measure.
liv) If h(T) = čüąŠ then Mmax(ąÉ, ąō) # 0.
(v) If the entropy map is upper semi continuous then MmaJX, T) is compact and non-empty.
Proof
(i) This follows since the entropy map is affine (Theorem 8.1).
(ii) If čå e A/max(X, T) is ergodic then it is an extreme point of M(X, T) (Theorem 6.10(iii)) and hence of A/mjx(X, T). Now suppose čå e A/max(A, T)
192
8 Relationship Between Topological Entropy jnd Measure Theoretic Entropy
is an extreme point of T) and p = ppx + (1 ŌĆö p)/j2 for some p e [0,1],
pup2 e M(X, T). Then, since h(T) = hŌĆ×(T) = phPl(T) + (1 - p)h,JT) (Theorem 8.1) and htli(T), 1ą│ą”2(ąó) < h(T) (Theorem 8.6). we must have pt. p2 e Mmail(X, T). Hence p = p: = p2 and p is an extreme point of M(X, T) and hence ergodic (Theorem 6.10).
(iii) Let p e Mmx(X, T) and let p = JE(x,n ┬╗uh(m) be the ergodic decomposition of p. By Theorem 8.4 li(T) = ąøąöąō) = \FAXT)liJT)di(m) and since hjT) < h(T) (Theorem 8.6) we have m e ą£čéą╗čģ(ąź, T) for čé-almost all m.
(iv) By Theorem f hoose pŌĆ× e M(X, T) with h^JfT) > 2ŌĆØ. Let
.┬½ j
^' = Z ^ąśą╗ąĄą£{ąź,ąó).
n= 1 z
Since
N 1 1
V = L y, V-* + v for some v 6 MX, T)
ąĖ ┬« 1 ŌĆö
we have
N ,
ąÖąöąó) > ┬Ż ŌĆö ft^fT) > N for each N (Theorem 8.1).
n = 1 ŌĆ£
Hence p e ą£čéčÅčģ(ąź, T).
(v) The set Mmax(X, T) is non-empty because an upper semi-continuous function on a compact space attains its supremum. The uppei semi continuity also implies Mmax(X, T) is compact because if pŌĆ× e Mmax(X, T) and pŌĆ× -* p e M(X, T) then ąøąöąó)> limsupn^ŌĆ×,/i(J7') = h(T) so that pe .ą£čéą╗čģ(ąøąō, T) Ō¢Ī Previous << 1 .. 71 72 73 74 75 76 < 77 > 78 79 80 81 82 83 .. 99 >> Next 