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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 31 32 33 34 35 36 < 37 > 38 39 40 41 42 43 .. 99 >> Next /ąĖ p 1
= II V
/┬╗! - 1
< II IV.
= -r II
= ┬½ŌĆ× + aP
T \rJ j by Theorem 4.3(x)
1 = 0
We then apply Theorem 4.9. Ō¢Ī
Theorem 4.10. // T.X -> AŌĆÖ /.Ō¢Ā mvhsiirc-prescnino and sJ is ufitote sub-algebra of 'A then (1 |ji)W(V"=o T~ŌĆśs/) decreases to h(T,sJ)
Pkooi We lirst show, by induciion, that
fn - 1 n -1 / J \
n[ . t-',v i = ąĮąĖ/i - X HV ŌĆó
,1 = 0 / čā = 1 V 1=1 /
bor /; = 1 it is clear, and if we assume it true for n = p then it also holds for n = p + 1 bccausc
//( V ąō )=/-/( \/ J- W v sJ
i = i
= // ^ V T ŌĆś.c/'j + II, .čü/ V ąō wj by Theorem 4.3(ii) = H^\J T~ŌĆśV ) -t- ą»(-čü/ V T~W^j by Theorem 4.3(x)
┬½ Wtf) čé ąō f/fv V/ r-'-A- ^lhe i"duc'ion
\ (j, /f assumption.
Thus the claimed formula holds for all n. From this formula and Theorem 4.3(iii) we have ą®/^ T~ŌĆśs/\ > nH{s*lf\/Ui so that
^4.5 Properties of ą(ąō. .čü/) and h{ ąō)
89
Hencc
D
We now show h(T) is a conjugacy invariant.
Theorem 4.11. Entropy is a conjugacy invariant and hence an isomorphism invariant.
Proof. Let T,: A',-Ō¢║A',, T2:X2->X2 be measure-preserving and let ąż:{ąø2,┬╗čī)-Ō¢║ (┬«i,i7ii) be an isomorphism of measure algebras such that 0771 = ąó;1ąż. Let ,g/2 be finite, s/2 čü ąø2, and = {^4 a,. Ō¢Ā Ō¢Ā, Ar).
Choose B, g ą╣?, such that ąÆ, = ąż(ąÉ,) and so that t] = {B,,... Br} forms a partition of (A, A,, ra,). Let ą╗/, = .c/(r;).
Now f|i = o ąóąōŌĆśąÆ,, (where qt e {1,... ,r}) has the same measure as f|*=o T2ŌĆśAqi since
čä('ąō] (čéą│ą╗čć)~) = čä(ą┐ ą¤ŌĆśąÉą▓1) = ąó;ŌĆśąż(ąÉą»)
\i = 0 / \i = 0 / 1= ą×
= "ą┐ ą┐%.= ą┐ (TrBj~.
1 = 0 t = 0
Thus ąØ(\/?-ą▓ TfW,) = //(V"=o which implies that h(Tl,j*l) =
/>(72, which in turn implies /i(7,) > h(T2). By symmetry we then get that/i(7,) = h(T2). Ō¢Ī
The proof of Theorem 4.11 also shows that if T2 is a factor of 7, (or a semi-conjugate image) then h(T2) < h(Tj).
After we have developed some properties of h(T,s-/) and h(T) we shall consider the problem of how to calculate h(T). These calculations and Theorem 4.11 will allow us to give examples of non-conjugate measure-preserving transformations.
┬¦4.5 Properties of h{T,sd) and h(T)
Recall that
/1(7, jrf) = lim - HI V T-ŌĆśsA.
n-ąŠąŠą¤ \i = 0 J
Theorem 4.12. Suppose ą▓/, ((! are finite subahjebras of ą¢ and 7 is a measure-preserving transformation of the probability space (A, Ji9, in). Then
(i) h(T.rj) < ąØą½).
(ii) h{T,sjv(g)<h(T,sJ) + h(T,V)-
VU 4 Entropy
. = - .ą╗ąō. . - . . i i.
! .┬╗ 1 ąō. T. J i < ąōąø T. I ŌĆö H( "/ '~t I.
: T. T~ Ō¢Ā:/.= ':( 7. :/
If ŌĆó > I. > \T. v'i = ąō ŌĆÖŌĆó ąō ; T-.-Ji ą│. ill Ij T is i.'jjj ŌĆó. > i wv/:
A.
lnY,*/i = h[T, \J T*/
Proof
1 1 \ 1" J
(i) -HI V T )< ^ //,ąō _c/) by Theorem 4.3 ┬╗ \, = o / " ŌĆ£o
1
= - ^ H(.c/) by Theorem 4.3 (xj ┬╗ ŌĆ£u
= ą®.ąō/).
/čÅ ŌĆ£ 1 \ /ąś ŌĆö 1 ą» ŌĆö 1
fii) Hi V ) = H V T~Wv\J T~irŌé
\i=0 / \i = 0 i = 0
< H^y T_,.a/^ + W^y
by Theorem 4.3(viiij.
(iii) If ą╗/ čü then
M - 1 #1 ŌĆö 1
V čé- 'čü-/ čü v t~ŌĆśv, n > i
< = 0 I = 0
so one uses Theorem 4.3(iv).
(iv) tf^y T-w'JzH^ V 7"ŌĆśŌĆś<
by Theorem 4.3(iv)
= ąĮ^čā ą│-^J+H^V T-\s/)/(y T~'(S
by Theorem 4.3(ii).
But by Theorem 4.3(vii),
4 x r *)/\$ r*)) ┬Ż 1 "MST_w
┬½ ŌĆö i
< X H(T-'sJIT~i(g) by Theorem 4 3(v) i = 0
ŌĆö nH(sJ!(Ōé) by Theorem 4.3(ix).
5-J.5 Properties of ąø( ąø .ą│/) and /;(ąó)
91
Thus.
ąØ ^ V ąó~[^<ąĮ(^čā ąó~ŌĆśvj + ą┐ąØ(ą╗//<┬Ż)-
(v) ąØ^ \/ ąó" ąōą¦čü/j by Theorem 4.3(čģ),
h(T.T~lsS) = h(T,sJ).
(vi) /i(V, V T~ŌĆśs-/^j = lim i H (\J T'^y'
1 /fc -mi ŌĆö 1
= lim -HI V 7'~ŌĆś-g/
ą»-*ąŠąŠ ^ \ i - 0
= lim (ŌĆöjjr-rwfv' Ōäó
┬╗i-a \ ą╗ čā ą║ + čÅ ŌĆö 1 \i = o
= ąø(ąó,ą╗ą×.
(vii) /i^7\ V T-WJ = h^T,\/ T-WJ by (v)
= h(T,.tf) by (vi). Ō¢Ī
Corollary 4.12.1. If srf/'Ōé are finite sub-algebras of ?A we have
\h{T.sJ) ŌĆö h(T/fi)\ < d(s#,W), so that h(T, ) is a continuous real-valued
function on the metric space (V,d) introduced in Theorem 4.5.
Proof. By (iii)
|h(T,s/) - ą”ąó,ą® < max(H(sS/V),H(V/sS))
<d(s/,V}- Ō¢Ī
We can deduce from Theorem 4.12 some simple properties of h(T).
Theorem 4.13. Let T be a measure-preserving transformation of the probability space
(i) For ą║ > 0, h(Tk) = kh(T).
(ii) If T is invertible then h(Tk) = [k\h{T) V/c e Z.
Proof
4 hntropy
This follows since
i'Kx r"\S 'rw))-'r.lH(k rŌĆØ
= kh(T,.'V).
Thus,
kli(T) = ą║ Ō¢Ā Min /i( -r, r./) = sup /1 j Tk, \J T-'.cA
xJ finiic ą╗/ čā i = 0 J
< sup /1 (ąóą║ąø) = h(Tk).
Also, h(T\s/) < li(Tk, \/f=o T~ŌĆśx/) = kh[T,s/) by Theorem 4.12(iii) and so, h{Tk) < kh(T). The result follows from these two inequalities.
(ii) It oufficcs to show that h(T~l) ŌĆö li[T) and all we need to show is that h(T~ *,.?/) = h(T,.(Ō¢Ā/) for all finite ą╗/. But
ąØ^ąŻ T!.jtf'J = II ^7'-'" - *' čā T.J^j by Theorem 4.3(x)
= w(\/ Ō¢Ī
Wc shall obtain more information on how h(T) behaves relative to natural operations on transformations when we have proved some results that make these calculations simpler. The following result allows us to understand when h(T,jaf) is /ąĄą│ąŠ (Corollary 4.14.1) and allows us to conclude that a non-invertible measure-preserving transformation T which is not mod 0 invertible (i.e. T~ must have (strictly) positive entropy (Corollary 4.14.3). ŌĆó Previous << 1 .. 31 32 33 34 35 36 < 37 > 38 39 40 41 42 43 .. 99 >> Next 