# An introduction to ergodic theory - Walters P.

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/и p 1

= II V

/»! - 1

< II IV.

= -r II

= «„ + aP

T \rJ j by Theorem 4.3(x)

1 = 0

We then apply Theorem 4.9. □

Theorem 4.10. // T.X -> A’ /.■ mvhsiirc-prescnino and sJ is ufitote sub-algebra of 'A then (1 |ji)W(V"=o T~‘s/) decreases to h(T,sJ)

Pkooi We lirst show, by induciion, that

fn - 1 n -1 / J \

n[ . t-',v i = ни/i - X HV •

,1 = 0 / у = 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 Г )=/-/( \/ J- W v sJ

i = i

= // ^ V T ‘.c/'j + II, .с/ V Г wj by Theorem 4.3(ii) = H^\J T~‘V ) -t- Я(-с/ V T~W^j by Theorem 4.3(x)

« Wtf) т Г 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 Щ/^ T~‘s/\ > nH{s*lf\/Ui so that

^4.5 Properties of (Г. .с/) and h{ Г)

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',-►A',, T2:X2->X2 be measure-preserving and let Ф:{Л2,»ь)-► (®i,i7ii) be an isomorphism of measure algebras such that 0771 = Т;1Ф. Let ,g/2 be finite, s/2 с Л2, and = {^4 a,. ■ ■, Ar).

Choose B, g й?, such that В, = Ф(А,) and so that t] = {B,,... Br} forms a partition of (A, A,, ra,). Let л/, = .c/(r;).

Now f|i = o ТГ‘В,, (where qt e {1,... ,r}) has the same measure as f|*=o T2‘Aqi since

ф('Г] (тглч)~) = ф(п П‘Ав1) = Т;‘Ф(АЯ)

\i = 0 / \i = 0 / 1= О

= "п п%.= п (TrBj~.

1 = 0 t = 0

Thus Н(\/?-в 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). □

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-‘sA.

n-ооП \i = 0 J

Theorem 4.12. Suppose в/, ((! are finite subahjebras of Ж and 7 is a measure-preserving transformation of the probability space (A, Ji9, in). Then

(i) h(T.rj) < НЫ).

(ii) h{T,sjv(g)<h(T,sJ) + h(T,V)-

VU 4 Entropy

. = - .лГ. . - . . i i.

! .» 1 Г. T. J i < ГЛ T. I — H( "/ '~t I.

: T. T~ ■:/.= ':( 7. :/

If • > I. > \T. v'i = Г ’• Г ; T-.-Ji г. ill Ij T is i.'jjj •. > i wv/:

A.

lnY,*/i = h[T, \J T*/

Proof

1 1 \ 1" J

(i) -HI V T )< ^ //,Г _c/) by Theorem 4.3 » \, = o / " “o

1

= - ^ H(.c/) by Theorem 4.3 (xj » “u

= Щ.Г/).

/я “ 1 \ /И — 1 Я — 1

fii) Hi V ) = H V T~Wv\J T~ir

\i=0 / \i = 0 i = 0

< H^y T_,.a/^ + W^y

by Theorem 4.3(viiij.

(iii) If л/ с then

M - 1 #1 — 1

V т- 'с-/ с v t~‘v, n > i

< = 0 I = 0

so one uses Theorem 4.3(iv).

(iv) tf^y T-w'JzH^ V 7"‘‘<

by Theorem 4.3(iv)

= н^у г-^J+H^V T-\s/)/(y T~'(S

by Theorem 4.3(ii).

But by Theorem 4.3(vii),

4 x r *)/$ r*)) £ 1 "MST_w

« — i

< X H(T-'sJIT~i(g) by Theorem 4 3(v) i = 0

— nH(sJ!() by Theorem 4.3(ix).

5-J.5 Properties of Л( Л .г/) and /;(Т)

91

Thus.

Н ^ V Т~[^<н(^у Т~‘vj + пН(л//<£)-

(v) Н^ \/ Т" ГЧс/j by Theorem 4.3(х),

h(T.T~lsS) = h(T,sJ).

(vi) /i(V, V T~‘s-/^j = lim i H (\J T'^y'

1 /fc -mi — 1

= lim -HI V 7'~‘-g/

Я-*оо ^ \ i - 0

= lim (—jjr-rwfv' ™

»i-a \ л у к + я — 1 \i = o

= Л(Т,лО.

(vii) /i^7\ V T-WJ = h^T,\/ T-WJ by (v)

= h(T,.tf) by (vi). □

Corollary 4.12.1. If srf/' are finite sub-algebras of ?A we have

\h{T.sJ) — 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/) - ЦТ,Щ < max(H(sS/V),H(V/sS))

<d(s/,V}- □

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 к > 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 r”

= kh(T,.'V).

Thus,

kli(T) = к ■ Min /i( -r, r./) = sup /1 j Tk, \J T-'.cA

xJ finiic л/ у i = 0 J

< sup /1 (ТкЛ) = h(Tk).

Also, h(T\s/) < li(Tk, \/f=o T~‘x/) = 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) — li[T) and all we need to show is that h(T~ *,.?/) = h(T,.(■/) for all finite л/. But

Н^У T!.jtf'J = II ^7'-'" - *' у T.J^j by Theorem 4.3(x)

= w(\/ □

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 /его (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). •

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