# An introduction to ergodic theory - Walters P.

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/i(T, x T2) = sup{/7(7, x 72,«):tf с &0/ finite}.

But if 4S is finite and ( £ then ¥ £ i, x for some finite sJy £ rJ2 £ 262. Hence, by Theorem 4.12(v),

/i(7, x T2) = sup{/i(7, x T2, x c $,, й?2; ja/,, л/, finite}.

We have

= -£(m, x m2)(Ck x Dj) • logfrn, x m2){Ck x D )

mO

4 Hmropy

where a г с the members of s( V< = o 'ХЛ J. and [D,] are die members

Remark. Theorem 4.23 readily extends to the direct product of any finite number of measure-preserving transformations.

^4.7 Examples

We shall now calculate the entropy of our examples.

(1) If —> <fX,SS,m) is the identity, then h(I) = 0. This is because li{I,.</) = lim;\/n)H\s/) = 0. Also, if Tp = I for some p Ф 0 then h(T) — 0 This follows smcc 0 = h{Tp) = |/;j ■ h(T) by Theorem 4.13. In particular any measure-preserving transformation of a finite space has zero entropy.

(2) Theorem 4.24. Any rotation, T[z) = a:, of the unit circle К has zero entropy.

Case 1: Suppose {a":n e 2} is not dense, i.e., a is a root of unity. Thus i/r = 1 for some p Ф 0; and T'\:) = apz = z so h(T) = 0 by example (1).

Case 2: Suppose {a":ne Z) is dense in K. Then {a":n < 0} is dense in K. Let <; = {Al,A2} where At is the upper half circle [1, — 1), and A2 is the

lower half circle [—1,1). For n > 0, T consists of semi-circles beginning at a " and —a~n. Since {a~":n> 0} is dense any semi-circle belongs to

= -\[Ck)>n:(DJ [logw.iQl t !og»j;(0;)J = — • !og/)),(Ct) —

inT.i-i h(T2).

□

i;4.7 Lx.implcs

101

V" = o T Hcncc any arc belongs to \/JL 0 T ns/{q). Thus, .id =

V<*= о aild so, h(T) = 0 by Corollary 4.1S.1.

(3) Theorem 4.25. Any rotation of a compact metric abelian group has entropy zero.

Proof

(a) Suppose X = K", the л-torus, and T{zb . ..,z„) = (a,z,,... ,anzn). Then T = Tx x T2 x • - ■ x T„ where T, .K -» К is defined by T^z) = atz. By example (2) /?(Т;) = 0 for all i so by Theorem 4.23

h(T) = V Л(Т^= 0.

i= 1

(b) General Case. Let T:G -* G be T(x) = ax. Let G = {yx,y2, • Let Hn = Kery, n ■ ■ ■ n Kery,,. Then H„ is a dosed subgroup of G and (G/HJ is the group generated by {y,,... , y„} (by 6 of §0.7.) Thus

(G/H„) = finite group x 2‘",

so

G/H„ = F„x K‘"

where F„ is a finite group and K‘n is a finite-dimensional torus.

The rotation T induces a map Tn:G/H„ —> G/Hn by Tn(gH„) = agHn. The map T„ is a rotation on G/H„, so that it can be written Tn = Tn l x Tn2 where T„, is a rotation of F„ and T„ 2 is a rotation of /C‘". Thus

h(T„) = h(T,Jf + h(Tn2) = 0

by example (1) and case (a) of this proof.

Note that \/„ £-/{G/H„) = where s/(G/H,,) denotes the (7-algebra consisting of those elements of that are unions of cosets of because y„ is measurable relative to f/(G/H„) and so every member of L2(m) is measurable relative to \Jn s/(G/H„).

Therefore if J$Q = , sJ(G/H„) then by Theorem 4.21

h(T) = sup h(T,V).

S J»n ’ finite

However, if *<? £ a?0 is finite then % £ ,z/{G/Hn) for some n and so h(T/6) < /i(T„) = 0. Thus h(T) = 0. □

Corollary 4.25.1. Any ergodic transformation with discrete spectrum has zero entropy.

This follows from Theorem 3.6. (Actually we have shown the result only when (X,SH,m) has a countable basis since the above calculation was for a metric group G.)

4 Emropy

(4) Endomorphism* of Compact (iron/>s. If -I is an endomorphism of ihc /i-ioms К" ото K" \\c shall show in Chapter 8 lhat h(/t) = £logi/.,| where the summation is over all eigenvalues of ihc matrix [/1] wilh absolute value giv.itcr than one.

Oi'ie can write down a complicated formula fyr ihe entropy of an endomorphism of a general compact metric abcliali^group. See Yuzvinskii [1].

(5) Ajlii,, Transformations. We shalWhow in Theorem Й.10 lhat when T = и ■ A is an affine transformation of K" then h(T) = li{A).

(6j Theorem 4.26. The lw «-.sided (/>0,. . . ,pk. {Уshift luis entropy ~Yj=o />, ‘ log/»,.

. Let Y = [0,1, . ,k — 1], .Y = ^ l°l T be the shift Let .4,- — = ij, 0 < i < к — I. Then с = {A0,..., is a partition

of A For ease of notation lei .г/ denote By the definition of the product .7-aljebra, A. we have

/

V t.cj = m.

By the Kolmogorov- Sinai Theorem (4.17),

h(T) = lim 1 v T~ \sJ v • • • v T~{n~usJ).

n -* an ^

A typical element of с(.г/ v T~1V v ■ • ■ v T~l"~ ‘’.V) is Aiti n T~lAit n n T-‘"-

t i ■ ^ i * ■ ■»-1 in — i i

which has measure pia • pit.....pin ,. Thus,

tf(.r/vr Vv - • -v Т-,,,_1’л/)

= ' />,„ ,) ■ logfPio ' ■/».„.,)

* - 1

= - I (Pf„ ■ • ■ A„-,)[logpio + • • • + logp,„. J

in. • .in -i=0 к - 1

= -«!/>.• logPi-i = 0

fc - i

Therefore, /?(T) = h{T,s/) — — £ p, logp,. □

i = 0

Remark. The 2-sided (i,i)-shifl has entropy log2; ihe 2-sided (j, 3, j-)-shifl has entropy log3. Thus these transformations cannol be conjugate.

(7) The 1-sided (/;0, - - ,4>k-J-shift has entropy — £=0 Pi' log/v The proof is very similar to the one in example (6) but Theorem 4.18 is used instead of Theorem 4.17.

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