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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 35 36 37 38 39 40 < 41 > 42 43 44 45 46 47 .. 99 >> Next /i(T, x T2) = sup{/7(7, x 72,¬´):tf —Å &0/‚Ç finite}.
But if 4S is finite and (‚Ç ¬£ 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 \$,, –π?2; ja/,, –ª/, finite}.
We have
= -¬£(m, x m2)(Ck x Dj) ‚Ä¢ logfrn, x m2){Ck x D )
mO
4 Hmropy
where a –≥ —Å the members of s( V< = o '–•–õ 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 ‚Äî> <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 –§ 0 then h(T) ‚Äî 0 This follows smcc 0 = h{Tp) = |/;j ‚ñÝ 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 –ö 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 –§ 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, ‚Äî 1), and A2 is the
lower half circle [‚Äî1,1). For n > 0, T consists of semi-circles beginning at a " and ‚Äîa~n. Since {a~":n> 0} is dense any semi-circle belongs to
= -\[Ck)>n:(DJ [logw.iQl t !og¬ªj;(0;)J = ‚Äî ‚Ä¢ !og/)),(Ct) ‚Äî
inT.i-i h(T2).
‚ñ°
i;4.7 Lx.implcs
101
V" = o T Hcncc any arc belongs to \/JL 0 T ns/{q). Thus, .id =
V<*= –æ 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 –ª-torus, and T{zb . ..,z‚Äû) = (a,z,,... ,anzn). Then T = Tx x T2 x ‚Ä¢ - ‚ñÝ x T‚Äû where T, .K -¬ª –ö is defined by T^z) = atz. By example (2) /?(–¢;) = 0 for all i so by Theorem 4.23
h(T) = V –õ(–¢^= 0.
i= 1
(b) General Case. Let T:G -* G be T(x) = ax. Let G = {yx,y2, ‚Ä¢ Let Hn = Kery, n ‚ñÝ ‚ñÝ ‚ñÝ n Kery,,. Then H‚Äû is a dosed subgroup of G and (G/HJ is the group generated by {y,,... , y‚Äû} (by 6 of ¬ß0.7.) Thus
(G/H‚Äû) = finite group x 2‚Äò",
so
G/H‚Äû = F‚Äûx K‚Äò"
where F‚Äû is a finite group and K‚Äòn is a finite-dimensional torus.
The rotation T induces a map Tn:G/H‚Äû ‚Äî> G/Hn by Tn(gH‚Äû) = agHn. The map T‚Äû is a rotation on G/H‚Äû, so that it can be written Tn = Tn l x Tn2 where T‚Äû, is a rotation of F‚Äû and T‚Äû 2 is a rotation of /C‚Äò". Thus
h(T‚Äû) = h(T,Jf + h(Tn2) = 0
by example (1) and case (a) of this proof.
Note that \/‚Äû ¬£-/{G/H‚Äû) = where s/(G/H,,) denotes the (7-algebra consisting of those elements of that are unions of cosets of because y‚Äû is measurable relative to f/(G/H‚Äû) and so every member of L2(m) is measurable relative to \Jn s/(G/H‚Äû).
Therefore if J\$Q = , sJ(G/H‚Äû) then by Theorem 4.21
h(T) = sup h(T,V).
S J¬ªn ‚Äô‚Ç finite
However, if *<? ¬£ a?0 is finite then % ¬£ ,z/{G/Hn) for some n and so h(T/6) < /i(T‚Äû) = 0. Thus h(T) = 0. ‚ñ°
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 –ö" –æ—Ç–æ 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 .
(5) Ajlii,, Transformations. We shalWhow in Theorem –ô.10 lhat when T = –∏ ‚ñÝ A is an affine transformation of K" then h(T) = li{A).
(6j Theorem 4.26. The lw ¬´-.sided (/>0,. . . ,pk. {–£shift luis entropy ~Yj=o />, ‚Äò log/¬ª,.
. Let Y = [0,1, . ,k ‚Äî 1], .Y = ^ l¬∞l T be the shift Let .4,- ‚Äî = ij, 0 < i < –∫ ‚Äî I. Then —Å = {A0,..., is a partition
of A For ease of notation lei .–≥/ 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 ‚Ä¢ ‚Ä¢ ‚Ä¢ v T~{n~usJ).
n -* an ^
A typical element of —Å(.–≥/ v T~1V v ‚ñÝ ‚Ä¢ ‚ñÝ v T~l"~ ‚Äò‚Äô.V) is Aiti n T~lAit n n T-‚Äò"-
t i ‚ñÝ ^ i * ‚ñÝ ‚ñÝ¬ª-1 in ‚Äî i i
which has measure pia ‚Ä¢ pit.....pin ,. Thus,
tf(.r/vr Vv - ‚Ä¢ -v –¢-,,,_1‚Äô–ª/)
= ' />,‚Äû ,) ‚ñÝ logfPio ' ‚ñÝ/¬ª.‚Äû.,)
* - 1
= - I (Pf‚Äû ‚ñÝ ‚Ä¢ ‚ñÝ A‚Äû-,)[logpio + ‚Ä¢ ‚Ä¢ ‚Ä¢ + logp,‚Äû. J
in. ‚Ä¢ .in -i=0 –∫ - 1
= -¬´!/>.‚Ä¢ logPi-i = 0
fc - i
Therefore, /?(T) = h{T,s/) ‚Äî ‚Äî ¬£ p, logp,. ‚ñ°
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 ‚Äî ¬£=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. Previous << 1 .. 35 36 37 38 39 40 < 41 > 42 43 44 45 46 47 .. 99 >> Next 