# An introduction to ergodic theory - Walters P.

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Remarks

(1) There is a minimal homeomorphism with h(T) = oo but Л,,('/") <

oo Vp e E(X, T), showing that statement (iii) fails if h(T) = oo (Grillenberger [1])

(2) Part (v) together with Theorem 8.2, shows that Mmax(X, T)>is nonempty when T is expansive. This also follows from Theorem 7.11 and the proof of the second part of the variational principle which gives a description of a measure with maximal entropy as a limit of atomic measures on separated sets.

The first example of a homeomorphism with Mmam[X, T) = 0 was given by Gurevic. We now give an example.

Choose numbers /?„ such that 1 < p„ < 2 but [i„ s 2. Let Tn:Xn-> X„ denote the two-sided /?„-shift (see §7.3). We know h(Tr) = log/?n. Suppose dn is a mctric on X„ and we can suppose dn(x, y) < 1 Vx, у e A'„. Wc define a new space X which will be the disjoint union of the X„ together with a “compactification” point and we shall put a metric on A' so that the subsets X„ converge to x„. Define the metric p an X by p(x, >) = (1 /n2)d„(x, y)

§8.3 Measures vn ith Maximal Entropy

193

if*, у £ X„. p(y. x) = />(.v, у) = УГ=И 1 fi1 if v £ X„. у £ Л'г and n < p, p{>.,r, x) = v e A'„. Then (X, p) is a compact metric space. The transformation T X -* X with T\x = T„ and 7(д-,) = > yj is a homeomorphism. If ^£ ЩЖ, Г) then // = i P,J‘n + (1 - /»„)<),., where p„ £ АДА',,. 7„) and Ри^0,£п= i P-,< 1- Hence if^e E(,Y, 7) then either/< e E(.V„. T„) for some il or p = <\„- Therefore lnT} = sup{htt(T)\pc E(X, 7)}=-sup,,-,, sup J/),,„(Tn)|ur e £(X„, T„\] = sup„>, h(T„) = log 2. If A/max(A', T)^ 0 then by Theorem 8.7(iii) A/max(X, 71 contains some ergodic measure p. Then p e ЩХ„, T„) for some n so li^T) = log[}„ < log2. Therefore \/irax(A. T) — 0

There are minimal homeomorphisms with h(T) < oc and Mm.ix( \. 7) = 0 (Grillenberger [1]). There are also difieomorphisms of compact manifolds with M,nJX, T) = 0 (Misiurewicz [l]). \ote that if 11(7 I = 0 then A/max(X, 7) = M(X, T).

There is a discussion in §20 of Den\er, Grillenberger and Sigmund [1] of necessary and sufficient conditions for A/niJJ1( A, T) / 0. In particular the following result of Denker is proved. The conditions h{T) < oo and A/max(X, T) ф 0 are equivalent to the existence of a sequence {«„}f of finite open covers of X with £7=, h(T,a„) < oo and Iirrv-л h(T, V5=i =0 “ HT).

The following is an entropy analogue of unique ergodicity.

Definition 8.3. A continuous transformation 7:X-» X of a compact metric space is said to have a unique measure with maximal entropy if МпЖх(Х, 7) consists of exactly one member. Such transformations are also called inirin-sicially ergodic (Weiss [J]).

Remarks

(1) If 7 is uniquely ergodic and M(X,T)= {^} then 7 has a unique measure with maximal entropy, because the variational principle gives ЛД7) = h(T) in this case.

(2) If/i(T) - со and 7 has a unique measure w'ith maximal entropy thcr 7 is uniquely ergodic, because if A/max(X, 7) = {p} and m £ M(X, 7) then ^/2+m/2(7)= oo so m = p.

(3) If V/mjx(X, 7) = {p} then p is ergodic. If h(T) = cc this follows from

(2) and „f /i(7) < oo it follows from Theorem 8.7(iii).

(4) There are two ways that 7 cart fail to have a unique measure v :,h maximal entropy; either A/mjx(X, 7) = 0 or A/injl(A', 7) has at least two members. One can easily obtain examples of the second type by taking a disjoint union of two compact spaces on which homeornorplnsms act. There are however minimal homeomorphisms of the second type. Pursier.berg’s example of a minimal homeomorphism 7’ of К2 which is not uniquely ergodic (see §6.5) provides such an example because h(T) = 0 and therefore MmJX,T) = M(X,T).

One way that unique measures with maximal entropy are useful is in constructing isomorphisms.

194 8 Relationship Bctveen Tcpologi«.jl Emropy .md Measure-Theoretic Entropy

Theorem 8.8. Let 7 :X, -» X, (i = 1. 2) be a continuous transom nation of a compact metrisable space and suppose 7, has a unique measure, mih maximal entropy. Suppose hm(WJ = hUl(T2). If ф:Хх~* X2 is a himeasurahlc bijecaon with ф - Tt = T2 ф then - ф~х = //2 iciud so ф is an isomorphism between the measure-preserving transformations 7, on (А’„^?(Л ,J. /i,)).

Proof. By Theorem 4.11 h^ ф->(T2) = hЦ1(ТХ so htlt „ .(T2) = ht,,{T2) so Hz □

In the proof we only used the fact that T2 has a unique measure with maximal entropy. The fact that Г, also does follows from the existence of ф.

In the next section we shall show that if T: G -* G is an affine transformation of a compact metrisable group then m e .V/mJX(G, T) where m denotes Haar measure. In this case it is known that if hm(T) < oo then Afmj,(G, T) = {hi) ИПin is ergodic (Berg [l] Сопге [1], Walters [4]I

We shall now prove this last statement in the special шее when T is the two-sided shift (which is a group automorphism)

Theorem 8.9. Let Y = {0, 1.....к - 1}, X = П-а. У a,ul k’1 T-A' * h<-’

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