486 XI Uniform Distribution and Ergodic Theory
(ii) For any givenz∈ X, the closure of the set(Tnz)n≥ 1 is a nonempty closed in-
variant subset ofXand therefore coincides withX. Thus for everyε>0 there exists
n=n(ε)≥1suchthatd(Tnz,z)<ε. This proves the proposition forp=1.
We suppose now thatp>1 and the proposition holds withpreplaced byp−1.
(iii) We show next that, for anyε>0, there exists a finite setKof positive integers
such that, for allx,x′∈X,
d(Tkx′,x)<ε/2forsomek∈K.IfBis a nonempty open subset ofX, then for everyz∈Xthere exists somen≥ 1
such thatTnz∈B. HenceX=
⋃
n≥ 1 T−nB.SinceXis compact and the setsT−nBare open, there is a finite setK(B)of positive integers such that
X= ∪
k∈K(B)T−kB.SinceXis compact again, there exist finitely many open ballsB 1 ,...,Brwith radius
ε/4suchthatX=B 1 ∪···∪Br.Ifx,x′∈X,thenx∈Bifor somei∈{ 1 ,...,r}
andx′∈T−kBifor somek∈K(Bi). Thus we can takeK=K(B 1 )∪···∪K(Br).
(iv) We now show that, for anyε>0andanyx∈X, there existsy∈Xandn≥ 1
such that
d(Tny,x)<ε, d(T^2 ny,x)<ε,..., d(Tpny,x)<ε.In fact, since eachTk(k∈K)is uniformly continuous onX, we can chooseρ> 0
so that d(x 1 ,x 2 )<ρimplies d(Tkx 1 ,Tkx 2 )<ε/2forallx 1 ,x 2 ∈Xand allk∈K.
By the induction hypothesis, there existx′∈Xandn≥1 such that
d(Tnx′,x′)<ρ,...,d(T(p−^1 )nx′,x′)<ρ.But the invariant setTXis closed, sinceXis compact, and soTX=X. Hence
TnX=Xand we can choosey′∈Xso thatTny′=x′. Thus
d(Tny′,x′)= 0 , d(T^2 ny′,x′)<ρ,..., d(Tpny′,x′)<ρ.It follows that, for allk∈K,
d(Tn+ky′,Tkx′)<ε/ 2 ,...,d(Tpn+ky′,Tkx′)<ε/ 2.For eachx∈Xthere is ak∈Ksuch that d(Tkx′,x)<ε/2. Thus ify=Tky′,then
d(Tny,x)<ε,...,d(Tpny,x)<ε.(v) Letε 0 >0andx 0 ∈Xbe given. By (iv) there existx 1 ∈Xandn 1 ≥1suchthat
d(Tn^1 x 1 ,x 0 )<ε 0 ,...,d(Tpn^1 x 1 ,x 0 )<ε 0.We can now chooseε 1 ∈( 0 ,ε 0 )so that d(x,x 1 )<ε 1 implies
d(Tn^1 x,x 0 )<ε 0 ,...,d(Tpn^1 x,x 0 )<ε 0.