Title
Uniformity of mixing transformations with infinite measure
Author(s)
TOMATSU, Shizuo
Citation
[岐阜大学教養部研究報告] vol.[17] p.[43]-[49]
Issue Date
1981
Rights
Version
岐阜大学教養部 (Dep. of Math. Fac. of Gene. Educ. Gifu Univ.)
URL
http://repository.lib.gifu-u.ac.jp/handle/123456789/47503
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
43
U niformity of mixing transformations
with infinite measure
ShiZUO T OM A T SU
Dep. of M ath. R lc. of Gene. E duc. Gifu U niv
(Rec61ved Oct. 5, 1981)
lntroduction. M iχing transformations on a conlpletely regular topologicaI rneasure space
, ゛ -
with infinite measure are not necessarly ergodic 〔7〕 .
T he purpose or this note is to treat
that some attached conditions of miχing transformations in order to imply the ergodicity.
Let (X s, μ) be a topological measure space of a completely regular topological space X,
thむ E field S of Borel sets of X , and a measure μ on S , whel゛e the measure μ assumed to
be non-negative, c
7-additive, locally finite and tight in the sense that μ(j ) こ sup{ μ{ 絹 ; K ⊂ 淮
尺 is compact} for every ノ1∈!!3よ A set j E S will be almost clopen if it characteristic function is μ- almost everyw here (in w hat fo110w s it denotes abbreviated a.e. ) continuous.
be the set of a11 clopen sets of 旺
lt is clear that 食 is a field but not E field.
L et 於
lt is known
that 託 is μ-metrically dense in 懇 〔7〕.
A n endomorphism T of χ is called m 加加g if it has the following properties 〔6, 7 〕.
(i)
T is continuous (a.e.), and measure preserving.
I
U四
(ii)
There is a sequence of 貨-subsets HkE 双 (だ= I , 2, 3, ‥.) of χ with finite measure such
that X
(iii)
肌 (a.e.)
There is a sequence of positive numbers pバ71= 1, 2, 3, ‥.) such that the miχing rela-
tion
11m 画 μい ∩T- 71B) = μ(j ) μ(召)
holds for any 介-sets j
and 召 included in
some H k
T-invariant SUbsetS.
L et T be a rneasure preserving transfmation on χ.
A 石 invariant
∞
93-set j
which has a S -subset D such that U
7` D こ j
( a.e. ) and O< μ印 ) < E for any small
positive number E. Such a set /1 1s called 7こosmoぴc and the seレ D i s called geneTetoT of A
A
皿 invariant 忠一set is called 71co se77 d 加e if it has no w andering set.
set is called 7こ-e7即 ぷ c if it has no non- triv ia1
7こinv ariant subset.
A
A
石 invariant S -
S - set D
iS called
7!
£r皿 j l加e if for any μ- non- null subsets £ and F of D there is an integer 7z such that μ(£ ∩
7¯¯ 司 ≠θ. Then it is clear that the followings.
L EMMA 1.
石 e7即 ぷc sd 油 ㎡so 7・-osmolfc.
L EMMA 2. T-o8motic 8et lx,hich has T- tm n81tilXe genemtor 18 T-etgodic.
44
Shizuo Tomatsu
PRooF.
i. e. , j
L et /1 be a T- osmotic set having 7! transitive generator D.
has non-trivia1 7乙invariant !B-subset 召.
elements which belong to j
lf /1 1s not r -ergodic,
L et C 二 j ¥ 召 (A ¥ 召 denotes the set of
but not 召) , then μ( C) ≠ a
L et £)1二 D n 召 and D2= 1) n C, then
clearlyμ(D1) ≠0, 試D2) ≠0, and D1, D2aregeneratorsof 召, C respectively. SinceD is 7しtransitive, hence there is a positive integer 71 such that μ(仙 n r -リ )2) ≠ θ, i.e. , μ(召 n C) ≠ a
lt
is a contradiction.
L EMMA 3、 T-osmotic sd 18 d 80 T-conseruat辿e.
L et j
be not T-Conservative, and its w andering set denotes { r £ } ; 71= θ, 土 j , 士 2,
X1
7
j
PRooF.
‥ . (where ? E = £ ). Now we put 卵y S-subset F of j
such that O< μ(F) < μ(£ ). Let F n
r £ = 凡 , then clearly 凡 s are mutually disjoint an(卜 7¯¯
¯ 凡 ⊂ 瓦.
Since μ( 0
r
凡
(凡 ) 二μ(F)く μ(£ ), hence we can put a !13-set G such that G⊂£ ¥ U T-゛ Fn. Then we have
-
∽
゛(;Cワ愉づOyS)⊂
?E¥? (r-凡)=?E¥凡=?£¥Ffo
l゛a
n
yir
lte
g
e
l`・,
∞
i.e., ( P G) n F= φor equivalently Gn ( 7¯
゛ F) 二 φ( φdenotes empty set). lt implies that U r F
-
is non-trivia1 7! invariant subset of A
∽
T his show s that /1 1s not 7こ osmotic.
M iχing transformations. The sequence { み }
of positive numbers in the definition of
miχing property(iii) is considered as some sort of dilution factor
and asymptotically inde-
pendent of・the choice of the 肌 ∈屁 in the definition of mixing property(ii) 〔6〕 .
lt is know
that the behaviour of the dilution sequence { 両 } of miχing transformation concerned to ergo-
dicity of the transformation 〔6, 7〕 .
N ow we show the followings with regared to the dilution sequence of miχing tr耳nsformations.
T HEoREM 1.
L d T be an inlJeTtible ・ 咄臨g tTanj oT・ (ltion goueTned by the 8equence { pj .
U theTe eχ18t T-o8motic 8et, then Σ pj = ∽.
71= 1
P RooF.
L et j
be a 7
戸 osmotic set, then there is a ∬ ∈託 in the definition of miχing prop-
」y
V such that pj > μ唐 n T`¯¯゛ 出 Z2 μ(H ) 2 for any integer 71≧ N o N ow , if w e suppose
oo , t h e l l
th el e
is
8 11
111t e g e l ゛
゛
≧
Ⅳ
s u ch
th a t
s u ch
th a t
Σ
μ ( 77 ∩
r
¯゛ H ) <
E
fol
a il y
く
一
几
y︰
︼
・
erty(ii) such that μ(Hn 川 ≠θ. From mixing relation(iii) there is a sufficiently large integer
p o siti ゛ e
m
n
u
m
b
e
[ε
. Le
tE<{ μ
唐∩
溺a
n
d尺
二
(Hn溺¥O(Hnド 珀,th
e
nμ
(幻>θa
n
dKnT
-゛K
m
-
φ for any integer 71≧n
/1 such th3t θ< μ( G) < 士
Consequently, 尺∩7¯
士l尺= φ for 71≧m. Let G be any S -subset of
μ( 幻 ・ lf G⊂ 瓦 th叩 μ( ( 0
≠ひ lt contradicts to 7しosmoticity of A
and GI = G n し
∩( 0
?
G) ∩ 尺) < μ( 目 j . e・・ μ(
Hence G is not contained in 瓦
∬∩ T¯゛ H ) ) , then μ( GI ) ≠ a
Since μ( 尺 ¥ 0
? 仙 )
り
Let D1= Gn 尺
> μ(幻 - 2mμ(D1)
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U niformity of miχing transformations with infinite measure
47
(/10n T- 召o) = μひo) μ(召o). Since μい on 7・- 召o) こ θ for any positive integer 71, hence μ(j o)
μ(召o) = θ, i.e., μ(A)) = θoΓμ(召o) = a lt is a contradiction.
・
み
que址 id un廿ormity and 8ahj ie8 the condition
5J
T HEoREM 3、 L et l ¯
` be an 伍むeTtibLe m臨伍g tT(1n8foTm (1tion gouem ed by { pj
. IL
f T
h s se-
1二 ゜ け hen theTe is (1 T- o8motic 8et.
PRooE. Suppose that there is no 孔 osmotic set. F rom L emma l r is not ergodic, hence
there is a non-trivia1 7! invariant S -set え
L emma 5 1mplies that /1qE屁.
L et 公= χ ¥ λ
M etrically denseness of 託 in S follow s that there is λoE 介 such that μ(λ 十 j o) く δ` for any
δ
`> θ
. Let Boこχ¥ j 0, j 1= λ∩λ0, λ2= 公∩λo, 瓦 = 公njjo, 瓦 こλnjjo, then it is clear that
g(λ2)< δ
, μ(烏)く δ
ヽ
. Sinceλぞ介, hence μ(λ2)≠θ
, μ(瓦)≠θ
. By theassurnptionthat 7こ
-in∞
variant set λ is not 710smtic, hence U
-
T7゛B2 1s non-trivia1 7! invariant subst of j , i. e. , 1et
cね
∞
&=λ¥ U P瓦, the
nμ
(ご)≠a The
reis∬∈託inthepro
pe
rty(ii) o
fm
ixingsuc
hthat,le
t
-
-
j = /1∩凧 召= 召∩凧 j o= j on凧 召o= 召0∩凧 /11= /11∩私 /12= /12∩H, B1= 召1∩H, B2= 召2∩μ
and C = & ∩ 凧 then μ(川 , μ( 召) , μい o) ↓μ(召o) , μ(痢 ) / μい 2) , μ(召1) , μ(召2)
and μ( C)
are a11
non-nu11。
ノ10n T-lj o= (山 ¥ T-1召2) U(血 ∩7-1j 2)⊃j l ¥ 7¯-1召2こj ¥ 召2∩7-1召2⊃C
shows that
μ(j on アーり 10) ≠θ. Similarly, μいo¥ 7-lj ) ≠θ. For, if μ(j o¥ 7-lj o) = θ, then j o⊂ T-lj o
(a.e.). Since 7 1smeasure preserving, hence j o= T-1/10(a.e.) i.e., j ois a non-trivial r-invariant 食-set.
By L emma 5 1t is impossible. lnductively, if μ( d ″
r几 40) ≠ θ and μ( U T- り10
)≠
θfo
ra
nin
te
g
e
rん
>j,th
e
n
,s
in
c
en7¯
¯り1
0⊇
恚¥UT
-JB
2⊃
C
,h
e
n
c
eべ∩7・-j
¥ 7-り O
i= o
j= l
j= 0
ゑ+ 1
k
k+ 1
ん
7¯- lj O⊃
m(︲ 心
7¯iy
10⊂∩7¯ij . Co
nse
q
ue
ntly,∩7¯¯ijo二∩T¯引0. Thus7¯¯i( ∩T-1Ao)二
then
j= o
i= o
k
7 - ij 0.
i= o
,
k
i= 0
E(
︲心
£
0
/1
)≠
θ
,S
im
ila
rly
,べ
(07¯り
1
0)¥7¯(゛¯
l¯
1
)/1
0)≠
θ
. Fo
r,ifべ
07-りo
¥T
-( 1
)y
lo
)=θ
,
k
1t follows that 7¯- 1( ∩ 7 ¯ij o) こ ∩ T- ij 0, i. e. , ∩ 7¯- i/10 1s a non-tri
i= o
via1 7こ invariant 託- set.
i= o
i= 0
lt impossible by L emma 5.
N ow , from the miχing relation(iii) oI T there is a positive integer y for any number E> θ
such that
ρ・μい0∩7-1召0) - μ(/10) μ(β0) < E for any integer
tz≧ 皿
Sequential uniformity of T follows,
p
N
十
k
μ
((U7
・¯り
1
0¥7・¯りo
)∩
r(″
4
¯
゛)召
o
)<″
(回T
¯り
lo
n
7¯りo
)イ
(B
o
)十
ゆ
(回T
-り
1
0¥7
T
-りO
)/μ
1
0)
fol any positive integer ん. Since
48
Shizuo T omatsu
μ
Oo
nr(N
十
い
Bo
)=μ
((071
う40)∩
7-(N
十
い
Bo
)十
万μ
((OT¯り10¥7・¯Mo
)∩T
べ
N
゛い
B
Q
)
サレ
Σp
N
十
ん
μ
((071
-Å
o
¥rん
40)∩
T
-(″)召
o
)<宍
μ
(OT
恍
40¥7・几
40)μ
印
o
)十
i= 1
瓦
ε
宍
μ
(O
T
犬
4
0¥T
-M
9
)/μ
い
O
)<宍
μ
(d7
・恍
4
0¥T
恍
4
0
)十
ε
p
N
十
k
μ
((OT
恍
4
0
)∩
r(″
¯
1
゛)£o
)>μ
い
o
か
(召
o
)一
万
μ
(OT
恍
4
0¥T
¯
り
1
0)μ
印
o
)-2ε
コ ー1
土
.
卜 1
、
Therefore,
Since
べ
07几
40¥7・-り
1
0=(071
-り2¥7-jべ
)U
(フ
プ
ね
2¥0T
¯哨
2)
9X
R
hence
j4μ
(OT
一
山
O
¥7
・-M
O
)十
jJ
;μ
(7
・-り
と
j2¥07¯
哨
2
)
(白7-咄
2)<μ
(志
)十
μ
(j1)-μ
(O=μ
い
O
)-μ
(0.
< μ(j 2) 十μ
Consequently,
らHべ
(OT
¯ツ
O
卜7¯(″)B
O
)>ノ
ル
1
0M
(剛二
悩
い
O
)-.り
(C
)h
心
O
)-2=
= μ( O μ(召o) - 2 ε> θ for sufficiently large integer Ⅳ
Since
μげ ー(″4 )j on T-(″ )βo) = μいon 召o) = θ,
hence
″
((OT
¯ら
40)∩
√
(″)召
0)さ″
((07¯り0¥7¯(j¯)j40)∩
7
・¯(″
¯
り
い
B
O
)
T herefore,
巧
V
十
ん
μ
(山
T- i/10 ¥ T- (N十k) 召0 > μ(O μ(召o) - 2 ε> θ
, i.e.,
i= 0
n
ん
7・¯jy10 ¥ r (N-χ
-1) 召0
/ 脳 (O g(£O)- 2ε}
i= 0
12ぺ乱<
8p Q1
7
r
lt follows that
9 1E
べ
OT
-り1
0¥T
-(″
4)j40)/脳
(C
か
(召
O
)-2ε
}
< Nμ(j O)バμ(C) g(且)) - 2ε
}
lt contradicts to
絞 1= (X)
T heorem 3 , 1emma 2 and lemma 3 follows that
8
91
EF
CoRoLLARY・ U (1 m臨流g tTan8foTmation l¯k s sequentid un廿oT・ ity and l ¯-tTansith)ity, then
ρ71= oo
impnes 伍at e昭odicity and consenXati折句 好 T.
or exanlplej t is easy to see that the mixing transformation defined by shift of nleasure
U niformity of miχing transformations with infinite measure
49
theoretical sample space constructed by an irreducible and aperiodic M arkov chain admitted
a non-trivial stationaryヽmeasure has the sequential uniformity and T - transitivity 〔 1, 3, 7〕 .
R eferenCeS
1 .
Chung,
K . L . : M (1Tkoむ cha匯8 加ith 8t(ltiona巧 tTansition pTobabUitie8、 Berlin- G6ttingen- H eidelberg : Sp-
rmger 1960.
2.
H almos, P . R. : LectuTes on ergodic theoTy.
3.
K akutani, S. and Parry, W . : I可 1711te measuTe pTeseTIXing tTan8foTmatio718 切ith
M ath.
Soc.
of Japan, 1956.
・ iχ泌g .
Bull.
A mer.
Math. Soc. 69, 752-756 ( 1963).
4.
K rickeberg, K . : M 18chenden TT(17tj oTmatione71 a可
M n nighJd tigkeiten une71dlichen M a8se8. Z . W ahr. v.
Geb. 7, 235-247 ( 1967).
5.
K rickeberg, K . : & ro g m注加g pn)pertie8 0j` M arkop (九c
山1 mith i吋 inite me(18uTe.
Symp.
Statis.
Proc.
F ifth Berkley
Prob. , 1966, V ol. 1, Part II , 43 1- 446.
6 . Krickeberg, K. : 尺eceM rea /訟 o m加加g 加 lopj oポc㎡ measz
zre spαce.
Lecture notes in M ath.
Vol.
89, Springer V . , 1969.
7.
P apangelov, F . : StTong Tatio a・ it8, R -々ecut rence a71d miχing pTopeTties oJ discTete paTa・ d eT M aTkolJ
pTocesses.
Z.
W har .
v.
G eb.
8 , 259- 297 ( 1967).
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