Title Quasi-Ergodicity of Transformations on a Topological Measure Space with infinite Measures.
Author(s) TOMATSU, Shizuo
Citation [岐阜大学教養部研究報告] vol.[12] p.[1]-[4]
Issue Date 1976
Rights
Version 岐阜大学教養部 (Dep. of Math. Fac. of Gene. Educ. Gifu Univ.)
URL http://hdl.handle.net/20.500.12099/47377
※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。
T he fact that in the space with infinite measures the miχing transformations may fail to be ergodic is shown by F . P apangelov 〔 5,6〕. M ore precisely, let 7・ be an invertible quasi -miχing tr ansform lation satisfyingΣ凪1< (x), thenT is notergodic. Butitisnotknownoutof thecase of Markov chainsthat the
converse of this result is established 〔 2, 6〕.
T hepurpose of thepresent paper istointroducea weaker concept than that of quasi- miχing and shows the relations including the aboveproblem between thenew concept and that of ergodicity of m easure pr eser v ing transformations in the space with infinte measures.
Let(χ, S, μ) be a top010gical measure space of a complete・regular topological space χ, the ,7- field S of Borel sets of X , and a measureμon 既 W here an measures arising in the sequel assumed tobe non- negative, (y-additive, 10cany finite, and tight in the sense that μ(A ) = sup{ μ( 聡 KC 爪 £ compactl for every /1E S8 . A set /1 C χ will be calledμ- almost cIopen if its characteristic function isμ- almost everywhere continuous, i. e. , its boundary is a nu11set.
A n endomorphism T of χ is called quasi-miχing if it has the fo11Qwing 皿 opertiesて 5, 6〕 ; ( i) T isμ-almost everywhere continuous, and measurepreserving・
(ii) There exist two measures μl and μ2 such that μ(Å) > O i㎡piies μ1い ) > O and ʻμ2(/1) > 0 , if /1 1s almost clopen relative toμ, μl and μ2.
(iii) There exist a sequence of almost clopen sets 坑 C χ with finite measure relativetoμ, μʼl and μ2 such that χ - U 隋 is a null set。
1
(iv)There exists a sequence of positive numders pnsuch thatthe quasi-miχing relation 11m A μい∩ T¯゛B) = μ1は) μ2(B)
Dep. 可 Math. F(1c. (λf Gene. Educ、 G面 UI面 . ( Received Sep., 30, 1976)
Quasi-E砲odicity of Transformations
on a T opologica1M easure Space with infinite M easures.
by shizuo T oM A T su .
(1)
instead of ( 1) ( quasi-mixing relation) in the definition of quasi- mixing. W e shaU ca117 1sgu si・e可 odic if it satisfies the above conditions (i), (ii) , (iii) and the relation (2) instead of ( 1) in (iv) .
W e first show the concept of quasi-ergodicity is weaker than that of quasi-miχing as the relation be- (2)
holds for an /1 and jEj included in some Hi which are almost clopen relative to μ, μl and μ2, W e introduce a concept weaker than that of quasi- miχing putting the relation
11m脊FIμ(/1nTB) ゜μ1(/1)μ 2召 ) …………
lim上 EI
四 71 1x4) ρ,μい∩ r 冶) = μ1い ) μ2(召)
PROR:g nONI . lI T is a qu(1si-・ 包流g tTan850T・ ation, then T 18 d 80 qua81-eTgodic.
PR(X:)F: lf there exisis a sequence { pj such that limp。Q(/1∩ f B) = μ1(j4) μ2( 召) , then
2 Shizuo T omatsu
L et j ) : = min { p j and ,ρ,r* = max { p j , then
o≦i ≦n- 1 0 ≧i ≧跨- 1
で瓦 石 4∩T ¯̀B } ≦ 十 ゛Σ ρ ・μ い∩T ¯ʻB)≦ ヅ*Σ μ (ノ 1∩rj)・
Thus, there eχists a positive number pSl such that
-yΣ ρ ,μ (j4∩7¯咀 )=ヴΣ μ (/1 ∩7・¯哨).
Consequently, we have
lim £ Σμ(j4∩ 7 一咀) = μ1い) μ2(£).
四 7z
Nextweshow thequasi-ergodicity hastheproperty tbata weak senseofnoeχistence ofnor卜
trivial invariant set.
P ROR:£ mON 2. £ d 7 6e α71 1nびerl泌Xe gzxαj ・ergoぷ c I γα7zべ/ormali叩 , 1yletz l加 ree冠j ls 710 71071・nuH T.inuaTiant 8et 切ith Jinite ・ e(18uTe.
lim& Σ し(/11) -2d = limp,ふu(4 ) -2d< X4(4 ) μ2(/11).
yx→|ʼ 7zl i≒4)| ” ” ʼ t4&jブ ・ ¨ ʼ ” ”
Since it is clear that μ(轟 ) ≧ μ(轟 ∩ 7 -浅 ) , hence
limp。{μ(轟) [≧μ1(瓜) μ2(轟)。 ダ
N ow we suppoes μG4) < O and put A;→ (x) , then μ1( j4) > 0, μ2(/1) > O and l兜 西 , = μ1い ) μ2(/1) /μ(声 T hen, in this case the quasi-ergodicity relation is descrided to the following ,form;
lim上Σ μ(£ n r -? 〉 = μ1(£) μ2(司 μい) /μ1い) μ2い) ………・(3)
Hj) ゜ O,there exist a positive integer k。 10r any positive number E such thllt μG4- /11) < E for
anyinteger k> k。。 Sincet ismeasurepreserving, henceμ (Tʼ y 1-T弓41) > Efor any positive
integer 7x. lt is clear from 71-invariantne卵 of /1 that μい - 7 已 4 ) = 0. T hus, we have μ(竃 - 7弓 11)
< 2£ for any positive integers 7x and ん ( ん> ん。), i. e. ,
μい1) - μ(轟∩ 7ʼ 瓦 ) = μ(瓜 - r ¯‰ ) < 2ε。
0 n the other hand, by the quasi-ergodicity relations (2) we have
霧- 1
1 汝jtミ
Consequentry,
1
P R(χ)FI L et r be a 71-invariant set with finite measure. L et A1 = j n (U H, ) then, since μ(χ -
i = l
∞
ツ
1 シ E !j ・χμ (4∩TBk)¬μ 1(j4゛)μ 2(日 ̀)μ (洵/μ 1(j4)μ 2(/1)
lt implieS μ1( 瓜 ) μ2( 瓦 ) = 0. 0 n the other hand, μ( 4 ) > O and μ( 瓦 ) > O implies μ1( j41) > O and μ2(玖) > 0. This is a contradiction. Therefore, μい ) = Oor (x).
P RoPosm oN 3. 1n the 8p(lce 切ith jk ite me(1suTe the qua81- eTgodicity impne8 the eTgodicity、
P RooF. L et (χ, S8 ,・μ) be a space with finite measure and let 7ʼ be a quasi-ergodic transformation on X , then as in the proof of proposition 2 we have ,
I匯-1ム χ瓜鳳∩戸几)か(又 )=肖凪)肖(凡)/肖(X)几(X)
for だ> ん。; where £ , F ES and 民 = £ ∩ (謳 ), 瓦 = Fn ( 心 Hi)
N ow , the uniqueりess of nleasure on χ implies there eχists the nleasure μosuch that μ。( X ) = 1, and
。 リ 恐 ¯ か 回μ 。(几nT ¯IFk )二 μ 。(鳳)μ 。(瓦)・
Consequently, at ん→ (x) we have
1 匯-yχ ) 。(£∩T¯ʻF)=几(£)心F).
T his shows that T is ergodic on χ,
ln the infinite case the quasi- ergodicity not necessarily follow s ergodicity.
P RoPosITIoN 4. Let T be (琲 流りeTtU) te quasi- e呵 o心c μ・n syormαぴ071 0 ( X , 亀 μ) . U T Xs ergd i c,
疏a Σ び,il = (x).
n= 1
P RooF. lf we suppoes Σ ρ;;1< (x) and l眺 H be a clopen set with non-null measure. T he quasi-
n= 1
ergodicity of t fonows the relation.
Since μい ) < (x), hence there cχists a positive integer A:l such that μ(j 1) > O and μ(玖 ) > O for A:> A;1. Since 召 is 711nvariant, thus 竃 and T -̀Bk are disjoint, i. e. , μ(竃 ∩ 7 -1Bk) = 0. F rom the relation (3) we have
for any clopen sets £ and F in sonie 坊 . L et 召 = X - j4 and 瓦 = 召∩ (白 瓦 )
aΣ宍
T hus we have
(G (Hnア¯ʻ劫) < 4μ 1(拍 μ 2(珀E fo r7 1> n 1………(4)
ことμ (亙 ∩T¯j劫> (デ トト幽 十 ……十 去)
>-yμ (ご田∩ず ¯̀拍).
上 至1μ(召n 竹山 ≦2 μ1(恥 μ2(拍 pj
71 μi)
for any integer 7z greater than suitable positive integer l 。. T he condition 勾 j < oo implies the followings; for any positive number E there eχists a suitable positive integer tzl(711> 71. ) such that
呈 上 11μ 田∩T¯j ) /2μ1(珀 μ2(槌< E for 71> 7x1.
m = n 7 7 1 1 = 0
1n the other hand, we have
Σ 瓜 H ∩ T¯̀出
every point of 召 is infinitely recurrent, and this means that 召=
for every j . By (4) μ(珀 = 0, it is a contradiction. Hence Σpj = (x)・
lt is not known that the converse of proposition 4 1s hold whenever for the quasi-miχing trans-
formations〔6〕. Butwe can show thatthefollowing relation.
P ROPoSlTr【oN 5. Let T be (lfl intJertible qu(181- eTgodic tTan8JoT・ (ltion on (χ, S , μ) .
」ぴ{ 言ぶ1 =(x ), X ん atm sX6ee 7go ぷ c .
O田nT¯j ) almosteve rywhe re
4 Shizuo T omatsu
N ow, since t is ergodic, hence χ is non-atmic and t must de conservative 〔1〕. But then alm ost
P nooE lf there eχists a r -invariant set yl of S such that O< μ(j4) . L et 召= 刄 j and
八= 召∩(心延). The n,bythequasi-e rgodic ityo I T wehave
-とΣ μ (瓦∩7万い≧̀jEμ 1(民)μ 2(瓦), 7 1≧7 1 .
for sufficiently large positive integer 71.. lt is clear that μ(玖 ) ≧ μ(瓦 ∩ T ¯ʻ瓦 ) for any integer i, thus we have
A;. 0 n the other hand, the facts that μ( 瓦 ) , μ1( 瓦 ) and μ2( 瓦 ) ʼare finite and μ( 瓦 ) > O implies μ1(八 ) > Oand μ2(玖 ) > O follow the relation; μ1(瓦 ) = Oor μ2(瓦 ) = O implies μ(瓦 ) = O for any
£ Therefore, μ(召) = 佃1y j(民) = 0. Thisshowsthatt isergodic.
ReferenCeS
1. P.R . H almos: LectuTe 071 eTgodic 伍eo巧 . M ath.Soc. of Japan, 1956.
2. S. K akutani and W . Parry: 垣 /池 Xe meo zxre preser㎡昭 XΓα覗 yor771alions 辰 1yx “μ& i,ぶ ʼ Bu11. A mer. M ath. Soc., 69, 752-756 (1963).
3. K . K rickeberg M ischende TTansJoT・ atione71 a可 M aTtnighJd t砥keiten unendlichen M asses. z. W a/1r. Ge&・, 7; 235-
247 (1967). ◇
4. K . K rickeberg: SIro昭 琲滅 昭 pr叩 er臨 s q/̀M a戌 oz, d a泌s 耀晶 17!yi砲 e measzxre. Proc. F ifth Berkly Symp. M ath.
Stat. Prob., 1966, vol. 1, Part H, 431-446.
5. K . K rickeberg: Recen! 7ʼes㎡ls 071 恨漉171g 171 10pologic㎡ measure 卯 ace. L ecture N otes in M ath. vo1. S , Springer v・, 1969.
6. F . P apan加 10v: & ro昭 7・α6 amμs, 沢- recr ra ce α冠 m滅 昭 F 叩 e戒 es q/゛ぷscrele pαΓa77zd er M arんou processes.
z . W a h r . G e b . , 7 , 2 3 5 - 2 4 7 ( 1 9 6 7 ) .
(m→ (x)), hence μ1(瓦 ) = Oor μ2(民 ) = O for any
μ (瓦) ≧言μ 1(Bk)μ 2(Bk )4ʼ!
Consequently,
μ (民) ≧主μ1(玖) μ2(玖) 主旨”祐1
2 m 。 、
Since m is not depend to ん・
