ConcerningthestrongorbitequivalenceofaCantorminimalsystem,wecan askthefollowingquestion・Withinanystrongorbitequivalenceclass,isthere aminimalsubshift？In{S31,weobtainthe駒llowingresult：FbreveryCantor minimalsystemitsstrongorbitequivalenceclasscontainsminima

within a strong orbit equivalence class

Fumiaki Sugisaki *

118 P.Sugisaki

thetopologicalsettingwehaveasimilarresult,thatis,theconceptsofstrong o r b i t e q u i v a l e n c e a n d t o p o l o g i c a l e n t r o p y a r e i n d e p e n d e n t ( [ S 1 ] , [ S 2 ] ) .

ConcerningthestrongorbitequivalenceofaCantorminimalsystem,wecan askthefollowingquestion・Withinanystrongorbitequivalenceclass,isthere aminimalsubshift？In{S31,weobtainthe駒llowingresult：FbreveryCantor minimalsystemitsstrongorbitequivalenceclasscontainsminimalsubshiftof allfinitetopologicalentropies・Inthispaperwegeneralizethisresultusingthe c o n c e p t o f t o p o l o g i c a l p r e s s u r e b y t h e f o l l o w i n g ( T h e o r e m 1 . 1 ) . F o r a t o p o l o g i c a l dynamicalsystem(X,T),denoteMIX)bythesetofBorelprobabilitymeasures onXandルi(X,T)bythesetofT‑invariantBorelprobabilitymeasuresonX.

LetC{X,K)denotethesetofallrealvaluedcontinuousfunctions.

Theorem1.1Supposethat(X,4>)isaCantorm伽加αIsystema冗〃eC(X,R),

e x p I s u p < ／ 抑 ^{r e M { x , 4 > ) X ¥ ≦ α ≦ ｡ 。 ( 1 . 1 ）}

α 仇髄.me測娩ereexistsaCa冗加rminimalsystem(Y,tp)stronglyorbitequiv‑

alentto(X,$)sueﾉithat

m/0‑*)=ioga,

ﾉ e7℃W‑)お〃ietopologicalpressure〃妙α泥de:X‑→Yisasti℃noorbit

Weremarkthatifノニ0,then1<a<ooandP(V>,0)isthetopologicalentropy o f t p . S o T h e o r e m 1 . 1 i s t h e g e n e r a l i z a t i o n o f [ S I ] , [ S 2 ] a n d [ S 3 ] . W e a l s o r e m a r k that(1.1)isthebestpossibleinequalitywhichacantake・Thereasonisthe following.Giordano,PutnamandSkaushowedthatan(strong)orbitequivalence map0:X→Ygivesabijection$:M(Y,if>)‑→M(X,<I>)definedby9(is)=〃。β ( T h e o r e m 2 . 2 i n [ G P S ] ) . U s i n g t h i s f a c t a n d t h e v a r i a t i o n a l p r i n c i p l e o f t o p o l o g i c a l pressure(seeTheorem9.10in[Wl]),wehave

P 他 ' ｡ ' ‑ り 言 愚 " { M 州 ル ‑ * d i / I i ; e M { Y , i l > ) ¥

> s u p ￨ / f o 9 ‑ * d v ￨ u e M { Y ^ ) ¥

= s u p U f d 9 ( v ) ￨ u e M ( Y ^ ) ¥

＝ " { 〃 " ￨ 似 E ﾙ * . * ) } .

Nowwegiveanoverviewofeachsectionbelow.Inthissectionbelowweintro‑

ducesomenotations,definitionsandconditionsconcerningBrattelidiagrams.In

§2,weconsidertherelationbetweenCantorminimalsystemsandsubshifts.We

willshowthatwheneveraproperlyorderedBrattelidiagramBsatisfiesProperty

1 . 5 , t h e n t h e a s s o c i a t e d B r a t t e l i ‑ V e r s h i k s y s t e m ( X z , 入 慮 ) i s t o p o l o g i c a l l y c o n j u ‑ gatetoasubshift(Theorem2.4).In3,wecalculateatopologicalpressureofa specialcaseofCantorminimalsystem・ByTheorem3.8,weonlycalculateapres‑

sureofasubshiftassociatedwithaBsatisfyingProperty1.5.In4,weintroduce twomodificationpropositionsofdiagramwhichpreservetheequivalencerelation onBrattelidiagrams.InProposition4.2weconstructabasedBrattelidiagram CusingagivendiagramB.InProposition4.5weconstructthedesireddiagram Bof(Y,ip)inTheorem1.1usingabaseddiagramC.Thesepropositionsplay

Notation1．2Basically,weusenotationsanddefinitionsin[HPS]and(GPSl.

SupposeB=(K,E,≧)isaproperlyordered(alsocalledsimplyordered)Bratteli diagram.SupposeAisasetandIAl(or#^4)denotesthecardinalityofA.

(1)Letr:E→Vdenotetherangemapands:E‑→Vdenotethesourcemap.

Namely,eE且､connectsbetweens(e)E脇‑,andγ(e)E賂．

( 2 ) L e t M ( ' ) = [ # r ‑ n w ) n s ‑ n v ル e v h , u E v h ‑ 』 d e n o t e t h e‑ t h i n c i d e n c e m a t r i x o f

B ( i ､ e ､ , M 無 ) i s t h e n u m b e r o f e d g e s c o n n e c t i n g b e t w e e n u E V h a n d t ﾉ G V , ‑ i ) . W e a l s o w r i t e B = { V , E , { M ^ " ) } , ≧ ) . L e t M か ) ＝ 1 M 蝿 ) 1 秒 E , , ; ､ ̲ , d e n o t e t h e u ' s

rowvectorofM^"^whichiscalleda冗如cidencevectorofu.For≧k,let j V / ( n , f c ) d e n o t e t h e p r o d u c t o f i n c i d e n c e m a t r i c e s M ^ ^ ^ M ^− i ) . . . M W . (3)SetXb={(e,)ieNICiGEi,r{ei)=s(ei+i)ViGN}.Wecallitthe伽一

伽茄elengtfisﾉpatﾉispaceofB.Forv^Vn,letP(v)denotethesetofall (finitelengths)pathsconnectingbetweenthetopvertexvqeVnandv.Then

l P ( U ) ￨ ＝ M 卿 6 1 ) h o l d s , P u t P ( 1 / ; ､ ) ＝ U o E V h P ( ひ ) . T h e r a n g e m a p r i s e x t e n d e d

toViVn),thatis,forp=(d.…,e")EP(脇),wedefiner(p)＝『(e")．

( 4 ) F o r X = ( e i ) i 6 N G X ^ o r a ; = ( e i , … , e n ) e V { V n ) , p n t z l . j 1 = ( c i . e t + i . … , e , ) andX(i,j￨=(ci+i,…,ej)．FbrpEP(略),se巾}B＝{zEXBlzI1,"l＝p}・

Wecallitthecylindersetofp.

(5)ForVGKiandeGr^{v),letOrder(e)denotetheorderofeinr^(v).If Pmin=(ei,e2,…)istheuniqueminimalpathinXb,thenOrder(cn)=1 forallnGN.IfPmax=(/i,/2,…)istheuniquemaximalpathinXb,then O r d e r ( ん ) ＝ ￨ r − 1 r ( / h ) ￨ f b r a l lE N S i m i l a r l y , O r d e r ( ･ ) i s d e f i n e d o n P ( 脇 ) ． I.e.,forpGV{Vn),Order(p)istheorderofpinVMp)).

(6)ForVGKi,wewriter^{v)={cj￨1≦j≦¥r‑^{v)lOrder(ei)=i}.Define

L i s t ( t ; ) = ( s ( e i ) , s ( e 2 ) , … . 5 ( e ￨ r ‑ i ( . ) ￨ ) ) G ( V n ‑ l ) " ' ‑ ' ( ' ' > "

WecallittheorderIistofv.

120 F・Sugisaki

(7)Foramonotoneincreasingsequence{tn}nez+CZ+withto=0,wesaythat

a B r a t t e l i d i a g r a m B ' = ( V , E ' , { M ' * " H ) i s a t e l e s c o p i n g ( o r c o n t r a c t i o n ) o l B t o { t " } , w h i c h w e w r i t e B ' ＝ ( B , { t " } ) , i f V ' " ＝ V La n d M ' ( " ) ＝ M ( t " ， t " − 1 + ' ） L e t ＆ " , f , ､ ̲ ､ + , ＝ { 麺 (" ̲ , ," l l z E X B } . T h e n t h e r e i s a b i j e c t i o n b e t w e e n E ' a n d E" ," ̲ , + , p r e s e r v i n g s o u r c e a n d r a n g e v e r t i c e s ・ W e c a l l { t n } a s e q u e n c e oftelescopingdep"蹄.Especially,wedefineBoddastelescopingBtoodddepths

｛ 0 , 1 , 3 , … } a n d d e f i n e S e v e n a s t e l e s c o p i n g B t o e v e n d e p t h s { 0 , 2 , 4 , … } ． (8)Let(Xb,入b)denotetheBratteli‑VershiksystemofB.Namelyi入B:XB‑→Xb

isalexicographictransformationdefinedbytheorder≧onE.

(9)ForBrattelidiagramsBandB',defineB〜B'providedthatthereexistsa BrattelidiagramBsuchthatBoddyieldsatelescopingeitherBorB',and

Remark1.3

(1)Let(X,T)denoteaCantorminimalsystem,C(X,Z)thesetofallinteger valuedcontinuousfunctions,C{X,Z)+={/GC(X,Z)¥ノ≧0}andBt=

｛ノー/or‑*IノEC(X,Z)}､Define

K(X,T)=C(X,Z)/Bt,K(X,T)+=C{X,Z)+/Bt.

In[Pu],Putnamshowedthatthetriple(K(X,T),K(X,T)+Al))isasim‑

pie,acyclic(i.e.K(X,T)^Z)dimensiongroupwiththe(canonicaldistin‑

guished)orderunit[1],where1=l*istheconstantfunction1・Herman, PutnamandSkaushowedin[HPS]thatthefamilyofCantorminimalsystems

showedthatKR(X,T)=Ko(V,E)(=meanstwodimensiongroupsareunital o r d e r i s o m o r p h i c ) , w h e r e ( V , E ) i s a B r a t t e l i ‑ V e r s h i k r e p r e s e n t a t i o n o f ( X , T ) andKo(V,E)isdefinedbytheinductlimitofasystemoforderedgroups

Kb(V;E)＝lim(ZlVh‑1l,Mh)＝Z1％l坐ZlVil坐Z￨雌￨些…．

Theyalsoshowedthatall(acyclic)simpledimensiongroupscanbeobtained inthis(dynamical)way.

(2)Itiseasytoseethat(V,E)〜(V',E')ifandonlyifKo(V,E)SKo(V',E).

Giordano,PutnamandSkaushowedin[GPS]thatBratteli‑Vershiksystems (Xb,,Xb,)and(Xb,,A^)arestronglyorbitequivalentifandonlyifBi〜Bo.

Definition1.4

(1)(distinctorderlist.)WesayK,hasdistincto㎡erhstsiffortﾉ,ひ′E脇，

L i s t ( v ) = L i s t ( i / ) i m p l i e s v = v ' ( o r e q u i v a l e n t l y , ひ ≠ t ﾉ ' i m p l i e s L i s t ( t ﾉ ) ≠ L i s t ( v ' ) ) .

(2)(Theminimal/maximalvertexproperty.)SupposeB=(V,E,≧)isaproperly orderedBrattelidiagram.WesayEnhastheminimal/maximalvertexproperty

i f t h e r e e x i s t U 耐 ↑ ひ 湿 e V n ‑ i s u c h t h a t f o r a n y e , ノ G E n w i t h O r d e r ( e ) = 1 andOrder(ﾉ)=￨r‑V(/)￨,thens(e)="肘ands(/)=t侭．

NowweconsideraproperlyorderedBrattelidiagramBofProperty1.5.Laterwe w i l l s h o w t h a t t h e a s s o c i a t e d B r a t t e l i ‑ V e r s h i k s y s t e m { X ^ , X s ) i s c o n j u g a t e t o a s u b s h i f t a n d i t s t o p o l o g i c a l p r e s s u r e i s c a l c u l a b l e .

P r o p e r t y 1 . 5 B = ( K , E , { M ^ } , > ) s a t i s f i e s t h e f o l l o w i n g p r o p e r t i e s . F o r a n y

neN,

( 1 ) M ^ i s a p o s i t i v e m a t r i x ( i . e . M i " J ≧ 1 f o r a l l u a n d v ) 、

（2）且､hastheminimal/maximalvertexproperty,

（ 3 ） ￨ 脇 ￨ ≧ 3 a n d u 品 i n ≠ ひ 品 a x , w h e r e t ﾉ 品 i n a n d U 品 a x a r e d e f i n e d i n D e f i n i t i o n 1 . 4 ( 2 ) ，

( 4 ) f b r e a c h 〃 E 典 , 〃 " l ̲＝ 〃 ( " 1 ‑ , ＝ 1 ，

U U m i n U U m n x

(5)Vnhasdistinctorderlists.(Inthecaseofn=1,weignorethisproperty.) 2．ConjugacybetweenCantorminimalsystemsandsubshifts

I n t h i s s e c t i o n w e c o n s i d e r a B s a t i s f y i n g P r o p e r t y 1 . 5 . W e w i l l s h o w t h a t ( X g , 入 圃 ） i s t o p o l o g i c a l l y c o n j u g a t e t o a s u b s h i f t . T h e d e t a i l s o f s h i f t s p a c e s a n d i t s t o p o l o g y , see[LMJin1and6.

Definition2､1

(1)Let(X,(t)denoteasubshift,thatis,Xisashiftspaceandaisshifttransfer‑

mation.ForxGXandi,j6Zwithi≧j,set

^M=^i^i+l…坊'^(M) XiZi+l…巧一'，

whicharecalledblocks(orwords)ofx.Set

Bn(X)={x<o,n)¥xeX},B(X)=Un^Bn(X).

SinceXisshiftinvariant,weseethatB"(X)＝{zI)｜zEX,j−j＝冗}and henceBn(X)isthesetofall(length)n‑blocksthatoccurinpointsinX.We callB(X)仇eIα u eQ/X､FbrBEB"(X)andj,jwithj−j＋1＝ ,put

m={xeX¥xuj,=B}.

122 F.Sugisaki

(2)(shiftoffinitetype)LetAbeanalphabet(afiniteset)andFbeasetof w o r d s w i t h a l p h a b e t A 、 F b r F , d e f i n e X ァ t o b e t h e s u b s e t o f s e q u e n c e s i n A z whichdonotcontainanywordinT.Wesayasubshift(X,<t)iss血峨of伽髄e type(SF乃ifXhastheformXtforsomeFandFisafiniteset.Wesayan SFTsubshift(Xｧ,cr)isM‑step(MGN)ifFconsistsofblockswithlength M+l.Wesayasubshift(X,a)isin℃ducibleifforanyu,wGB(X),there existsvGB(X)suchthatuvwGB(X).

Remark2､2SupposeXCA*.ByTheorem6.1.21in[LM],

Xisashiftspaceiff3J"suchthatX=XtiffXisshift‑invariantandcompact.

Definition2.3(SubshiftassociatedwithB)SupposeB=(V,E,≧)isa properlyorderedBrattelidiagram・Letr:X*U(UieNW))‑→V(Vi)denote atruncationmap,thatis,γz=xiwherex=(xi,X2,…)．

( 1 ) D e f i n e a s h i f t i n v a r i a n t s u b s e t X o q C V i Y i f " t o b e X ｡ ｡ ＝ { ( 丁 入 息 " z ) " e z l z E X g ｝

OnecanshowthatXqoiscompact.Let(Toodenotetherestrictionofshiftto

Aoo‑

(2)DefineafinitedirectedgraphGk=(V,)arisingfromVCVk)asfollows.Define aedgesetE=V{Vﾙ)andavertexsetV={i{p),t(p)¥pG},wherei(p) ( t ( p ) , r e s p . ) i s t h e i n i t i a l ( t e r m i n a l , r e s p . ) v e r t e x o f p s a t i s f y i n g t h a t

p9E8，t(p)＝i(9)iff

ItiseasytoseethatGkisairreduciblegraph.LetXkdenotetheedgeshift X<VI‑e‑,

X k ＝ X c 脂 ＝ { z ＝ ( z f ) i E z E P ( 唾 ) z l t ( : r) ＝ j ( z 愈 + , ) f b r a l l j E Z } ． (See[LM]:Definition2.2.5.)Let&kdenotetheshiftonXk・Itiseasytosee that(Xk,ケた)isa1‑stepshiftoffinitetype.DefineXk=^k(Xk),wherethe mapTTfc:Xk‑→V(Vi)*isdefinedby

**:(…X‑i.XoXl…)=(…(γお‑l).(γ〃o){rxi)…)．

LetびんdenotetheshiftonXk‑

F i r s t w e c o n s i d e r t h e r e l a t i o n s h i p b e t w e e n { X g , 入 直 ) a n d { X ･ ･ , ぴ ｡ ｡ ) ．

( 2 . 1 ）

T h e o r e m 2 . 4 S u p p o s e B = ( V , E , を ) i s a p i ℃ p e 伽 o r d e r e d B r a t t e l i d i a g r a m s a t ‑ 蛾ﾉ畑Property1.5.Then(X*,入)istopologica"yconjugateto(Xoo,ぴ｡｡)．

Proof.WewriteA=入0forshort.Definettoo:X息→Xooas 汀｡｡垂＝(γ入"s)nez.

Wewillshowthat汀ooisaconjugacy.ClearlytTooissurjective､TTooO入＝グーo汀函

holdsbecause

( T o o 入 勿 ) n = γ 入 " 入 勿 ＝ γ 入 " " a ; = ( 7 『 ｡ ｡ z ) n + l = ( C o c T l o o z ) 沌 ．

ThereforewewillshowthatTTooisinjective.Wecalltheargumentbelowtheone‑

to‑oneα噸秘ment.

Theone‑to‑oneargument､Chooseanyz＝(垂i),y＝(眺)EX息withz≠yandfix them・Itsu田cestoshowthatthereismEZsothat丁入mz≠丁入"VIfrz≠γり，

theclaimwouldhavebeenproven.Thereforeassumethatthereis/>1sothat

お 【 1 , 1 ] = y [ i , i ) a n d a : [ i , / + i j ≠ " [ 1 , 1 + 1 ] ( X [ 1 M = ( ^ l > ^ 2 , … , { ) ) . S u p p o s e n < 0 i s t h e m a x i m u m n u m b e r s o t h a t ( 入 " 麺 ) 1 , , 1 + , } l i e s i n t h e m i n i m a l p a t h i n P ( γ ( 毎 【 + i ) ) . T h i s i m p l i e s t h a t O r d e r ( ( 入 " a O n ‑ i ) = 1 a n d ( 入 " 毎 ) [ M ‑ 2 , ｡ ｡ ) = a : [ / + 2 , o o ) ‑ T h e n w e c o n s i d e r

t h e f o l l o w i n g t w o c a s e s :

( i ) ( 入 " a O [ i , i ] = ( A " y ) i i , i ] ,

()(^x)lU]≠(入"!ﾉ)m‑

Inthecaseof(i),wenotethatOrder((¥"y)i+i)=1becauser(入.x)i=r(入"y)i=

vi..andProperty1.5(2)and(4)．Letu=r(入"x)i+iandv=r(¥"y)t+i.

T h e n 秘 ≠ v b e c a u s e o f x n f + i ] ≠ y ¥ i , i + i ] ‑ S i n c e V J + i h a s d i s t i n c t o r d e r l i s t s , thereexisteEr‑1(私)，ノEr−1(ひ）andtheminimumnumberl＜ ′＜

m i n ( ￨ r ‑ * ( u ) ￨ , ￨ r* ( u ) ￨ ) s u c h t h a t s ( e ) ≠ * ( / ) a n d O r d e r ( e ) = O r d e r ( ﾉ ) = n ' .

L e t n = n + Y Z L i N ^ ( * ( c i ) ) l w h e r e e * G r ‑ i ( w ) w i t h O r d e r ( e i ) =T h e n s ( 入 " ) / + ! = s ( e ) a n d s ( 入 免 り ) i + i = s ( f ) . T h i s i m p l i e s t h a t ( 入 * z ) m ≠ ( 入 免 Z ﾉ ) m ‑

B o t h t h e c a S e ( i ) a n d ( i i ) i m p l y t h a t t h e r e e x i s t s Ⅳ E Z s u c h t h a t ( 入 " z ) [ M # ( 入 j ^ 2 / ) [ U ] h o l d s . B y r e p e a t i n g t h i s p r o c e d u r e , w e g e t T 入 ､ 〃 ≠ 丁 入 m y f b r s o m e

meZ.Sowefinishtheproof.

Definition2.5ForvGV¥Vq,definewords(orblocks)Con(v)andrCon(v)as C o n ( t ; ) = P 1 P 2 … P ¥ v ( v ) ¥ > r C o n ( v ) = ( t p i ) { t p 2 ) … { t P ¥ v ( v ) ¥ ) ,

where{pi￨Order(pi)=i,1≦j≦¥vm=v(v).

124 P.Sugisaki

Remark2.6UsingCon(･)andTCon(･),weseethat

昨{…〃'ｮ{僅嚇ご芸蒐鰯皇君:制 蕊‑{…〃'ｮ琶鵜鰯繁謡酬鵡}

S o f J J f f c , け た ) a n d ( X k , ぴ k ) a r e r e n e w a l s y s t e m s w i t h t h e g e n e r a t i n g l i s t { C o n () ￨ U E V k h { r C o n ( v ) I t ﾉ 6 1 4 } r e s p e c t i v e l y ( s e e [ L M ] , 1 3 . 1 ) .

Weconsidertherelationshipbetween(Xk^ak)and(Xk,<7k)‑Thefollowingtheo‑

r e m i s i m p o r t a n t s o a s t o c a l c u l a t e t h e t o p o l o g i c a l p r e s s u r e o f { X q , 入 息 ）

Theorem2､7SupposeB=(V,E,≧)isaprope吻orderedBrattelidiagramsat‑

isfyingProperty1.5.ThenforanyA;GN,(Xk,ケ上）α九diXk,<7k)α犯topologicα"y c o n j u g a t e .

Proof.WewillshowthatthemapTr/tisaconjugacy.ClearlyTrjtissurjective andTTkoo'k=o"fco汀齢Sowewillshowthatifkisinjective.Supposex=

M,z'=(x'i)GXksatisfiesthatz≠z'andxoissomeminimalpathinV(Vk).

Ifra;o≠丁勿'0,thenwehavebeendone.ThereforeweassumeTz0=Tx'n.Then thereexist{n,},{'i}CZand{vi},{v'i}CVjtsuchthatforanyiGZ,

a ^ I n i , n , + o = C o n ( v i ) , x ' [ n ' i , n ' i + , ) = C o n ( v ' i ) , n o = 0 ,' 0 ≦ 0 <' ル

Here,letusconsiderthefollowingthreecases:

( i ) ひ ' 0 ≠ 妙 0 ，

(ii)v'o=vqandn'o≠0, (iii)v'o=Vqand'0=0.

I n t h e c a s e o f ( i ) a n d ( i i ) , t h e r e e x i s t s / w i t h I < l < k s u c h t h a t ( x o ) n , / ) = ( a : ' o ) [ i , i ] a n d ( a ; o ) [ i , / + i ) ≠ { x ' o ) l i , i + i ] ． S o w e u s e t h e o n e ‑ t o ‑ o n e a r g u m e n t i n T h e o r e m 2 . 4 andobtainrx^≠Tx'mforsomem.Inthecaseof(iii),byz≠x'thereexists

/GNsuchthat

ofbranyjwithljl＜I,Uf＝u'f(therefbrei＝'iholds),

I f V / ≠ v ' i , b y P r o p e r t y 1 . 5 ( 2 ) , { x n , ) ¥ i , k ] a n d { x ' n j ) u k ] a r e t h e m i n i m a l p a t h i n V i v i ) a n d V { v ' i ) r e s p e c t i v e l y ( a n d h e n c e { x n i ) [ i , k ‑ i ] = ( a ; ' n / ) ( i . f c ‑ i ) ) a n d { X n , ) [ l , k ] ≠ （ 毎 ' n / ) [ i , f c ] ‑ S o u s i n g t h e o n e ‑ t o ‑ o n e a r g u m e n t i n T h e o r e m 2 . 4 , w e

haveTzm≠丁勿'mforsomem・Uv‑i≠v'‑i,basicallybythesameargumentwe

Inthecasewherexoisnotsomeminimalpath,wemayconsidersomeminimal pathXninsteadofxq.Thereforewehaveaconclusionthat汀kisinjective.

3．Calculationoftopologicalpressure

T h e a i m o f t h i s s e c t i o n i s t o c a l c u l a t e t h e t o p o l o g i c a l p r e s s u r e o f a B r a t t e l i ‑ V e r s h i k s y s t e m i n a s p e c i a l c a s e . F i r s t w e i n t r o d u c e t h e d e f i n i t i o n o f t o p o l o g i c a l p r e s s u r e . T h e d e t a i l s o f d e f i n i t i o n s a n d n o t a t i o n s a r e w r i t t e n i n [ W l l .

3.1.Definitionsandpropertiesoftopologicalpressure

Definition3.1Let{X,T)beatopologicaldynamicalsystem.(I.e.Xisa compactmetricspaceandTisacontinuoustransformationonX.)ForノE

C{X,R)andneN,put低ノル)=E封/CTx).Fore>0,put

｡川｡1−"{悪鱈州………forA"I,

Q ( r ,, 倉 ) = ' 慨 p : ' ・ g Q 凧 ( r ,, ) ,

p(r,/)=iimQ(r,ﾉ,ey

ThenitiseasytoseethatP(T,f)existsbutcouldbeoo・ThemapP(T,．）：

CiX,R)‑→RU{00}iscalledthetopologicalpressu泥ofT.

WhenTisanexpansivehomeomorphism,wecancalculateP(T,f)asthefollowing

{A"}足‑..ofmembersofa,thesetn墨‑..T‑"A"containsatmostonepointof

X.Define

剛加‑"{遷恕｡'州伽"……璽門｝

Theorem3.2([Wl]:Lemma9.3,Theorem9.6)LetTbeanexpansive

P ( 叩 = 蝿 : l o g , 蝿 ( r , 八 α ) = 磯 寿 ¥ o g P N { T J , a l

Inthecaseofasubshift(X,ぴ)withalphabetA,α＝{Ial8laEA}isgenerator

fora.Moreoverweseethat

。V封。‑iα＝{{Bl8llBEB鯉(X)}andhenceV封｡*Qisafinitecoverof

X,

。 s i n c e ｛ ( B 1 8 − 1 l B E B 沌 ( X ) ｝ i s a d i s j o i n t f i n i t e c o v e r ( i ､ e , B ≠ B ' i m p l i e s [ B ] J " ^ n [ B %^ = 0 ) , i t h a s n o p r o p e r s u b c o v e r .

SobyTheorem3.2wehavethefollowing.

126 F・Sugisaki

Proposition3．3Supposethat(X,a)isasubsh蛾α ノEC(X,R)jspote7l"α

ん泥c虎on.Then

帥ﾙ典器唾(ふ雫鮮'…）

‐磯寿1噸(､黒副雫淵州訓）

3．2.TopologicalpressureofBratteli‑Vershiksystems

InthissubsectionweassumethatBsatisfiesProperty1.5.Firstwecalculatethe

t o p o l o g i c a l p r e s s u r e o f ( X k , ケ ん ) w i t h r e s p e c t t o s o m e s p e c i a l p o t e n t i a l f u n c t i o n s . Definition3.4SupposeBisaproperlyorderedBrattelidiagram・Wesaythat/

i s a s i m 此 ん 冗 c t i o n o n X q b a s e d o n V { V n ) i f f o r a n y x , 〃 ' E X B w i t h z I 1 , " l ＝ z ' 1 , , " 1 ， /(*)=ﾉ(毎')holds,ThenfbrpEP(Wjwecandefineノ{plB＝ノ(z)ifzEIplB．

Remark3.5Sinceeachcylinderset[pisisaclopenset,/isacontinuousfunc‑

tion.

Fbr9EC(X9,R)andkEN,let9kdenoteasimplefUnctionbasedonP(砿）

satisfyinglimん一・.Ofc=gasthesupremumnorm.Wedefineacontinuousfunction OfconXktobe

§ k { x ) = g k [ x o U ,

where〃＝(z")EXkandhence§kisasimplefimctiononXk．

Lemma3.6Inthes"秘α虎o αbove,uﾉehave P(けた,5k)=logafc,

whei℃αkistﾉiemaximumpos""esolutiono/仇eequα"o冗/brz9"enby

二淵−1，

UE脇

whereTOu)=exp

Proof.ByTheorem2.7,(A*,or*)is1‑stepirreducibleSFT.LetAbetheadja‑

cencymatrixofthegraphGkdefinedby

4，．妻{;:闘総』，ル

LetDbeadiagonalmatrixdefinedbyD＝e9脂{PlB．PutS＝AD・Let入S＝

max{￨A￨:入isaneigenvalueofS}.AisanirreduciblematrixandsoisS.Then

u s i n g L e m m a 4 . 7 i n [ W 2 ] , w e h a v e P ( ケ k , 5 / b ) = l o g 入 s a n d t h e r e e x i s t s a n e i g e n v a l u e AsuchthatAs=入Nowwewillshow入s=αk.ByPerron‑FrebeniusTheorem ( S e e [ W l ] : p l 6 , T h e o r e m 0 . 1 6 . ) , A s i s a n e i g e n v a l u e a n d i t s e i g e n v e c t o r i s p o s i t i v e .

L e t O b e t h e r i g h t e i g e n v e co r o f 入 s , W e w r i t e e a s e ＝ ( 8 " ) E R r) 1 , w h e r e

8 ツ ー ( 8 p ) p E p ( ｡ ) . ( T 一 人 s ) 0 = 0 f o l l o w s t h a t

。 一 入 s 0 p + e t o M t O a = 0 , w h e r e r ( p ) = r ( q ) a n d O r d e r ( p ) + 1 = O r d e r ( q ) .

● 一 入 s 8 p + , e * * k k 0 = 0 , w h e r e O r d e r ( p ) = p ( r ( p ) ) ￨ a n d q i s t a k e n o v e r

Orderfa)=1.

Theseareequivalentto

。 8 , ＝ 入 : 侭 d " ( p )l e x p ( ‑ E 9 〃 1 9 ) 0 , , w h e r e 『 ( p ) ＝ 『 ( 9 ) , O r d e r ( 9 ) ＝ 1 a n d

p ' i s t a k e n o v e r p ' E P ( 『 ( p ) ) w i t h l ＜ O r d e r ( p ' ) ≦ O r d e r ( p ) .

● E , e " ' ' 1 息 8 ｡ ＝ 入 r ( 『 ( ， ) ) l e x p ( ‑ E p , 9 k { p ' } g ) 8 , , w h e r e O r d e r ( p ) ＝ 1 , 9 i s

takenoverOrder(g)=1andp'istakenoverp'GVirCp))with1<

O r d e r ( p ' ) ≦ O r d e r ( p ) .

Thenwehave

鰻言撰;蓋言淵峠三八黙；器普)鴎‑雲P(L>9kW]8)"

wherep'istakenoverp'EP(γ(p))withl＜Order(p')≦Order(p)and9'istaken overq'6VMq))with1<Order(q')≦Orderfq).Soweha八'e

忌畿=1.

Sowefinishtheproof.

Lemma3､7

P(グ｡｡Pot)=KmP{ok,9ko師F')(3.1)

Proof.Firstwewillshow

n^=x･･･

kEN

FbrkENandUE脇十,,thewordTCon(ひ)correspondstoconcatenatedwords rCon(wi)rCon(u2)…γCO､(秘")，

where(wi,U2,…,u"）＝List(u)．ThenbyRemark2､6X1コ麺．…and

xexooifffbranynEN，thewordz(−m,lappearsinapointofX｡｡，

128 F・Sugisaki

w e w i l l s h o w z I ‑ " , " l E B 2 " + 1 ( X b . ) . I t s u 髄 c e s t o s h o w t h a t z I ‑ " , " l a p p e a r s i n TCon(v)forsomevertexv.SupposethatNGNsatisfiesmin{￨P(t;)￨￨vG15v}>

2 .Form≧Ⅳ,DefineAm,BmandCmas

A m = { v G V m ￨ X [ ‑ n , n ] a p p e a r s i n T C o n ( v ) } ,

m={(u,v)G晩×豚n￨Z(‑n,n]appearsinaconcatenai Cm={(u,v)e賂×賂IrCon(u)rCon(v)GB(Xm+i)}‑

appearsinaconcatenatedwordTCon(w)TCon(v)},

SincexGn^tiXk,BmnCm≠0holdsforanym≧Ⅳ.SupposeAm=0for anym.If(u,v)GB^DCvwith(u,v)≠(ひ脇>^min)>thenthereexistwGVWi ande,/Gr*(w)suchthats(e)=u,s(/)=tﾉandOrder(/)=Order(e)+1.

ButthisimpliesthatwGAn+iandhenceBnHCn={(ひ船^min)}'NOW,for anyyGV)v+2,TCon(y)containsthewordrCon(u^)rCon(〃船).Becauseby Property1.5(2),anyconcatenatedwordTCon(ti)rCon(v)withu,vGVWicon‑

t a i n s r C o n ( ひ 総 J r C o n ^ J a n d r C o n ( t ﾉ ) c o n s i s t s o f c o n c a t e n a t i o n s o f r C o n ( t y ) ' s

(wGVn+i).ThereforeyGi4Ar+2holdsandhenceitisacontradiction.Therefore

" m ≠ O f b r s o m e m a n d : E ( ̲ " , " l a p p e a r s i n γ C o n ( ひ ) f b r s o m e v e r t e x U ． DefineﾉinGC{Xq,R)basedonP(Ki)tobe

ノin{x)=max{0(y)￨yG[pis)if^G[pis‑

Thenweseethatlim"一｡｡IIﾉIn‑pi1=0andhencelimfc‑｡o¥¥hn‑gn¥¥=0.By Theorem9､7(iv)in[Wl]

¥ P ( < r k , 9 k O T T . ^ ) ‑ P ( < 7 f c , h k o 汀 億 ' ) l ≦ I I ﾉ i k ‑ 9 k ¥ ¥ → 0 a s f c ‑ → C O ・

Thereforewewillshow

P(｡｡｡,5.曙)=limP(。雌,内晦｡汀‑')

Clearly

P f a , ん l ) ≧ P { ^ 2 ^ 2 0 7 r r ' ) ≧ P i e r s , h z ． 汀 亙 ' ) ≧ … ≧ P ( ぴ ｡ ｡ . 5 . 汀 三 ）

becauseXiコX2コ…。Xooandﾉinismonotonedecreasingwithrespectton.

ByProposition3､3foranye>0,chooseiVsatisfying

詩'･噂(､ふ!雫岬嶋…)<'ぃ｡噌吟

ByHfegN‑Xfe=‑Xoo,thereexistsKGNsuchthatforanyk≧KandxGX*,

Bn{Xo｡)=BviXk)andhk(x)<q{x)+‑e.

UsingProposition3.3again,foranyk≧Kwehave

岬…崎･(ふ釧雫緋…､"）

〈か"(・ふ雲鮮……鵬）

<P(｡｡｡,9.汀三)+e.

Theorem3､8SupposethatB=(V,E,≧)isaproperlyorderedBrattelidiagram sα"wmgProperty1.5,gisapotentialんnctiononX*and{g^}isasapienceof simpleん冗ctionsonXbasedonV(Vn)/'oreacﾉZsα"吻inglimn一｡｡Il5‑ffn￨￨=0.

Supposeanistheuniquepositiveso此"o加Q/仇eequα"o /orxgivenby

二淵……‑．麺(為州）

α冗dliman=aexists.ThenP(A,ff)=logα・

Proof.ByTheorem2.4,入8andぴ｡oareconjugateandhenceP(Ag,g)=P(ぴ｡｡,9 t t ‑ M . B y T h e o r e m 2 . 7 , ケ & a n d o ‑ f c a r e c o n j u g a t e a n d h e n c e P ( ケ k , 9 k ) = P { < r k , 9 k o t i y * ) . T h e r e f o r e b y L e m m a 3 . 6 a n d 3 . 7 w e h a v e

P ( 入 B ' 9 ) = l i m P ( 。 臆 , § k o 派 F * ) = l i m P ( ケ 態 , p f c ) = U r n l o g o ! * = l o g a .

4．ThemodificationofsimpleBrattelidiagrampreservingequivalence

relation

Inthissection,wegivetwomodificationsofdiagramspreservingtheequivalence

relationofBrattelidiagrams(seeNotation1.2(9))．Thefirstmodificationis

g i v e n s i m p l e B r a t t e l i d i a g r a m B = { V , E , { M ^ } ) a n d a s e q u e n c e o f t e l e s c o p i n g d e p t h s { t " } " E z + , C ＝ ( W ; R { ﾉ V ( " H ) i s c o n s t r u c t e d b y t h e f o l l o w i n g : ( W e c a l l t h e constructionbelowt旅Uertexam吻oma加冗.)

ThevertexamalgamationconstructionofC.Defineanequivalencerelation

onverticesof(B,{})as

u〜U(u,ひE脇"）−

^ifn=0,

Usingthisequivalencerelation,weconstructWby

Wn=VtJ〜．

130 F.Sugisaki

F b r 鯵 E W h ‑ 1 a n dE W h , d e f i n e ﾉ V f c > a s

j 噸 ＝ Z M 総 '蝿 ‑+ ' ) ， w h e r e u E " ・

vE韮

(Inthecaseof ＝1,weputuoEt"owhereWb＝{tuo},妬＝{Uo}.）Notethat

Remark4､1

( 1 ) W e g i v e a n e x a m p l e o f ( s t a t i o n a r y ) B r a t t e l i d i a g r a m s s a t i s f y i n g t h e c o n d i t i o n s above､FbranyEN,sett"＝泥,脇＝{1,2,3,4,5,6}andWh＝{,,t"2,"3}・

IncidencematricesM^andN^aredefinedby

"卿‑ルー側"御‑職}M釧一{湖陀，

Thenweseethat1,2ewi,3,5Guﾉ2and4，6E 3

(2)Inthisexample,W2≠^3butN&}=N$

P r o p o s i t i o n 4 . 2 S u p p o s e B = ( V , E , { M ^ } ) i s a s i m p l e B r a t t e l i d i a g r a m a 卸 。 {t"｝jSasee冗ceq/telescop卿｡es"jSﾉｼ町thα"M(t","−1+')もα花DOS‑

""em 7､ices・SupposeCistﾉiediagramconstructedabove.The冗娩e/'ollowing

statementshold:

(1)/brα ENa sEN,＃{tUEWhルー'(")￨≦s}<2*, (2)foranyvGuﾉGW,¥V(v)¥=¥V(uﾉ)¥,

(3)forany0≦r＜1,仇e花函sjsKENsuch仇E"EwhrlP(u')￨＜1ﾉorα〃

≧K,

(4)5〜c.

Proof.(1)Since

｛ M M t " ，" − 1 + ' ) ￨ u E v I " } 亡

｛偲釧置側鵬帳l好弓見叫Wn‑1+1…｝

and

（州伽賃"}言(見附…"￨州）

wehave

細川『Ⅷ‑州{(蝿測'峰釧￨好十（

Thenwehave

8

拶 { " e 鵬 ' ' ' 一 l ( " ) ' ≦ s ] ≦ D { ¥ V t n Z l ¥ ‑ l ) = (

j＝￨暁､‑,1

w h e r e w e u s e d t h e f b r m u l a ( 圏 ) ＝ ( ¥ ) − ( " ？ ' ) ．

S

￨ 脇 九 一 , ) 〈 2 ， ，

s − 1

1畷‑ll‑l

(2)Inthecaseofn=1,¥V(v)¥=M^=A#&,=¥P(w)¥holdsforany

uE EWi、SupposethatfbranyuEzEWh‑,,￨P(u)￨＝￨P(毎)lholds・

ThenfbruEwEWh,wehave

州琴｡昂州峨…‑雪見(雲州叫…）

=Ei^)i^"‑^=Epwi*無ノ=¥V(w)¥.

諺 E W f B ‑ , ひ E エ エE叱り−1

(3)Putp隅＝min{￨P(z)￨｜zEWh−,}・BythesimplicityofBand(2)above,it iseasytoseethatp^i",ismonotoneincreasingwithrespectton.Using(1),

wehave

逗 蔽 ' 戸 ( " ) ' ≦ E 『 , 湾 ' 癖 ‑( ｡ ) ' < 量 『 , 渦 ｡ × 2 ｡ ＝ ' _{，−2γP調→0} 『 ' 綱

山 E 1 " ｳ 8 m E W h s ＝ 1

( 4 ) W e w i l l c o n s t r u c t a B r a t t e l i d i a g r a m 8 ＝ ( ' , 白 , { 血 ( " ) } ) s o t h a t a v e n c o r r e ‐ s p o n d s t o ( B , { t " } ) a n d 8 b d d c o r r e s p o n d s t o C ・ F b rE N , w e p u t 喝 郷 ‑ 1 = W n , 喝 " ＝ V I " a n d d e f i n e t h e i n c i d e n c e m a t r i x 血 ( " ) a s

ハ " 卿 − 1 ) ＝ M M り , t " − 0 + ' ) ， w h e r e u E ⑩ ，

必瀞)＝珊，where6鮒＝1if妙E ,and噸＝Oift 任切・

WewillcheckthatM^^‑D=M<*‑'‑+i>andM(2+i.2)=w(n+i)

( M < ) = N W = [ 1 ] f o r c o n v e n i e n c e ) .

j雌"'2"−1)＝E蝿靭>i雌ｰ')＝E噸M#"，蝿‑+'）（E"）

WZWn E眠り

^Wn‑l+l)(.,^jg切→Af(w‑+i)=M''‑'‑>),

）

132 F・Sugisaki

ﾊ " 鮮 + l , 2 n ) ̲ Y ^ A " 卿 + 1 ) 蝿 誓) ＝ Z 雌 … ，癌 十 ' ) 螺 ） （ # E " ）

秒 E V h T j E 暁 、

‑E蝿…'"+'>=*&+>(vtew).

UE垂

Remark4､3

( 1 ) I n t h e e x a m p l e o f R e m a r k 4 . 1 , M ^ i s t h e f o l l o w i n g .

( 2 )

血(1)＝ 胤仙‑￨蕊}"…=^23456

^(EN)．

SupposeBandCareBrattelidiagramssatisfyingProposition4.2.Thenthere

isanontomapの:E'‐→F,whereE'＝U経,Et風, "‑,+listheedgesetof { B , { t n } ) , s u c h t h a t

・の(Et",t"‑,+,)＝島，

・fbranyeEE',s(e)ES(の(e))andr(e)Eγ(の(e)),

ofbranyteweWn,のgivesabijectionbetween{eEEtn,t恥‑,+,｜γ(e)＝

v } a n d r ^ { u ﾉ ) ,

●fbranygERande,e'Eの−1(9),s(e)＝s(e')．

Thenwecandefineamapの^:V(Vu)‑→V(Wn)as

^ ( l . t n l 骨 の ( z 1 , , t 1 1 ) の ( z ( t 1 ， t 3 1 ) … の ( a ^ ( t n ‑ . . t n l ) ‑

Weseethat

(i)の"iss町ective,

(ii)therestrictedmapの¥‑p(y¥isabijectionbetweenP{v)andV{w)where

Vew,

( i i i ） f b r a n y p E P ( W h ) a n d z I 1 , t n l , z ' 1 , , t " l E ( の 沌 ) ‑ ' ( p ) , z 1 , , t " ‑ , 1 ＝ 毎 ' [ l . t n ‑ l l

holds.

Usingの"'s,wedefine<:Xq‑→Xras

や((z")"EN)＝(y")"EN等fbranyEN,の'"(3[l,u)=y(l,n]‑

Thenwecanshowthatやisbijective町thefbllowing・By(i)やissuIjective・

F o r a n y f i x e d y G X と , t h e n u m b e r o f p a t h s i n P ( Ⅵ ､ ) c o r r e s p o n d i n g t o y ( , , 宛 l v i a の( i ､ e ､ , ￨ ( の " )' ( y ( , , " l ) ￨ ) i s ＃ { U E 脇｜ ひ E r ( y " ) ｝ b e c a u s e o f ( i i ) . H o w e v e r ， b y ( i i i ) s o u r c e v e r t i c e s o f e a c h e d g e i n ＆ ､ + , , f " + , c o r r e s p o n d i n g t o Z ﾉ n + i v i a の a r e a s a m e v e r t e x ・ T h e r e f b r e c o n s i d e r i n g p r e i m a g e o f y I 1 , 冗 十 , l v i a の 沌 + i w e c a n c h o o s e u n i q u e l y t h e p a t h i n P ( V I n ） c o r r e s p o n d i n g t o g I 1 , " l v i a の n ． T h i s

means<pisinjective.

( 3 ) y > p r e s e r v e s t h e c o f i n a l r e l a t i o n . I . e ,

z≠x'6XqandVn≧tⅣ,z"＝z' 一V汎≧Ⅳ,P(〃) ＝や(釘').

Therefore,ifweassignanyproperorder≦B,≦conB,Crespectively,<pisan orbitequivalencemap・Moreoverif≦Band≦csatisfies^(x^in)=yminand v t e m a x ) = j / m a x . < p i s a s t r o n g o r b i t e q u i v a l e n c e m a p .

(4)SupposeノisasimplefimctiononXBbasedonP(VIn‑,).ThenfoP−isa simplefunctiononXcbutnotbasedonV(Wn‑i)ingeneral・Indeed,/ou>*

isbasedonVCWn‑i)ifandonlyif加}b=地'lBfbranyp,P'EP(VI"̲,）

withの"(p)=#"(p').However,focp'*isbasedonV(Wn).Weregard/asa simplefunctionbasedonV(V*)by

/(*)=/Iph,‑,i]8if^gbis.peWJ.

B y t h e c o n d i t i o n ( i i i ) , の * ( * [ l * . l ) = の( 毎 ' ￨ i , t ) ) i m p l i e s X [ i ^ . , ] = 毎 ' [ M n ‑ l l ‑

Therefore

ノ 。 や − 1 ( " ) ＝ 加 1 , ," ̲ , l l B i f 〃 E I の " ( P ) ] c

doesnotdependonachoiceofpEP(脇,､)andisasimplefunctionbasedon P ( W h ) ．

Hereweintroducethe"converse"constructionofthevertexamalgamation,which i s c a l l e d t h e i ﾉ e r t e x s p l i t t i n g .

ThevertexsplittingconstructionofB.SupposeC=(W,FJN^})isa s i m p l e B r a t t e l i d i a g r a m . S u p p o s e B = ( V , E , { M ^ } ) s a t i s f i e s

o脇＝U"Ewn脇,ujasdiSjointunionand脇,≠0,(I.e,,wesplitulintoM,"｜

v e r t i c e s i n K , . )

･fbrany ｡E鴎,",蝿飛=M),

・fbrt",zEWhwithl"≠z,then秘")≠蝿")fbruE1/handひE脇,z

・ f b r a n yE 鵬 , " , Z 秒 E , h ‑ , , 雪 j 嘘 ) ＝ j V 鯛 ．

Remark4､4Inthecaseofthevertexamalgamationconstruction,Cisuniquely determined.However,inthecaseofthevertexsplittingconstruction,thereisan ambiguityofanumberofverticesandhenceBisnotuniquelydetermined.

ー

Proposition4．5B〜c.

Proof.ThisfollowsProposition4､2byputtingB＝Band{t"｝＝Z+、

Remark4,6

134 F・Sugisaki

( 1 ) S u p p o s e B a n d C a r e s i m p l e B r a t t e l i d i a g r a m s s a t i s f y i n g t h e v e r t e x s p l i t t i n g construction.BysimilarargumentsofRemark4.3(2),weha八'eabijection の:蝿‑→Xcpreservingthecofinalrelation.SupposethatBandCaresimple BrattelidiagramssatisfyingthevertexamalgamationconstructionandBand B h a v e p r o p e r o r d e r s ≦ b , a n d ≦ 息 r e s p e c t i v e l y s a t i s f y i n g

や(Xmin)=ViXmin)and¥>(Xmax)=^(^max)‑

T h e n ( p * o i p i s a s t r o n g o r b i t e q u i v a l e n c e m a p b e t w e e n ( A s , 入 B ) a n d ( ^ . 入 息 ) . (2)Letの"：P(路)‑→VCWn)beanontomapwhichinducesaconjugacy<p(see Remark4.3(2))andﾉibeasimplefunctiononXcbasedonV(Wn).Thenwe

s e e t h a t f o r a n y i , 毎 ' E X g w i t h の " ( 毎 1 , , , 1 ) ＝ の * ( * i i . l > = 9 .

ん 。 ｡ ( 曇 ) ＝ ﾉ Z ｡ ｡ ( 毎 ' ) ＝ ﾉ i [ q } c ‑

〜

ThisimpliesthatfbranyU,u'E脇，

*｡伽u‑Eん｡伽]b=E%Jc.

p E P ( 秒 ） P E P ( U ' ） q E P ( 画 ）

5．ProofofTheorem1．1

5.1．RequirementsofasimpleBrattelidiagramfor(Y^ip).

ByTheorem9.7in[Wl],foratopologicaldynamicalsystem(X,T)andpotential function/6C(X,R),

h(T)+infノ≦P(TJ)≦ん(T)+supノ

andsoP(T,ﾉ)=ooiぼん(T)=oo.Inthecaseofa=oc,thereexistsaCantor m i n i m a l s y s t e m ( Y , t / j ) s t r o n g l y o r b i t e q u i v a l e n t t o ( X , < f > ) s u c h t h a t ﾉ i ( i p ) = o o ( s e e [ S 2 ] ) . T h i s m e a n s

W , / O = 0 .

Soweonlyconsiderthecasewhereaisfinite.LetB=(V,E,{M^H,≧)be aproperlyorderedBrattelidiagramwhichisarepresentationof(X,<p).Sowe identify(X,<}>)with(Xb,入b)‑Fromthesimplicityofdiagram,wemayassume t h a t a l l M ^ ' s a r e p o s i t i v e m a t r i c e s . W e o n l y c o n s i d e r w i t h i n a s t r o n g o r b i t equivalenceclassof(X,<f)).SoapplyingProposition4.2toB,wemayalsoassume

that

W,sEN,＃{ひE脇llr−1(U)￨≦s}≦2',

0≦Vr<1,3KeNs.t.Vn≧k,y>i^>i<i.

UE晩3

( 5 . 1 ）

( 5 . 2 ）

C h o o s e a n y d e c r e a s i n g s e q u e n c e { e n } n e N s a t i s f y i n g 0 < ^ n < n + i < j n

andfixit.NowwewillconstructaproperlyorderedBrattelidiagramB=

( V , E , { M M } , ≦ ) w h i c h i s a r e p r e s e n t a t i o n o f ( V , ^ ) . F i r s t , a p p l y i n g t h e v e r t e x amalgamationconstructionto(B,Un})where{,}issomesuitabletelescoping depths,wehaveabasedBrattelidiagramC=(W,剛Ⅳ<">})withC〜B(see Proposition4.2).Second,applyingthevertexsplittingconstructiontoC,wehave BwithB〜C(seeProposition4.5

5.2.Preliminary

Inthissubsection,wewillintroducesomelemmas.

Lemma5､1SupposethatN,A,QeN切鋤A≧3,REZ+and1<r<2satisfy

（1）Ⅳ−2=(A‑2)Q+Rand0≦R<A‑2,

(2){r‑l)Q>A,(2‑r)Q≧1and立二幾二型>1.

Zｿie冗仇e/'ollowingineq秘α"如加Ids.

￥j(n,)eN*‑*￨誉職‑"‑…．}ご(止苦塾‑r)"

P r o o f L e t { I i } 告 3 b e a s e t o f n o n ‑ n e g a t i v e i n t e g e r s w i t h J i 〈 ( γ − 1 ) Q . D e f i n e { " * } f a i * C N a s

。｡{鵜￨畳:→

T h e n w e c a n e a s i l y v e r i f y t h a t { " i } s a t i s f i e s E 省 2 " , ＝ L ‑ 2 a n d b y c o n d i t i o n

(2),1≦ t<rQholdsforeachi.Moreoveritiseasytocheckthatthemap (Jl.fe,…,^‑l)‑*(m,"2,…,n,4̲2)isinjective.Let[1denotetheGauss s y m b o l ( i . e . [ x ] i s t h e i n t e g e r p a r t o f x ) . S o w e g e t

籍JMgn^‑2i芸禰‐N‑…｡}量州倒‑>¥li<(r‑ﾘ。｝

≧ ( [ ( r ‑ り ・ . j j r f ‑ s ^ m r ‑ l ) ( N ‑ 2 ‑ R ) ^ y ‑ * ≧ ( { 止 二 幾 二 4 1 ）

≧ ( 竺 幾 二 型 − 1 ) " ‑ ( 止 二 幾 二 且 ‑ 『 ) "

Sowefinishtheproof.

A − 3

136 F.Sugisaki

Wewillusethefollowingnotations.

* < ‑ = 越E鴫一， m w , M < : ' = ￨ i ￨ .

L e m m a 5 . 2 S u p p o s e t h a t { V , E , { M ( " H ) i s a s i m p l e B r a t t e l i d i a g r a m w i t h p o s i ‑ tivematricesa冗dNeNisgiven.Thenther℃exists{culweViv‑i^**ﾉI0<Cu<1

sucﾉIthat

c 鯉 ≦ i n f { 蝿 W + ' ' ' ^ ) I V G V N + k , k G N } . （ 5 . 3 ）

蝿 』 V + / . . N ) < ￨ ￨ ^ ( i V ) ￨ ￨ x 蝿 N + k , N + i ) ^ h e r e ￨ ￨ M ( ^ ) ￨ ￨ = V M i i ' ) . ( 5 ‑ 4 )

皿,U

Also,thefollowinginequalityholds.

叫僻'")〉蝿"+k,N+l)×駅m/,y>(5.5)

F r o m ( 5 . 4 ) a n d ( 5 . 5 ) , w e g e t M i J ' ‑ ^ * ' ^ ^ > c w h e r e C u = ( m i n / g v ^ m / J ^ ^ ) / ￨ ￨ M ( ^ ) ￨ ￨ .

Itisclearthat0<Cu<1foralluGVn‑i.Therefore(5.3)holds.

L e m m a 5 . 3 F o 『 ｡ " n G N , ( ^ ) " < n ! < ( 2 a r ^ ' .

P r o ･ f . I f n = 1 , t h e i n e q u a l i t y h o l d s t r i v i a l l y . I f n ≧ 2 , t h e n e " = E r = o ^ > S ‑

Thereforetheleftpartoftheinequalityholds・Next,wecancalculate

1･創州一言1．帥慧獣+1･"雲州lo馴州‑(n+1).

Sincelog(n+1)>1for ≧2,weget

logn¥<{n+2)log(n+2)‑(n+1)一log(n+1)

< ( n + 2 ) { l o g ( n + 2 ) ‑ 1 } = l o g ( ( n + 2 ) / e ) " + ^ .

Sotherightpartoftheinequalityalsoholds.

LetノbeafunctionofXb・ForXeXbandmGN,put

S { f , x , m ) =

龍 ' ( い )

Lemma5,4SupposeB={V,E,≧)isaproperlyorderedBrattelidiagram,fis asimpleん抑c"･no"XB伽sedo"P(Wv)．Fbrα"yβ＞exp(sup{〃d似｜似E ﾙ！(XB,入B)}),仇e'℃existsⅣ'>NsueﾉIthatforany ≧Ⅳ'α九dvGK.,

β l P ( u ) ￨ ＞ e x p

ProofSupposethislemmaisfalse、Thenthereareinfinitelymany冗'sandU E Vj,sothat

Q ¥ ‑ P { V n ) ¥ ≦ e x p

今exp(logβ−S(ﾉ¥Xn,¥P{Vn)¥))≦1, where⑳"EXBisintheminimalpathofP(u").Define〃 EルUXb)as

,P(Wn)￨‑l

似聡=両可雲 …

C h o o s e s u b s e q u e n c e { r i i } s o t h a t { S ( / , 釘 ", ￨ P ( U ") ￨ ) ｝ i s c o n v e r g e n t a n d い " ， ｝ i s convergentintheweak*topologyon人4(Xb)‑Letfj.=limi‑ooHm.ByTheorem 6,9in{W11,weseethat似EM(XB,入B)and

， ( 加 瀬 " K ) l ) = 御 鋤 一 か U { i → ｡ ｡

Thereforewehave

。 " ( l o g β ‑ / 伽 ) ≦ ，

T h … " ｡ i " 鼠 β > e " ( " p { 〃 ' " e " ( x 職 入 圃 ' } )

5．3．TheconstructionofabaseddiagramC.

I f { t n } i s d e c i d e d , w e c a n c o n s t r u c t C b y t h e v e r t e x a m a l g a m a t i o n c o n s t r u c t i o n . Then,wedefinetp:Xb‑→XbasRemark4.3(2)andasimplefunction/nonXb basedonP(脇")as

A(z)＝min{/(y)lyE(plB｝wherezE{plBandpEP(脇")．

(SetV(Vo)=0and[%=Xb‑Thenﾉb(z)＝min{鮒)IvEXB}.)Weseethat

。｛ん}ismonotoneincreasingandlim"一｡｡IIノー九11=0,

●ん‑,oや−1isasimplefimctiononXbbasedonP(Wh)(Remark4､3(4))，

●fbranyUET〃EWh，

E 九 一 l b ' 1 , ,鰯 ̲ 1 , 1＝ E 九 一･ や ‑ ' { 9 1 c ^{（ 5 . 6 ）}

P E P ( U ） q E P ( w )

(SeeRemark4.3(4).Putpn01=0.)‑

138 F･Sugisaki

Definer"Mas

『蝿(tu)=exp勲伽‑ills㈹

wherevGwGWn‑Now,wewilldecide{tn}byinduction.

The1ststep・Putto=0.ApplyingLemma5､4toﾉoandB,thereexistsUGN

s a t i s f y i n g

(･告")卿釧'…,(易加[i.o]]･)…仰ﾙ州州州ル

(鵠)'診(訓>，

f o r a l l v G V i , . ( T h e s e c o n d p a r t o f i n e q u a l i t y a b o v e h o l d s b e c a u s e m i n v g v i [ P i t ﾉ ) l ismonotoneincreasingwithrespecttoti.)Wefixti.ThenwecanconstructWi

a n d N t o o f C b y t h e v e r t e x a m a l g a m a t i o n c o n s t r u c t i o n . S i n c e ￨ P ( i u ) ￨ = ￨ > ( v ) ￨ h o l d s f o r v G t o , t h e f i r s t p a r t o f i n e q u a l i t y a b o v e i s e q u i v a l e n t t o ( a + k e i ) >

TiHforanywGWi.Let{A^GN￨wGWAsatisfy

$ ' > 2 … { ( α 芸 普 型 ' M } ，

whereV*,^={vGVi,￨vGty}.Thenthereexistsauniquenumberai>a+Ei

suchthat

愚織黒=1.

Chooseanyen>ai−aandfixit.

Then‑thstep.For冗≧2,supposethe(丸−l)‑thstepdataaregivenbythe following:ForanywGWn‑i,

P n ‑ l ‑ l ) ( 蓋 芸 ﾗ ) ¥ n * > ) ¥ > 2 ,

( D " ‑ , ‑ 2 ) ( α ＋ e " ̲ , ) l P ( " ) l 〈 ( A− 1 ) − 2 ) r " ̲ 1 M ， ( D " ‑ , ‑ 3 ) ' 1 / I " ̲, u , ￨ 〈 A 断 − 1 ) − 2

C h o o s e r ^ G R s a t i s f y i n g ( 5 . 8 ) a n d f i x i t .

…"蝿(;(鴨畿誌￨叫）側

F b r a n y f i x e d t＞ t " ‑ , , w e c a n t e m p o r a r i l y c o n s t r u c t W h a n d Ⅳ ( " ) b y t h e v e r t e x amalgamationconstruction.DefineQxwGNandR垂､〃GZ+tobetheunique

numberssuchthat

jV期−2＝(A聯 ')−2)Q霊"＋R露 andO≦/U<4r'>‑2.(5.9)

DefineBx,CxwandD‑was

峠､…揺裂;綜赫"(剛‑劃晶噸）

伽‑{(蝿測'鵬…'￨悪緬鋤‑叫

｡"‑{峻州…'噌卿轄害卿‑"≦…善"｝

NowwewillshowthatClaim5､5holdsfbrsu田cientlylarget ．

Claim5､5Foγα九yzeWn,

（ ' ) r " { z } 〈 ( α ＋ ; g " ) ' P ( 錘 ) ' ， ( 2 ) B 毎 r " I z l ( α ＋ E " ̲ , ) ‑ ￨ P ( 露 ) l ＞ 1 ， (3)fora孔ywWn‑i,￨CxJ<U>ｪto;

（ 4 ） M 恋 , 諺 ￨ r " { 錘 } ＜ ( α ＋ g 河 ) I ｱ ( 麺 ) ￨ ，

( 5 ) E , 6 w r 誕 畿 碧 掃 計 < 1 ,

( 6 ) ( 釜 詩 3 ) l " < * > l > 2 .

(1)/n̲iisasimplefunctionbasedonP(昭八‑1).SoapplyingLemma5､4toA−1

（崎榊)順"伽…｡(〃帆山）

forallveV*.Thereforeby(5.7)and¥V(i)￨＝￨P(毎)lwhereUEz,wehave

『心'<(時雨¥l*<零川

( 2 ) S i n c e

恥'−．期｡"(晶州｡雌…割)…

≧ルP53fn‑2[q[l,t…I州>…

140 F･Sugisaki

Ⅱ （ r " ‑ , M ) 噸 ，

E脚ら､−，

＝＝

wehave

… i x ( ^ ‑ + 2 ) ( a 黒 等 ¥ V ( w ) ¥ J N鮒 島 r " { 麺 ｝

( α ＋ E " ‑ , ) l P ( ｴ ) 1

w h " e 島 ＝ { ( ( 鮒 ) ‑ 2 ) / e ) ， Ⅱ ． 直 w ｝̲ 伽 Q 壷 " + 2 ) / e ) 蝿" }' C h o o s o

asmallnumbere>0satisfying(5.10)andfixit(See(5.8)).

（ 鵜 劣 識 ' 一 霞 参 ^{（ 5 . 1 0 ）}

SinceminzEwhQz ‑→ooastn‑→CO,町(5.9)fbrasu缶cientlylarget"，

M票諾裂裂ｧ(｡,,参(鵜望満]̲二剛

FromLemma5,2,thereexists{c luE脇"̲,｝withO＜c ＜lsuchthat

蝿 # ，唾 ‑+ ' ) ≧ c of b r a n yE 1 / 1 , ､ ､ P u t 砥 W ＝ j V 4 3 / j W ) . F b r U E z , w e

have

* < > = ‑ 帳 謡 憲 異 亦 逗 篭 蝶 芸 巽 ， （ 5 . 1 2 ）

‑E*&"^‑1*"≧E趣三CIM.

uE 狸E

Then0<d*,<1and(5.12）isindependentoft andzEWh．As

l i m " → ｡ ｡ 丸 ' /＝ 1 ， i t f b l l o w s t h a t ( 凡 ) ' / j V A " ) ‐ → 1 a s t ‑ → o o . T h e r e f o r e b y

( 5 . 1 0 ) , ( 5 . 1 1 ) a n d ( 5 . 1 2 ) , t h e f o l l o w i n g i n e q u a l i t y h o l d s f o r s u f f i c i e n t l y l a r g e

t 兜 ：

側 " " 芦 品 ( ( 鵜 望 識 ' ‑ ) …

Finallyweget

砥W‑c'

( ( ｡ 黙 坐 修 , , ) 叩 " 参 鋤 瓢 ( 鵜 皇 満 ' ‑ ^＞1．

(3)Itiseasilyseenthat

lq"￨＝

１ト ー叩 の四町

Ⅳ恥

（ )<(卿)'脇緬‑"'一 ^●

^{( 5 . 1 3 ）}

andr,areconstant,

TwiUixw≧1and

wehave(『妙−1)Q霊〉A"'),(2−

＞1fbrsu価cientlylarget ．Therefbre byLemma5.1,weseethat

，｡雲",≧((軸器砦''一綴.)劉噸"･剛

Using(D"‑,‑3),weobtainthefbllowinginequalityfbrsu髄cientlylarget"：

，ﾙｰﾄ』<((筈鶴‑"‑輪)岬" 剛

B y ( 5 . 1 3 ) , ( 5 . 1 4 ) a n d ( 5 . 1 5 ) , w e g e t t h e i n e q u a l i t y o f ( 3 ) .

( 4 ) A s ￨ r ‑ i ( v ) ￨ = M i ' " ^ ≦ 耐 f 価 ' t 侭 ‑ l + ' ) ＝ l r − 1 ( z ) l h o l d s f b rE 畷 , ､ , a ; , b y u s i n g

( 5 . 1 ) w e h a 八 ' e

l v I " , 霊 ￨ ≦ ＃ { U E 1 / I " l l r − 1 ( U ) ￨ ≦ ¥ r ‑ H x ) ¥ } ≦ 2 ￨ r ‑ ( x ) ￨

By(D"‑,‑1)andClaim5､5(1),fbranyt ＞T(whereTisdefinedinthe proofofClaim5.5(1))wehave

（ α + 躍 ) ' 頓 に Ⅲ > r " 1 忽 ' " 黒 ̲ ( 無 ) ' 頓 州 ，

〉 r " I 露 l Ⅱ 2 1 v 期 ＝ r " ( z l 2 I r' ( 霊 ) 1 ．

W^Wn‑i

T h e r e f b r e w e g e t M " , " ￨ r " I 毎 1 ＜ ( α ＋ g " ) ￨ ｱ ( 垂 ) 1 ．

( 5 ) C h ･ o s e a n y n u m b e r 叩 w i t h 0 < り < ^ n ‑ i a n d f i x i t . P u t r = 窯 誓 . T h e n

O＜r＜1.By(5.2)fbrasu伍cientlylargetn，

Trl^'t")!<1.

UE暁冗

Clearly¥Wn¥≦IV*￨.ByProposition4.2(3),wehave Vrl^WI<1.

諺E砿、

S i n c e ¥ V ( x ) ¥ ‑ → 0 0 h o l d s a s t n → 0 0 , f o r s u f f i c i e n t l y l a r g e t n , 2 ( a + n ) ^ ｱ ( : = ) l <

（ α ＋ E " ＋ 叩 ) l P ( 韮 ) ￨ h o l d s ・ T h e r e f o r e w e h a v e t h e i n e q u a l i t y o f ( 5 ) .

(6)Sinceg"+，isindependentwithrespecttot andlP(z)｜‐→ooholdsas

t n → ｡ o , ( 釜 鴛 ﾅ 面 ) l ^ ( = = ) l > 2 f o l l o w s . ^□

142 F.Sugisaki

P u t t n s a t i s f y i n g C l a i m 5 ､ 5 . T h e n w e c a n d e f i n e A ^ G N a s

( A ' ) − 3 ) r 弧 { 砥 } ≦ ( α ＋ g " ) ￨ P ( 諺 ) ￨ 〈 ( A r ) − 2 ) r " { z } 〈 A # ) r " { 鰯 l 〈 2 ( α ＋ g " ) ￨ P ( 垂 ) ｜

（ 5 . 1 6 ） becauseofClaim5.5(1).Sowehavethen‑thstepdatabythefollowing:Forany

zEWh，

( D 狐 ‑ 1 ) ( 詳 窯 論 ) P ( * ) I > 2 ,

( D " ‑ 2 ) ( α ＋ g 禰 ) ￨ P ( 垂 ) ￨ 〈 ( A # ) − 2 ) r n I z l ， ( D 漉 ‑ 3 ) l V I 魂 , 垂 ￨ 〈 A r ) − 2 ．

5.4.TheconstructionofB.

I n t h i s s u b s e c t i o n w e w i l l c o n s t r u c t B = ( V " , E , { M ^ } , ≦ ) s a t i s f y i n g P r o p e r t y 1 . 5

andforeachnGN,

． +綱 く α " < … 卿 ‑ i a n d ェEWru T ) ! 為 踏 1 ‑ 1 . ^{（ 5 . 1 7 ）}

Theconstructionof脇.ForxGWn,weset

｜ 脇 , 露 ￨ ＝ 婚 ) ． ^{（ 5 . 1 8 ）} Bythecondition(D"‑3),Mu,￨≧3holds､Let＊EWh(**EWhresp)denotethe vertexsatisfyingthattheminimalpathXminXb(themaximalpathXmaxXg r e s p . ) g o e s t h r o u g h s o m e v e r t e x i n 畷 " , 掌 ( 脇 癒 , * * r e s p . ) . W e c a n c h o o s e a n y d i s t i n c t verticesu船inE脇,*andUHiaxE脇,態*(becauseofl脇,u,￨≧3)andfixthem.Then P r o p e r t y 1 . 5 ( 3 ) h o l d s .

TheconstructionofM^KInthecaseofn=1,forvGVi,wedefine

^VVn^WWa

whereVo={vq}andWq={wq}.Inthecaseof汎≧2,weconsiderthefollowing c o n d i t i o n s w i t h r e s p e c t t o M ^ :

( c . O ) I f x . x ' G W n w i t h x ≠ 錘 ' , t h e n M j ^ ≠ 蝿 ? ¥ w h e r e v G K , , x a n d v ' G V ? , 霧 , ． ( c . l ) F o r a n y v , U ' e 典 ｡ 窓 , 刷 施 ) ̲ 秘 ？ )

(c.2)ForanyvGVn,x,

（､ ) ) u E 鴎 ̲ , . " E D 諺 幽

whereD^^isdefinedby

〜

Dz山＝

and

SJuﾉ)=

(c.3)螺鼠,=蝋設=1ibrany"E脇

I t i s e a s y t o c o n s t r u c t M ^ " ' s a t i s f y i n g t h e c o n d i t i o n s ( c . l ) , ( c . 2 ) a n d ( c . 3 ) a n d

theseconditionsimplythatBsatisfiestheassumptionsofthevertexsplitting

constructionandProperty1.5(1),(4).Nowwewillshowthatwecanconstruct i t s a t i s f y i n g a l s o t h e c o n d i t i o n ( c ､ 0 ) .

S u p p o s e t h a t M ^ ^ ^ s a t i s f i e s o n l y t h e c o n d i t i o n s ( c . l ) , ( c . 2 ) a n d ( c . 3 ) . I t i s c l e a r t h a t i f i V i " ^ ≠ N i ' ' ¥ t h e n M i " > ≠" ) w h e r e Ⅷ E V ; 腿 , " a n d り E 鴎 , a ; . I n g e n e r a l , z ≠ z ' e W h d o e s n o t i m p l y j V 4 " ) ≠ n ! . ? ' ( s e e R e m a r k 4 . 1 ( 2 ) ) a n d s o w e w i l l s h o w t h a t f o r a n y x ≠ 懇 ' e W h w i t h 雌 " ) ＝ j V 4 ？ ) , w e c a n r e c o n s t r u c t 蝿 " ) a n d M ^ r ^ s a t i s f y i n g M ^ T ^ ≠ 蝿 ？ ) f b m e 魚 , 露 a n d' e V h , 錘 , B y t h e c o n s t r u c t i o n o f

i V < " ¥ w e s e e t h a t

＃{sewwvf)=雌)}≦Ⅱ￨C.,￨.(5.19)

⑩EW冗一，

A s M i " ^ a n d 必 『 ^ ' s a t i s f y t h e c o n d i t i o n ( c . 2 ) , b y C l a i m 5 . 5 ( 3 ) a n d ( 5 . 1 9 ) w e h a v e

拶{swwv:凧)=蝿")}≦ⅡP.tx.1≦Ⅱ¥DxJ.(5.20)

w E W h ‑ , u J E W h ‑ , ･

Therightpartoftheinequality(5.20)meanswhatthemaximumpossiblevalue f o r i n c i d e n c e v e c t o r s i n N ' ^ " ‑ i I s a t i s f y i n g t h e c o n d i t i o n ( c ､ 2 ) i s ・ T h e r e f o r e , w e c a n

c h o o s e i n c i d e n c e v e c t o r s s a t i s f y i n g M v ≠ 蝿 n )

Theconstructionof>.Wewillcheckthatwecanconstruct>on"withthe

propertythateachEnhastheminimal/maximalvertexproperty(Property1.5 (2))andeach脇hasdistinctorderlists(Property1．5(5))．FbrzEWh,define Distfo)6Nas

o帥'一(殻爵!Ⅱ豊論

whereUEVh,霊andV;:−，＝脇̲,/{U耐,ひ臓}･Distfo)meansthemaximal

possiblenumberoforderlistsof0E脇,露satisfyingProperty1．5(2)．Suppose m≠垂,秘GVnwandvGK*,*.Bythecondition(c.O),ifweassignanyorderon r * ‑ ( u ) , r V v ) r e s p e c t i v e l y , L i s t ( u ) ≠ L i s t ( v ) a l w a y s h o l d s . T h e r e f o r e V n c a n h a 八 ' e

distinctorderlistsifandonlyif

ー

Dist(x)≧脇灘

即切目胃呂の民具目︒胃昌＄ず兵員こ︶

園目ご葛①箸琶普ｏ言吾農吾のｏｇ８目口旨冒堅豊里①日︵ק蔀︶ず具〆邑８号牙切 言のｏｏｐＱ旨︒目印具弓彦の︒﹃の日﹄﹄ロの津冒の︒ ×国︲←×何画切罰の目色烏合②︵二・弓彦の口 金Ⅱｅ１房◎も ×国︲←×回一切伊切言◎眉◎３審２巳く巴の目︒①目四℃冨冒﹃の①目︵×も︶色目・

︵気色．ロの言の昏目昌目の書画且垣ｏｐｘ画画切

曾Ⅱデー﹄・もＩ﹄・写Ⅱデー戸ｏ︑岸秒且写Ⅱ︑ＯＳＩ﹄ｏやⅡ︑ｏ守再・

弓冨目寄一吻画の冒号昏冒§◎冒︒昌轟ｇｍ８ｏ園も︵宗︶色目冒昌一一浄 三一ⅡＰ旨︒園の︲

葛ｌ８

ｏぐの﹃︾切言ｏのデー﹄ＯＳＩ壷︑色切目豆の言昌昌◎口︒巨序ｇｍ８ｏｐも︵弓峯さ同色星Ｓｍ弓誹 的且ｅ︾ご応示・巳誤﹃の胃くの

弓冨葛︲吾爵己ｇｇｅ高︲里言︺吾の切言農︵Ｑ十句葛︶一寸︵圏︶一八一隷妻繍言菖亙．弓冨﹃の胃① 吾閏の①×厨房巨冒菖巨のＱ冨考毒壷Ｑ＋句菖八Ｑ菖八Ｑ＋ｍ葛Ｉ停め崖︒言計画喜

刻ｊ一宗雪箇弓菖宜．

ロ︵言ｂＨ賃一十画︶青︶息亀十噌一課︲﹄・ｇ−ｊ子１く目Ｉ垣

芦︾ｍ一室ゴー﹄

Ⅱ叩国噺弓三目一Ｖ岸

巽言の愚ごｍ宗間．︵重の宮︑の吾の胃言冨ご﹃富Ⅳ今︾言①口皇ＩいＶ︵也菖冒匡の．︶

弓匿の吊昏扇

豆呉臼︶Ｖ︵︒＋旬ご︶｜も︵周︶一県亙︲樟十四Ⅳ一宗言Ｈ一

ヶの○画旨のの︒︷︵函．﹄・︶四国且︵ｍ・樺の︶︒

弓昏①ａ彦の︒天︒由命・弓︶．国望宙・岳︶︾命・民︶騨口Ｑｏ盲冒︺働函︵９︶言の琴画くの

胃騨ご筒閣且富国︒の一雪の号の︒胃三ｍ旨の管巴辱．盟邑︒の尋︵劇︶切農恩の 吾の８且昼ｏ目

︵ｏ・里画冒且︵︒．ご雪巨の言︑ｇ巴目餌函宙︶騨目旦伊の日日秒画輿署の彦智くの

︵ロ異圏︶−号夢一圏一〃︵︵融冨︶︲画︶旦綴副︶︲噛

：急１重︲︲イ︲一寸︵Ｈ二 程心心

︵Ｑ十町葛︶言︵日︶一

同︒葛専Ⅱ門野亭ⅡＨ酢︲﹄・も︲﹈写

も︑寸言︶画ｍも︵己︑︶○ｍも︵９︶

V V

ロ︵︵量率︶＋暑価︶尋電十園︲

賃︑ぐ率！﹄

︵︵鼠菖︶ 四︶ヘ︒︶摸冒︾︲蝉

ｓ小訊忍︵Ｑ＋ｎコ−緯︶一℃︵卿︶一

図

卜Ｌ︵Ｑ己言︵間︶一

図

０

一宗・閣言冨亙

II

八房

×︵Ｑ＋旬再︶Ｉ一寸︵Ｈ︶一

×︵Ｑ＋阿冨Ｉ﹄︶Ｉ一寸︵制︶一

︵・画﹄︶

1■■

(seeRemark4､6(2)).Sowedefiner"(｡)(UEI4)as

r"(ひ)＝exp

ー

thenfortﾉE脇,",r"(ひ)＝r"Mbecauseof(5.21)．

followingconditions:ForeachnGN,

ThereforeBsatisfiesthe

("…蝿…‑剛愚鴇餅‑(言豊総硫‑m）

ー

( 2 ) B s a t i s f i e s P r o p e r t y 1 . 5 .

ApplyingTheorem3.8to5,wehave

P(沙,ノoO‑')＝P(入9,9)＝limlogα"＝logα・

F i n a l l y b y T h e o r e m 2 . 4 , ( Y , i p ) i s t o p o l o g i c a l l y c o n j u g a t e t o a s u b s h i f t .

References

[BH]M.BoyleandD.Handelman,EntropyVも応usOrbitEquivalenceForMinimal Homeomom航sms,PacificJ.Math､164(1994),1‑13

[ D y l ] H . D y e , O n g r o u p s o f m e a s u r ℃ p r e s e r v i n g t r a n s f o r m a t i o n s I , A m e r . J . M a t h 81(1959),119‑159

[ D y 2 ] H ・ D y e , O n g r o u p s o f m e a s u r e p r e s e r v i n g t r a n s f o r m α " os / / , A m e r . J . M a t h 85(1963),551‑576

[ G P S ] T . G i o r d a n o , I . F . P u t n a m a n d C . F . S k a u , T o p o l o g i c a l o r b i t e q u i v a l e n c e a n d C*crossed抑℃血cts,J.ReineAngew.Math.469(1995),51‑111

[HPS]R.H.Herman,I.F.PutnamandC.F.Skau,OrderedBrattelidia‑

grams,dimensiong7℃ups,andtopologicaldynamics,Internat．J.Math.3

（ 1 9 9 2 ) , 8 2 78 6 4

[ J e ] R . J e w e t t , T h e p 泥 砂 α l e n c e o f u n i q u e l y e r g o d i c s y s t e m s , J ・ M a t h . M e c h . 1 9

（ 1 9 6 9 / 1 9 7 0 ) , 7 1 77 2 9 ．

[ K r l ] W . K r i e g e r , 0 冗 冗 o n ‑ s i n g u j a r 加 冗 s f o r m a i i o n s o / α m e a s u r e s p a c e s , I , I I , Z . Wahrsch.Th.11(1969),83‑97;98‑119.

[Kr2]W・Krieger,Onergodicflowsandisomom航smoffac加汚,Math.Ann.223

（ 1 9 7 6 ) , 1 9 ‑ 7 0 ．

[LM]D.LindandB.Marcus,A冗伽troductiontoSymbolicDynamicsandCoding,

CambridgeUniversityPress,1995