Continuous maps of the circle with finitely many periodic points

Pub . Ma . UAB N° 25, Juny 1981 CONTINUOUS MAPS OF THE CIRCLE WITH FINITELY MANY PERIODIC POINTS Jaume Llib e Secció de Ma emá iques, Uni e si a Au bnoma de Ba celona, Bella e a, Ba celona, Spain . Rebu 1'1 de Juny del 1981 Abs ac . Le be a con inuous map o he ci cle in o i sel . The main pu pose o his pape is o s udy he p ope ieso he uns able mani oldassocia ed o a pe iodic poin o . Le 2( ) deno e he nonwande ing se o . Suppose has ini ely many pe iodic poin s . Then, using he uns able mani olds associa ed o pe iodicpoin s o , h ee heo ems a e p o ed p o iding comple e answe s o he ollowing h ee ques ions : (1) Which a e he possible pe iods o he pe iodic poin s o ? (2) Which is he alue o he opological en opy o ? (3) I 2( ) is ini e, which a e he poin so sl( )? §l . In oduc ion Le S 1 deno e he ci cle and CO(S 1 ,S 1 ) deno e he space o con inuous maps o S 1 in o i sel . Fo e CO(S 1 ,S 1 ) le O( ) deno e he nonwande ing se o , and le P( ) deno e he se o posi i e in ege s which occu as he pe iod o some pe iodic poin o . Ou main esul s a e Lije iulluwing (see §2 o de ini ions) : THEOREM A . Le e C O (S1 ,S 1 ) and suppose ha has ini ely manype iodic poin s . Then he e a e in ege s m >,1 and n >,O, such ha P( ) = {m,2m,4m, . . .,2nm} . THEOREM B . Le e C O (S 1 ,S 1) and suppose 2( ) is ini e . Then S2( ) is he se o pe iodic poin s o . THEOREM C . Lé e C O (S 1 ,S 1 ) and - suppose ha has ini ely manype iodic poin s . Then he opological en opy o is ze o . THEOREM D . Le e C 0 (S i ,S 1 ) . Suppose has ini ely many pe iodic poin s, and all pe iodic poin s o a e ixed poin s o . Then S2( ) is he se o ixed poin s o . A map eC0 (S 1 ,S 1 ) is a Mo se-Smale endomo phism o he ci cle i i sa is ies he ollowing p ope ies (see [3] o mo e de ails) : (1) is a con inuouslydi e en iable map . (2) q( ) is ini e . (3) All pe iodic poin s o a e hype bolic . (4) No singula i y o is e en ually pe iodic . Fo a Mo se-Smale endomo phism o he ci cle i was p o ed, by Block in [3] and [41, ha Theo ems A and B hold . Theo ems B,C and D we e p a con inuous map o a closed in e al in o i sel . The p oo s o Theo ems B and D can easily . an a bi a y in e al . Suppose Q( ) is ihi e, hen he o bi o any x e sa( ) is ini e . This implies ha x is e en ually pe iodic (i .e . some poin in he o bi o x is pe iodic) bu does no imply ha x is pe iodic .  I is possible o some e C 0 (S 1 , S1 )  o ha e poin s x e P( ) which a e e en ually pe iodic bu no pe iodic . In he p oo o Theo em B, we show ha his canno happen when Q( ) is ini e . We also no e ha o e C 0 (S 1 , S1 ), 2( )  may no be he closu e o he se o pe iodic poin s o . See [2] o an example : An example was gi en, by Block in [6], o a con inuous map , o a compac ,connec ed, me izable, one-dimensional space, o which 2( ) consis s o exac ly wo poin s, one o which is no . pe iodic . We conclude his sec ion wi h he ollowing heo em . THEOREM E (p o ed by Block in [4] ) . Le m and n be in ege s m >,1, n >,O . The e is a map e C O (S 1 ,S 1 ) such ha P( ) = {m,2m,4m,, 2 n m} . In ac , Block p o ed ha he e is a Mo se-Smale endomo phism o he ci cle wi h P( ) = {m,2m,4m_ . .,2 n m} o any in ege s m, 1 and n,0 . §2 . P elimina y de ini ions and esul s Le X be a opological space, and CO (X,X) deno e he se o con inuous maps o X in o i sel . Fo any posi i e in ege n, we de ine j~' induc i ely by 1 = and n = ° n-1 . Le ,? deno e he iden i y map . Le p e X .  A poin p  is cal led a ixedpoin o i (p) = p . Le Fix( ) deno e he se o ixed poin so . We say p is a pe iodic poin o , i p is a ixed poin o n o some posi i e in ege n . Le Pe ( ) deno e he se o pe iodic poin s o . I p is a pe iodic poin o , he smalles posi i e n wi h n (p) = p is called he pe iod o p . Le P( ) deno e he se o posi i e in ege s whichoccu as he pe iodo some pe iodic poin o ' . Fo any p e X we de ine he o bi o p by o b(p) ={ n (p) : n= 0,1,2, . . . } . The o bi o any pe iodic póin will be called a pe iodic o bi . We say a poin p e X is e en ually pe iodic i o b(p) is ini e (o equi alen ly i some elemen o o b(p) is pe iodic) . A poin p e X is said o be mande ing i o some neighbo hood V o p, n (V)n V= 0 o all n > 0 . The se o poin s which a e no wande ing is called he nonwande ing se and is deno ed 2( ) . Le X be a compac opological space . Fo e C0 (X,X) le en ( ) deno e he opological en opy o (see [1] o a de ini ion) . Le a and b be wo dis inc poin s o S 1 . We will use he no a ion (a,b) ( espec i ely [a,b]) o deno e he open ( espec i ely closed) a c om a coun e clockwise o b . Simila ly, we will de ine he a cs (a,b] and [a,b) . The poin a ( espec i ely b) is called he Ze ( espec i ely igh ) endpoin o he a c . Le X deno e an a bi a y in e al o he eal line . Le e C~(X,X) ( espec i ely e C0(S 1 ,S 1 )) and le p be a pe iodic poin o . We de ine he uns abZe mani old k?" (p, ) and one-sided uns abZe mani olds O (p, ,+) and O (p, ,-) as ollows . Le xeW ú (p, ) i o e e y neighbo hood V o p, x e n (V) o some posi i e in ege n . Le x e W u (p, ,+) i o e e y closed in e al ( espec i ely a c) K wi h le endpoin p, x e n (K) o some posi i e in ege n . Le x e W u (p, ,-) i o e e y closed in e al ( espec i ely a c) K wi h igh endpoin p, x e n (K) o some posi i e in ege n . In Lemma 1, we compile some p ope ies o he uns able mani old . See [6] o p oo s . Al hough p oo s a e gi en o a mapping o a closed in e al, hey can easily be modi ied o a mapping o he ci cle o o a mapping o an a bi a y in e al . LEMMA 1 . Le X be ei he an a bi a yin e al o he eal line o he ci cle, and le eC0 (X,X) . i)  Le p e Fix( ) . Then O(p, ), O(p, ,+) and S u (p, ,-) a e connec ed . Lé p e Pe ( ) . ii)  P1' (p, )  = AA~' (p, ,+) U k P (P, , -)  . iii) I p 1 = p and o b(p) = {P1, . . .,P n } ,  hen 0(p 1 , ) = ["' (P V . ) U . . . U''~''(pn, i ) (0 (p, )) = 0 (p, ) . )  Le J = 0 (p, ) and le J deno e he closu e o J . I he se J - J is nonemp y, hen any elemen o J - J is pe iodic . i) Suppose n( ) is ini e . Le x e s2( ) and suppose x 0 Pe ( ) . Then o some p e Pe ( ), he e exis s z e O (p, ) such ha (z) =p and z 0 Pe ( ) . LEMMA 2 . Le X be ei he an a bi a y in e alo he eal line o he ci cle . Suppose e C O (X,X) and (p 1 - . .,p n ) is a pe iodic o bi o . I (p i ) =pj,  hen (0(p i, )) _ [  (p j , ;  ) . F oo . Le x e W u (pi, n ) . We shall show ha (x) e W u (p J . , n ) . To p o e his, le V be any neighbo hood o pj . The e is a neighbo hood 1 , 1 o p i , wi h (W)cV . Now o some m> 0, x e nm (14) . Hence (x) e ( nm (W)) = nm ( (W))e nm (V) . Since V was a bi a y, (x) e W u(pj , n ) . Thisp o es ha (W u (pi, n ))cl . l u (pj, n ) . By enumbe ing we may assume ha (pi) = pi+1 o i= 1, . . .,n-1 and (p n )= p 1 . The e o e n (W u (P1, n )) c n-1 (W u (p 2 , n )) c . . . c (1J u (p n , n )) c W u (P 1 , n ) . By i ) o Lemma 1, we ha e ha n (W u (P 1 , n)) = W u (P 1 , n ) . Hence (W u (P n , n )) = W u (P 1 , n ).  O .E .D . The ollowing Lemma is a simple consequence o Bolzano's Theo em . LEMMA 3 . Le e C C (IR,R) . I K is a closed in e al . such ha K c (K), hen has a ixed poin in K . Le e C O (S 1 ,S 1 ) andle X be a subse o S 1 .  Le S 1 =1R / Z and le p : IR -- " S 1 be he na u al p ojec ion . Since p is a co e ing map, i g is he es ic ion o o X he e exis s a con inuous map g : X - IR such ha g = pog . F om now on o a gi en con inuous map g : X -} S 1 , g : X - 62 will deno e he con inuous map such ha g =p-g . The ollowing lemma ollows immedia ely om Lemma 3 . LEMMA 4 . Le e C 0 (S1 ,S 1 ) and suppose Kc S 1 is a closed a e such ha ei he Kc (K) and (K) ? S I o K c ?(K) . Since 5 1 = R/Z , we may assume K c (0,1) . Then has a ixed poin in K . §3 . Some esul s o e C O IS 1 ,S) wi h ini e pe iodic se We shall use he wo ollowing Lemas, which a e p o ed in [6] (see Lemma 6 and Theo em 7 o [6]) . LEMMA 5 . Le X be an a bi a y in e al o he eal Zine, and le e CO (X,X) . Suppose Pe ( ) is . ini e, and p e Fix( ) . Le x e  [, U (p, ) .  I x> p,  hen x e Py~ (p, , +) .  I x < p,  hen x e  kP (p, , -) . LEMMA 6 . Le X be an a bi a y in e al o he eal Zine, and le e CO (X,X) . Suppose Pe ( )is ini e, and p e Fix( ) . I x e [0 (p, ) and (x) = p,  hen x= p . By a pa i ion o S 1 , we mean a ini e se o poin s o S1 , {xl, . . .,xn} such ha o i= 1, . . .,n-1, (xi,xi+1)!1{xl, . . .,xn} =p . THEOREM 7 . Le e C 0 (S 1 ' 5 1 ) . Suppose Pe ( ) is ini e and { p1' . . . , P n } is a pe iodic o bi o wi h pe iodn >, 2 .  I 0 (pi, ) 4 S 1 and j ~¿ i,  henpj 0  0 (pi , , ) . P oo . Suppose pi and pJ a e dis inc elemen s o {pl, . . .,pn} wi h pJ e W u (pi, n ) . By Lemma 2, we ha e ha o each k = 1, . . .,n, Wu (pk, n ) con ains an elemen o {p l . . . . ,pn} -{pk}' By enumbe ing, we may assume ha {pl, . . .,pn) is a pa i ion o S 1 . By i) o Lemma 1, ei he p2 e W u (p l , n ) o p n e Wu (p l , n ) . Wi hou loss o gene ali y we can suppose ha p2 e Wu (p l , n ) . Le J =W u (p l , n )U W u (p2, n ). We sepa a e he p oo in o wo cases . Case 1 . J ~ S1 . The e o e J is a closed a c . By i ) o Lemma 1, n (J) =J . Le g be he es ic ion o n o J . Then Wu(p i , n ) = Wu(pi,g), o i=1,2 . O cou se, ei he p l eWu (P2,9) o p3 e W u (P2,9) Suppose p 1 eW u (P2,9) . By Lemma 5, P2 e W u (P 1 ,9>+) and p 1 e Wu(P2,9,-)- Since [pl,p2]e W u (P1,g), i ollows om Lemma 6, ha o all x e (P1,P2), g(x) belongs o some a c o he o m (p 1 ,y) . Because P2 e Wu(P1,9,+),  o some x e  (P1,p2),  9(x)=P2 .  Le z= in {x e_(P 1 ,P2) : g(x) =P2} . Then z e (P 1 ,P2) and g(z) = p2 . Le a e (p l ,z) . Then he o m  [b ,P 2 ] .  Since p 1 e W u (P2,g, - ) 0 . This implies ha p l e gm+1([a,z]) . con aining p 1 and P2, 9 m+1 ([a,z] ) :D[a,z] . By Lemma 4, 9 has a pe iodic poin in [a,z] . Since a was an a bi a y poin wi h a e (p 1 ,z), g has in ini elymany pe iodicpoin s . This is a con adic ion, and so p 1 ¢ W u (P 2 ,9) . Hence p3 e Wu (P2,g) . Tha is, p3 e-Wu(P2, n) . g([a,z]) con ains an a c o p 1 e g m ([b,P 2]) o some m > Since gm+1([a,z]) is an a c By he same a gumen , i ollows ha p i+1 e W u (p i , n ), o i=l_ . .,n-1, and p 1 e I-l u (p n , n ) . Then [pi,pi +1 ]cW u (Pi, n ), o i = 1, . . .,n-1, and [P n ,p 1 ]e W U (P n , n ) . By iii) o Lemma ha IJ u (pi, ) =S 1 , o i= 1, . . .,n, a con adic ion . Case 2 . J = S1 . Since WU (Pi, )~ S 1 , by iii) o Lemma IR . By i ) o Lemma 1, n (J) = J . Le h be o J . Then 14 u (p i , n ) =W u (pi,h), o i = 1,2, and he iden ic o he abo e case . Q .E .D . 1, we ha e 1, J is homeomo phic o he es ic ion o n p oo is LEMMA 8 . Le e C 0 (S I ,S I ) and Ze (p 1 ,- ,p n ) be o bi o wi h pe iod n > .2 . Suppose Pe ( ) is ini e i,,u(p1, )=S1 . I (p i , pj )(Í {p1, . . .,pn}=O'x  e(pi ,p j ) a pe iodic and and x 1 Pe ( ), hen ei he x e  o x e h (p~, n ) . P oo . Suppose x  14 u (pi, n ) and x 0 W u (p j , n ) . By ) o Lemma 1, x 0 W,(Pi, n) hecause x 0 Pe ( ) . The e o e W u (p i , n ) # S1 . By Lemma 2, W u (Pk, n )~ S 1 o k=l, . . .,n . Since W U (P 1 , ) = S1, by iii) o Lemma 1, x e Wu (Pk, n ) o some k e {1, . . .,n} - {i,j} . Le J= W u (Pk , n ) . By i ) o Lemma 1, n (J) = J . Le g be he es ic ion o n o J . - Then Wu( .Pk, n) = W u (pk,9) . By Lemma 5, ei he x e Wu (Pk,g,+) o x e Wu(Pk,g,-) . Wi hou loss o gene ali y we may assume ha x e Wu(Pk,g,+)=Wu(Pk, n,+) . Then p i e Wu(Pk, n,+) . Le m be he numbe o elemen so he pe iodic o bi {pl, . . .,Pn} con ained in W u(P k , n ). By Lemma 2, W u (pi, n ) con ains he same numbe o elemen so {P l "." P n } . Then, by i) o Lemma 1, pk e W u (p i , n ) because x 0 IJ u(p i , n ). The e o e Wu (Pk, n ,+)c Wu (p i , n ) . Hence x e W u (p i , n ), and we ge a con adic ion . Q .E .D . LEMMA 9 . (p o ed by Li and Yo ke [8]) . Le I be a cZosed in e al and le e C O (I,I) .  Suppose he eexis wo cZosed in e aIs L and R such ha L URc (R), Rc (L) and z (L n R) n R= ~ . Then o e e y m =1,2, . . . he e exis s a pe iodic poin in R wi h pe iod m . THEOREM 10 . Le e C 0(S 1 ,S 1 ) and suppose Pe ( )is ini e . Le {p1, . . .,pn} be a pe iodico bi o wi h pe iod n>, 2 . I k , u(p l , ) = S 1 , he ollowing holds o some m e {n,n/2} . i)  I (pi , P~) n {p 1 , . . "pn} =o'  hen ( [p i , pjj) = [p i , pjj, and k([pi,pj]) ) (pi , pj )=d, o any k e {1, . . .,m-1} . ii) Pe ( )=Pe ( ) . ([x,y])z[x,y], a con adic ion . Q .E .D . THEOREM 13 . Le e C 0(S 1 ,S 1 ) . Suppose Pe ( ) = Fix( ) _ {P1, . . . , p } and (S 1 ) =S 1 .  Then  U  Wu (pk, ) = S1 . 1,<k< P oo . We de ine W=  U  bl u (P,, ) . Suppose > 1 and W# S 1 . 1,< k, We claim ha S 1 -W has mo e han one connec edcomponen . To p o e his, suppose S 1 - W has only one connec edcomponen . By ) o Lemma 1, S 1 -W=(p i ,p j ) wi h (p i ,p j )(1Fix( )=0 . F om i ) o Lemma 1 i ollows ha (W)=W . Then, since (S 1 )=5 1 , ( [P i ,PJ])=> [p i ,Pj] . By Lemma 12, (p i, Pj)CW u (pi, ,+)uw u (P i , , - ) cw, a con adic ion . This es ablishes he claim . Le (p i ,p j ) and (PI,Pk) be wo dis inc connec ed componen s ó S 1 -W . I is clea ha (p i ,p j )(1 Fix( ) =0 and (p l ,p k )(1 Fix( ) =0 . F om Lemma 12 i ollows ha ([p i ,p j ] )ay [p i ,pj ] and ([Pl'Pk]) ~5 [PI,Pk] . Then ([Pi,PJ])=>[P .7,pi]n [PI , Pk] and simila ly ([PI,Pk])D [pi,PJ] . Hence 2 ([p i ,p j ])D [Pi,P j] . By Lemma 12,  (p i ,p j ) e W u (Pi, 2 ,+)U Wu(P j , 2 ,-)c W, a con adic ion . Now, suppose =1 and W#S 1 . We may assume ha he e exis s a neighbo hood V o p= p 1 such ha -1 (p)(1V= {p} . O he wise, he e  is  an a c [x, y] such ha  pe  [x,y] ,  ( [x,y] ) = {p}  and ([a,b]) ¢ {p} o e e y a c [a,b] wi h [x,y] c (a,b) . Le X deno e he quo ien space o S 1 ob ained by iden i ying al] poin so [x,y] o he single poin p, and le g : X - X be he quo ien map o ob ained by his iden i ica ion . Then g e i ies he hypo heses o he heo em and he e exis s a neighbo hood V o p such ha g-1(p)1)V={p} . We sepa e e he p oo in o i e cases . Case 1 . Suppose ([p,x]):)[p,x], o some x su icien ly close o p . This implies ha he e exis s y su icien ly close o p such ha y e (p, (y)) . The e o e W U (p, ,+) =S 1 , a con adic ion . Case 2 .  Suppose ([x,p]) n [x,p], o some x su icien ly close o p . Simila ly, W U (p, ,-)= S 1 , a con adic ion . Case 3 . Suppose ([p,x])c [p,x], o some x su icien ly close o p . Then ([x,p]) :)[x,p] . By case 2, we ha e a con adic ion . Case 4 . Suppose ([x,p])c [x,p], o some x su icien ly close o p . Then ([p,x]) - - >[p,x] .  By case 1, we ha e a con adic ion . Case 5 . Suppose ( [p,x] )c [a,p] and ( [y,p] )c [p,b] o x and y su icien ly close o p, and o some a,b e S1 - {p} . Hence, by he abo e cases we ha e a con adic ion o he map 2 . Q.E .D . COROLLARY 14 . Le e C 0 (S 1, S 1) . Suppose Pe ( ) ={p 1 . . .,p } and (S 1 ) =S 1 . Then  U  0 (p k , ) = S1 . 1,<k,< P oo . Le n be he p oduc o he pe iods o al] he pe iodic poin so . Then all he pe iodic poin s o a e ixed poin s o n . By Theo em 13,  U  W'(Pk, n) =Si . Since W u (Pk, n ) c 1,k, Wu (Pk, ), U Wu (Pk, ) = S 1 . Q.E .D . 1, k, THEOREM 15 . Le X be an a bi a y in e al o he eal line, and le e .C 0 (X,X) . I Pe ( ) is ini e hen o some in ege n >,O, P( )={1,2,4, .,2 n} ., This heo em is con ained in a heo em o Sha ko skii (see [6], [9] and [10]) which says he ollowing . O de he posi i e in ege s as ollows : 3,5,7, . . .,2-3,2.5,2-7, . . .,4-3,4-5,4,7, . . ., 8-3,8-5,8 .7, . . .,8,4,2,1 . Then i m is o he igh o n and has a pe iodic poin o pe iod n, hen has a pe iodic poin o pe iod m . THEOREM A . Le e C O (S1 ,S 1 ) and suppose Pe ( )is ini e . Then he e a e in ege s m >,1 and n> .0, such ha P( ) _ {m, 2m,4m, . . ., 2nm} . Case 2 . The e is a ixed poin p o wi h W u (p, ) = S1 . We ep esen S 1 as he in e al [0,1] iden i ying he poin s 0 and 1 o he poin p . Le g : [0,11 --+ S 1 be he na u al map de ined by his iden i ica ion . By Lemma 11, he e exis s a map h : [0,1] -= [0,1] such ha og=g-h . The e o e P( )= P(h)= {1,2,4, . . .,2n} o some in ege n>,0 . 124 P oo . We sepa a e he p oo in o h ee cases . Case 1 . The e is a pe iodic poin p o wi h pe iod >,2 and W U (P, ) =S 1 . By Theo em 10, Pe ( ) =Pe ( m )= U ij Pe ( m j ), whe e m e { , /2} and mj is he es ic ion o m o [p i ,p j ] wi h Pi,pj e o b(p) and (pi,pj)n o b(p) =0 .  By Theo em 15, o e e y ' j he e is an  in ege n(ij), 0 such ha P( mj)={1,2,4, . . .,2n(ij)} .  Le n be he g ea es elemen o {n(ij)} . Then P( ) = {m,2m,4m,, 2 n m} . Case 3 . Fo e e y pe iodic poin p o we ha e ha W , (p, ) ~S1 . Le g e CO(S 1 ,S 1 ) and le X be a subse o S 1 such ha g(X)c X .  F on now on gIX will deno e he es ic ion o g o X . I (S 1 )~ S 1 , le J = (S 1 ) . Then P( ) =P( 1J) . By Theo em 15, he e is an in ege n>,0 such ha P( 1J) ={1,2,4  2n } . Hence, he heo em is p o ed . The e o e, we shall assume ha (S 1 ) =S 1 . Le p be a pe iodic poin o wi h pe iod and le J be a connec edcomponen o W u (p, ) . Since W u (p, ) # S 1 , J # S1 . By iii) and i ) o Lemma 1, (J) =J . F om Theo em 15 i ollows ha P( lj)= {1,2,4  . .,2s } o some in ege s,0 .  Because (J)=J, P( ,J) = {1,2,4  2 } whe e = s i s , 1, and e {0,1}i s= 0 . Fo each connec edcomponen o Wu (p, ) we ha e an in ege >, 0 . Le (p) be he g ea es in ege associa ed o some connec ed componen o 4J u (p, ) . Then P( ,Wu(P, )) = {1,2,4, . . .,2 (p)} . Hence P( lW U (p, )) = { ,2 ,4 , , 2 (p) } . Le m be he smalles elemen o P( ) and le pbe a pe iodic poin o wi h pe iod m . IJe claim ha P( lW u (p, )U Wu (q, ))= {m,2m,4m, . . .,2 m} o any pe iodic poin q o such ha W u (p, )(1W u (q, ) ~ O, and o some in ege = (p,q) . We shall p o e his claim . By Co olla y 14, he e a e pe iodic poin s q such ha W u (p, )n Wu (q, )~ 0 . Le q be such a pe iodic poin wi h pe iod k . By ) o Lemma 1, he se s P( jW u (p, )) = {m,2m,4m, - 2 (p)m} and P( lW u (q, )) = {k,2k,4k, . . .,2 (q)k} in e sec . Then,  since k >,m, we ob ain ha k .=2 am, o some in ege a>,0 . The e o e, i (p,q) is he g ea es elemen o { (p),a + (q)}, he claim is p o ed . By he same a gumen and by Co olla y 14, he heo em ollows . Q .E.D . §5 . P oo o Theo em B LEMMA 16 . Le e C0 (S 1 ,S Z ) . Suppose S1( ) is ini e and [,  (p, ) $S 1 o al 1 p e Pe ( ) .  Then 2 ( ) =Pe ( ) . P oo . Suppose x e Q( ) and x0 Pe ( ) . By i) o Lemma 1, o some pe iodic poin p l , he e exis s z eW u (pl, ) such ha (z) =p, and z is no pe iodic . Le n be he pe iod o pl and le o b(p l ) ={P l ` .,p n } . By iii) o Lemma 1, z e W u (Pk, n ) o some k e {1, . . .,n} . No e ha n (z) e(P l " ."p n} and (by i ) o Lemma 1) n (z) e W u (Pk, n ) . We sepa a e he p oo in o wo cases . Case 1 . pl is a pe iodic poin wi h pe iod n>,2 . Then, by Theo em 7, n(z)= pk . Le J =W u (P k , n ). By i ) o Lemma 1, n (J)= J . Le g be he es ic ion o n o J . Then z eWu (Pk, n ) =W u (Pk ,g), and g(z) =pk . By Lemma 6, z = Pk . This is a con adic ion,because z is no pe iodic . Case 2 . pl is a ixed poin . Then n =l, and (z)=p, The p oo is iden ic o he abo e case . Q .E .D . THEOREM 17 (p o ed by Block in [6]) . Le I be an a bi a y in e alo he eal line . Le e C O (I,I) and suppose 2( ) is ini e . Then Q( )=Pe ( ) . LEMMA 18 . Le e C0(S 1 ,S 1 ) . Suppose Q( ) is ini e and O(p 1 , ) = S 1 o some pe iodic o bi {p1 " pn } wi h n >,2 . Then si ( ) = Pe ( ) . P oo . By Theo em 10, Pe ( ) = Pe ( m)= UiJ Pe ( m j .) and s2( ) =si( i ) = V id sl( m i ), whe e m e {n,n/2} and m~ is he es ic ion o m o- [p i ,p j ], i (pi,pj)(1{pl, . . .,pn} o . By Theo em 17, P( m~)= Pe ( m~) . Hence s2( )= Pe ( ) .  Q .E .D . LEMMA 19 . Le e C 0 (S 1 ,S 1 ) . Suppose 2( ) is ini e and k~ (q, ) ~¿S 1 o any q e Pe ( ) wi h pe iod g ea e han 1 . I 0(q, )=S 1 o some q e Fix( ), hen 2( )=Pe ( ) . P oo . Suppose ye s2( ) and y 0 Pe ( ) . By i) o Lemma 1, o some pe iodic poin p, he e exis s z e W U (p, ) such ha (z) = p and z is no pe iodic . Le n be he pe iodo p . We sepa a e he p oo in o h ee cases .  ' Case 1 .  pis a pe iodic poin wi h pe iod n ; 2 . '  Since W U (P, )~ S 1 , by he same a gumen used in he p oo o case 1 o Lemma 16, we would ha e a con adic ion . Case 2 . p is a ixed poin wi h W U (p, ) # S1 . Now, we should ha e a con adic ion by he same a gumen used in he p oo o case 2 o Lemma 16 . Case 3 . pis a ixed poin wi h Wu (p, ) =S 1 . By he p oo o case 2 o Theo em A, he e a e wo con inuous " maps g : [0,1] --, S 1 and h : [0,1]  -= [0,1] such ha -g= g-h . By Theo em 17, 2(h) = Pe (h) . Then n( ) = Pe ( ) .  Q .E .D . THEOREM B . Le e C 0 (S 1 ,S 1) and suppose Q( ) is ini e . Then 2( ) = Pe ( ) . Theo em B ollows immedia ely om Lemmas 16, 18 and 19 . §6 . P oo so Theo ems C and D THEOREM 20 (p o ed by Block in [5]) . Le X deno ean a bi a y in e al o he eal line, and le e C 0 (X,X) . Suppose Pe ( ) = Fix( ) is ini e . Then P( ) = Fix( ) . THEOREM D . Le e CO (S 1 ,S 1 ) . Suppose Pe ( ) = Fix( ) = {p 1 " ",p } . Then S2 ( ) =Fix( ) . P oo . We sepa a e he p oo in o wo cases . Case 1 . The e is a ixed poin p o wi h W u (p, ) =S 1 . By he same a gumen used in he p oo o case 3 o Lemma 19 and by Theo em 20, we ha e ha P( ) = Fix( ) . Case 2 . Fo e e y ixed poin p, W u (P, )# S 1 . l (S 1 )~S 1 , le J= (S 1 ) . Then, by Theo em 20, Q( )=S2( 1J) Fix( 1J) = Fix( ) . Hence, he heo em is p o ed . The e o e, we shall assume ha (S 1 )=5 1 . F om Theo em 13 i ollows ha > 1 . Le p be a ixed poin o . By i) o Lemma 1, W u (p, ) is connec ed . By i ) o Lemma 1, (W u (p, )) =Wu (p, ) . F om Theo em 20 we ha e ha S2( lW u (p, )) = Fix( lW u (P, )) . Then, by Theo em 13, S2( ) = Fix( ) .  Q .E .D . LEMMA 21 (p o ed by Adle , Konheim and McAnd ew in [1]) . Le be a con inuous map o a compac opological space and Ze n be a posi i e in ege . Then en ( n ) = n-en ( ) . LEMMA 22 (p o ed by Bowen [7]) . Le be a con inuous map o a con pac me ic space and suppose S2( ) is ini e . Then en ( ) = 0 . Now, he p oo o Theo em Cis iden ical o he p oo o Theo em A o [51 . We include i he e by i s b e i y . THEOREM C . Le e C O (S 1 ,S 1) and suppose Pe ( ) is ini e . Thenen ( ) = 0 . 128 P oo . Le n be he p oduc o he pe iods o al] he pe iodic poin s o . Then Pe ( n ) = Fix( n ) . By Theo em D, 2( n ) = Pe ( n) . In pa icula , 2( n ) is ini e . Hence en ( n ) = 0, by Lema 22 . Thus, by Lemma 1, en ( ) = 0 .  Q .E .D . .4cknowledgemen I am indeb ed wi h C . Simó o his sugges ions abou a i s e sion o his pape . Re e ences 1 . R . Adle , A . Konheim and M . McAnd ew,TopoZogicaZ en opy, T ans . Ame . Ma h . Soc . 114 (1965), 309-319 . 2 . L . Block, Di eomo phisms ob ained om endomo phisms, T ans . Ame . Ma h . Soc . 214 (1975), 403-413 . 3 .  , Mo se-Smaleendomo phisms o he ci cZe, P oc . Ame . Ma h . Soc . 48, (1975), 457-463 . 4 .  , The pe iodic poin s o Mo se-S7nale endomo phisms o he ci cZe, T ans . Ame . Ma h . Soc . 22 6 (1977), 77-88 . 5 .  , Mappings o he in e al wi h ini ely many pe iodic poin s ha e ze o en opy, P oc . Ame . Ma h . Soc . 6 7 (1977), 357-360 . 6 .  , Con inuous maps o he in e al wi h ini e nonmande ing se , T a  Ame . Ma h . Soc . 240 (1978), 221-230 . 7 . R . Bowen, TopoZogicaZ en opy and Axiom A, P oc . Sympos . Pu e Ma h ., ol . 14, Ame . Ma h . Soc ., P o idence, R .I ., 1970, pp 23-41 . 8 . J . Li and J .A . Yo ke, Pe iod h ee implies chaos, Am . Ma h . Mon hly 82 (1975), 985-992 . 1 9 . A .N . Sa ko skii, Coesis ence o cycles o a con inuous map o a Zine in o i sel , Uk ain . Ma . Z . 16 (1964), 61-71 . (Russian) MR 28 * 3121 . 10 . P . S e an, A heo em o Sa ko skii on he exis ence o pe iodic o bi s o con inuous endomo phisms o he eal Zine, Co an . Ma h-Phys . 54 (1977), 237-248 .