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
.