Publicacions
Ma emá iques,
Vol
37
(1993),
305-316
.
Abs ac
SIMPLE
AND
COMPLEX
DYNAMICS
FOR
CIRCLE
MAPS*
LLUÍS
ALSEDÁ
AND
VLADIMIR
FEDORENKO
The
con inuous
sel
maps
o
a
closed
in e al o
he
eal line
wi h
ze o
opological
en opy
can
be
cha ac e ized
in
e ms
o
he dy-
namics
o
he
map
on
i s
chain
ecu en
se
.
In
his
pape
we
ex end
his
cha ac e iza ion
o
con inuous
sel
maps
o
he
ci cle
.
We
show
ha ,
o
hese
maps,
he
chain
ecu en
se
can
exhibi
a
new
dynamic
beha iou
which
is
speci ic
o
he
ci cle
maps
o
deg ee
one
.
1
.
In oduc ion
.
The
aim
o
his
pape
is
o
ex end he
cha ac e iza ion
o
he
complex
and
simple
in e al
maps
(in
he
sense
o
posi i e
o
ze o
opological
en opy
espec i ely)
o
ci cle
maps
.
We
shall s a
by
s a ing
his
cha -
ac e iza ion
o
in e al
maps
o
comple eness
(see
Theo em
A)
.
To
do
i
we
ha e
o
in oduce
he
app op ia e
no a ion
.
Le
be a
map
om
a
opological
space
X
in o
i sel
.
We
shall
deno e
by
'
he
map
o
o
.
. .
o
n
imes
(i
n
=
0
we
se
n
=
Id)
.
Le
now
be an
in e al
map
( ha
is,
a
con inuous
map
om
a
closed
in e al
I
o
he
eal
line
in o
i sel )
.
We
say
ha
has
a
ho seshoe
i
he e
exis
n>
0
and wo
closed
in e als
Il,
12
C
I
wi h
pai wise
disjoin
in e io s
such
ha
h
U
12
C
n
(I1)
and I
l
U
12
C
n
(I2
)
.
The
abo e
condi ion
was
used
o
he
i s
ime
by
Sha ko skii(see
[13])
and
has
been
used
widely
in
he
s udy
o in e al
maps
(see
[9], [4]
and
[12])
.
The
name
o
ho seshoe
was
gi en
o
his
condi ion
by
Misiu ewicz
*This pape
was
w i en
du ing
a
s age
o
he
second
au ho
a
he
Cen e de
Rece ca
Ma emá ica
.
He
was
also
pa ially
suppo ed
by
he
Fund
o
Fundamen al
Resea ch
o
he
S a e
Commi ee
o
Uk aine
on
Science
and
Technology
g an
numbe
1/356
.
The
i s
au ho
has
been
pa ially
suppo ed
by
he
DGICYT
g an
numbe
PB90-
0695
.
306

LL
.
ALSEDÁ,
V
.
FEDORENKO
in
[9]
.
An
in e al
map
ha ing
a
ho seshoe
was
called
u bulen
by
Block
(see
[4])
and
in
[12]
a
simila
no ion
was
called
an L-scheme
.
We
no e ha
he
se
o in e al
maps
ha ing
a
ho seshoe
is
open and
dense
in
he
space
o
all
in e al
maps
(we
suppose
his
space
endowed
wi h
he
opology
o
he
uni o m
con e gen e)
.
Thus,
in his sense,
he
p ope y
ha
a
map
has
a
ho seshoe
is
gene ic
.
I
is
well
known
ha
a
map
has
posi i e
opological
en opy
i
and
only
i i
has
a
ho seshoe
(see
[9])
.
In
o he
wo ds, he
exis en e
o
ho seshoes
cha ac e izes
he
complex
in e al
maps
.
Now
we
in oduce
he
necessa y no ions
o
cha ac e ize
he
simple
in e al
maps
.
Le
S be
a
closed
in a ian
se o
an
in e al
map
.
We
say
ha
S
spli s
in o
So
and
Sl
i
So
and
S
I
a e closed
nonemp y
subse s
o
S
such
ha
(S,)
n
(S2)
=
0
(whe e
(Si)
deno es
he
con ex
hull
o
Si,
i
=
1,
2),
Sl
U
S2
=
S,
(Sl)
=
S2
and
(S2)
=S
i
.
We
also
say
ha
S
spli s
k
imes
i
S
spli s
in o
So
and
Sl
and
each
o
hem
spli s
(k
-1)
imes
unde 2
.
The
se
S
is
said
o
be simple
i
ei he
i
is
a
ixed
poin
o
i
spli s
k
imes
o
each
k
G
109
2
Ca d
S
(see
[6])
.
Rema k
1
.1
.
F om
he
abo e
de ini ion
i
ollows
easily
ha
each
simple
se ei he
consis s
en a
unique
pe iodic
o bi o
does
no
con ain
any
pe iodic
o bi
.
Recall ha
i
is
a
con inuous
map
om
a
me ic
space
X
in o
i sel
he
se
o
chain
ecu en poin s
o
is
deno ed
by
CR( )
and
is
de ined
o
be
he
se o
all
x
E
X
such ha
o
each
e
>
0
he e
exis s
{xi}?
o
wi h xo
=
x,,,
=
x
and
¡
(xi)
-
xi+11
G
E
o
i
=
0,
1,
2,
. .
.
,
n-
1
.
The
ollowing
heo em
cha ac e izes
he
complex
and
simple
in e al
maps
(see
[6])
.
Theo em
A
.
Each
in e al
map
sa is ies
one
and
only
one
o
he
ollowing
wo
condi ions
.
(a)
has
a
ho seshoe
.
(b)

The
chain
ecu en
se
o
is
he
union
o
all
simple
se s
o
.
To
ex end
Theo em
A
o
ci cle
mapswe
ha e
o
e o mula e
he
abo e
no ions
in
his
con ex
.
We
shall
ep esen he
ci cle
S
1
as
he
se
{z
E
(C
:
Iz1
=
1}
.
Any
con inuous
map
om
S
1
in o
i sel
will
be
called
a
ci cle
map
.
We
no e ha
he
no ion o
a
simple
se
and
o
ho seshoe ex ends
na u ally
o
ci cle
maps
by
simply
eplacing
closed
in e als
by
closed
a es
o
he
ci cle
( ha
is,
subse s
o
S
l
which
a e
homeomo phic
o
closed
in e als o
he
eal
line)
.
SIMPLE
AND
COMPLEX
DYNAMICS
FOR
CIRCLE
MAPS

30
7
We
no e ha
i
an
in e al
map
has
a
ho seshoe
h,
12
hen
we
always
ha e
ha ,
o
each
i,
j
E
{1, 2},
he e
exis
a
closed
in e al
I~
C
I
i
such
ha
n
(I
.~)
=
I,
.
Howe e ,
his
is
no he
case
i
we
a e
alking
abou
ho seshoes o
ci cle
maps
.
Indeed,
i
h,
12
C
S
1
is
a
ho seshoe
o
a
ci cle
map
g
i
may
happen
ha
gn
(II)
=
S
1
in
such
a
way
ha
gn
¡In (I
1
)
is
injec i e
and
gn(a)
=
gn(b)
E
In (I
2
)
whe e aand
b
deno e he
wo
endpoin s
o
h
.
Then,
clea ly,
does
no
hold
ha
o
each
i,
j
E
{1,
2}
he e
exis
a
closed
a c
Ij
C
I2
such
ha
gn
(I
.~)
=
h
.
Le
be
a
map
om
a
opological
space
X
in o
i sel
and
le
x E
X
.
We
say
ha
x
is
a
pe iodic
poin o
i
n(x)
=
x
o
some
n
>
0
.
The
smalles
n
wi h
he
abo e
p ope y
is
called
he
pe iod
o
x
.
I
x E
X
is
a
pe iodic
poin
o
o
pe iod
n
hen
he
se
{x,
(X)
....
,
n
-I
(x)}
will
be
called
a
pe iodic
o bi
o
o
pe iod
n
.
In
he sequel
we
shall
deno e
by
Pe ( )
he
se
o
pe iods
o
all
pe iodic
poin s
o
.
Also,
i
x E
X
we
shall
deno e
by
w
x
( )
he
omega
limi
se
o
x
which
is
de ined
o
be
he
se
o
all
accumula ion
poin s
o
{ '
(x)
:
n
>_
0}
.
We
will
also
use
he
no a ion
w( )
o
deno e
UxEXwx
(
)
.
The
mai i
esul
o his
pape
is
he
ollowing
.
Theo em
1 .2
.
Each
ci cle
map
sa is ies
one
and
only
one
om
he
ollowing
heee
condi ions
:
(a)
has a
ho seshoe
.
(b)

The e
exis
n
>
0
such
ha
w
x
(
n)
is
a simple
se
o each
x
E
S'
.
(c)
Pe ( )
=
0
.
We
no e
ha ,
as
o
in e al
maps,
Condi ion
(a)
o
he
abo e
heo em
is
gene ic
in
he
space
o
ci cle
maps
endowed
wi h
he
opology
o
he
uni o m
con e gen e
and
is
a
c i e ion
o
posi i e
opological
en opy
(see
[9])
.
On
he
o he
hand,
i is
known
ha
Condi ion
(b)
wi h
n
=
1,
in
he
case
o
an
in e al
map,
is
equi alen
o
Condi ion
(b)
o
Theo-
em
A
(see
o
ins an e
Theo em
2 o
[7])
.
Howe e ,
o ci cle
maps
i
is
no
.
Theo em
1 .2 is
s a ed
in his
way
o
simplici y
bu
in
Sec ion
4
he
opological pic u e
o
he
chain
ecu en
se in his
case
will
be
desc ibed
in de ail
.
Finally,
he
dynamics
o a
ci cle
map
sa is ying
Condi ion
(c)
o
he
abo e
heo em
can
be
oughly desc ibed
as
ollows
.
The
map
has
a
unique
w-limi
se
which
is
minimal
(Le
.
i
has
no
closed
in a ian
p ope
subse )
and
he
es ic ion o
o
i
is
semi-conjuga e
o a
o-
a ion
o
he
ci cle
by an
i a ional
angle
.
A
de ailed desc ip ion
o
he
dynamics
o
such
a
map
can
be
ound
in
[3]
and
[11]
.
308

LL
.
ALSEDÁ,
V
.
FEDORENKO
2
.
De ini ions
and
p elimina y
esul s
.
In
his
sec ion
we
will
in oduce
he
necessa y
no a ion
o
p o e
The-
o em
1
.2
.
Also
we
will
p o e
a
lemma
ha
will
play
a
key
ole in
ha
p oo
.
Le
be
a
ci cle
map
.
As
usual,
ins ead
o
wo king
wi h
i sel
we
shall
use
a
li ing
o
.
A
con inuous
map
F
:
I[8
-->
R
is
called
a
li ing
o
i
e
o
F
=
o
e,
whe e
e(x)
=
exp(27 ix)
is
he
na u al p ojec ion
om
1[8
o
S
1
.
We
no e
ha
i
F
is
a
li ing
o
hen
F
+
m
is
also
a
li ing
o
o
each
m
E
7G
and
ha
Fn
is
a
li ing
o
'
.
Also,
he e
exis s
an
in ege
d
such
ha
F(x
+
1)
=
F(x)
+
d
o
each
x
E
R
.
This
numbe
d
is
called
he
deg ee
o and
is
deno ed
by
deg(
)
.
I
is
no
di icul
o
see
ha
deg( n)
=
deg(
)n
.
We
say
ha
a
poin
x
E
R
is
pe iodic
(mod
.
1) o
pe iod q o
F
i
Fq(x)
-
x
E
7L
bu
F
1
(x)
-
x
¢
7L
o
j
=
1,
2,
. . . ,
q
-
1
.
Clea ly,
x
is
a
pe iodic
(mod
.
1)
poin
o
F
o
pe iod
q
i
and
only
i
e(x)
is
a
pe iodic
poin
o
o
pe iod
q
.
Le
F
be
a
li ing
o a
ci cle
map
.
In
he
sequel
we
shall
deno e
by
Pe (F)
he
se
o
pe oods
o
all
pe iodic
(mod
.
1)
poin s
o
F
.
Clea ly,
Pe ( )
=
Pe (F)
.
Le
be a
ci cle
map
o
deg ee
one
and
le
F
be
a
li ing
o
.
Fo
x
E
R
we
de ine
i s
F- o a ion
numbe
as
lim sup
Fn
(x)
-
x
n
-
oo
n
and
deno e
i
by
PF(x)
.
We
no e
ha ,
since
has
deg ee
1,
PF(x)
=
PF
(X
+M)
o
all
m
E
7G
.
Also,
i
x
is
a
pe iodic
(mod
.
1)
poin
o
pe iod
q
o
F
hen
PF
(x)
=
F9(x)
-
x
E
q
The
se
{PF(x)
:
x
E
l[8}
=
{PF(x)
:
xE
[0,1)}
is
deno ed
by
LF
.
I o in
[8]
p o ed
ha
LF
is
a
closed
in e al
(pe haps degene a e
o
a
poin )
o
R
.
Thus,
in
he sequel
LF
will
be
called
he
o a ion
in e al
o
F
.
The
o a ion
in e al o a
li ing
o a
ci cle
map
o
deg ee
one
cap u es
a
lo
o
i s
dynamical
p ope ies
and
plays
a
undamen al
ole
in
hei
s udy
(see
o
ins ance
[10]
and
[2])
.
Le
F
be a
li ing
o
a
ci cle
map
o
deg ee
one
.
We
de ine
(see
[1])
F
.
(x)
=
sup{F(y)
:
y
<
x}
.
SIMPLE
AND
COMPLEX
DYNAMICS
FOR
CIRCLE
MAPS

309
I
is
no
di icul o see
ha
F
u
is
non-dec easing
and
ha
i is
a
li ing
o
a
ci cle
map
o
deg ee
one
.
Mo eo e ,
F
<
Fu
.
Now
we
a e
eady
o
s a e
and
p o e
he
lemma
we
a e
looking
o
.
Lemma
2
.1
.
Le
be a
ci cle
map
and
le
F
be a
li ing
o
.
Then
one
o he
ollowing
p ope ies
hold
:
(a)
has
a
ho seshoe
.
(b)

The e
exis
q
E
N,

p
E
7G
and
.
I
=
[x,
x
+
1]

C
l[8
such
ha
(Fq
_
p)
(J)
C
J
.
(c)
Pe ( )
=
0
.
P oo
.
I
deg( )
¢
{-1,
0,1}
hen
Cea ly
has
a
ho seshoe
.

Thus,
(a)
holds
.
Assumenow
ha
deg( )
=
0
.
Then,
F(x
+
1)
=
F(x)
o all
x
E
R
.
Hence,
F(R)
=
F([0,1])
=
[a,
b]
.
I
b
<_
a
+
1
hen
we
se
q
=
1,
p=
0
and I
=
[a,
a+
1]
;
and
(b)
holds
.
Thus,
assume
ha
b
>
a+
l
.
Le
c
E
R
be
such
ha
F
(c)
=
a
.
Clea ly
F(c+
1)
=
a
and
he e
exis
d
E
(c,
c+
1)
such
ha
F(d)
=
b
.
Se
I
l
=
e([c,
d])
and
12
=
e([d,
c
+
1])
.
Clea ly
Il
and
12 a e
a cs
o
S
1
and
(I1)
=
(I2)
DS
1
D
I
l
U
12
.
Thus,
has
a
ho seshoe
and
(a)
holds
.
Now
we
conside
he
case deg( )
=
1
.
F om
[10]
i
ollows
ha
Pe ( )
=
0
i
and
only
i
LF
=
{a}
wi h
a
0
Q
and
ha
has
a
ho seshoe
i
LF
is
non-degene a e
.
Thus
we
only
ha e
o
conside
he
case
LF
=
{p/q}
wi h
q
E
N
and
p
E
7G
ela i ely
p ime
.
F om
[5]
(see
also
[1,
Theo em
3
.7
.20
and
i s
p oo ])
i
ollows
ha
he e
exis s
x,
a
pe iodic
(mod
.
1)
poin
o
F
o
pe iod
q
and
o a ion
numbe
p/q, such
ha
Fi
(x)
=
F,,
(x)
o
all
i
>
0
.
Se
P={F'(x)+m
:

i=0,1,2,
. .
.,q-1,
mEZ}=
=
e

1({ 2(e(x))
:

i
=
0,1,
2,
. . . ,
q
-
1})
=
=
{
. . .
x_2, x_1,
xo, xl,
X2
.
. . .
}
wi h
(xi, xi
+
1)nP
=
0
o
all
i
E
7G
.
Se
also
G=
Fq-p
and
G
=
Fú-p
.
Since
F,,
is
non-dec easing
so
is
G,
4
and, hence,
G
<_
G
because
F
<
F

.
The e o e,
i
y
E
R,
z
E
P
and
y
<
z
hen
G
(y)
<
G
.
(y)
<
G
.
(z)
=
G
(z)
=
z
.
Mo eo e ,
i is
no
di icul
o see
ha
LG
=
{0}
.
Se
J =
[xo,
xo
+
1]
.

We
no e
ha
J
C
G(J)
because
G(xi)
=
xi
o
all
i
E
7L
.
Thus,
G
¡
(J)
C
G'+1(J)
o
all
i
E
N
.
Le
K
be
he

31
0

LL
.
ALSEDÁ,
V
.
FEDORENKO
closu e
o
U
°°
1
G'(J)
.
Clea ly
K
is
a
closed
in e al
such
ha
J
C
K
C
(-oo,
xo+1]
and
G(K)
C
K
.
I
K
is
no
bounded,
hen
he e
exis
zE
J
and
m
E
N
such
ha
G'(z)
<
xo
-1
.
The e o e,
he e
exis s
z
E
J
such
ha
G'''(z)
=
z
-
1
.
Tha
is,
z
is
a
pe iodic
(mod
.
1)
poin
o
G
wi h
o a ion
numbe
-1/m
.
This
con adic s
he
ac
ha
LG
=
{O}
.
Thus,
K
is
o
he
o m
[a,
xo
+
1]
wi h
a<
xo
.
I
a
E
P
hen
we
ake
I
=
[a,
a+
1]
.
Since
a
+
1
E
P
we
ha e
ha
G(I)
C
(-oo,
a+
1]
.
On
he
o he
hand,
G(I)
C
G(K)
C
K
.
Thus
G(I)
C
I
and
we
a e
done
.
Now
assume
ha
a

P
.
Then
he e
exis s
i
<
0
such
ha
a
E
(xi,
xi+l)
.
Since
G(xi)
=
xi
he e
exis s
a
ixed
poin
o
G
in
[xi,
a]
.
Le
y
be
he
sup emum
o
he e
ixed
poin s
.
Since
G(a)
> a
we
see
ha
o
each
z
E
[y,
ca]
we
ha e
ha
G(z)
>
y
.
Thus
he
se
S
=
{y
E
[xi,
ca]
:
G
(y)
=
y
and
G(z)
>
y
o
each
z
E
[y,
a] }
is
non-emp y
.
So
we
se
,3
=
in
S
and
K
=
[,0,
xo
+
l]
.
Since
xi+1
E
P
and_xi+l
<_
xo
we
ha e
ha
G([/3,
ca])
C
P,
xi+1]
C
K
.
The e o e,
G(K)
C
K
.
Now,
i
,0
=
xi
we
se I
=
[/
0,
0+
1]
and
we
p oceed
as
abo e
o
ge
G(I)
C
I
.
So
we
may
assume
ha
0
:7~
xi
.
I
G([xi, i])
C
[-00,3]
hen
we
shall
show
ha
G(I)
C
I
wi h
I
=
[,3,
_
_(
3
~-
1]
.
To
see
his
i
is
enough
o
p o e
ha
i
z
<
~3
+
1
hen
G(z)
<
~3+1
because
I
C
K
and
G(K)
C
K
.
We
shall
p o e
i s
ha
i
z
<_
0
hen G(z)
<
0
.
I
z
E
[xi,
N]
hen
his
ollows
by
he
assump ion
.
I
z
<_
xi
hen,
since
xi
E
P
we
ha e
ha
G(z)
<_
xi
<
~
.
Hence,
since
G
is
a
li ing
o
e
which
has deg ee
one,
o
each
z
<_

+
1
we
ha e
G(z)
=
G(z
-
1)
+
1
<_ ~3
+
1
and
we
a e
done
.
Thus,
we
may
assume
ha
he e
exis s
E
[xi

C3]
such
ha
G( )
>Ñ
.
Since
xi
=,L
~3,
by
he
de ini ion
o
l,
he e
exis s
zE
[xi,
~3]
such ha
G(z)
<
xi
.
I
<z
hen
he
in e als
[ ,
z]
and
[z,,(3]
o m a
ho seshoe
o
G
.
So
F
has
a
ho seshoe
and
(a)
holds
.
I
z
<
hen
he e
is
a ixed
poin
o
G
in
(z,
)
.
Le
y
be
he
sup emum
o
he e
ixed
poin s
.
We
ha e
y< <0
and
G(z)
>
y

o
all
z
I
also
G(z)
>_
y
o
all
z
E
[

3]
hen
y E
S
which
con adic s
he
ac
ha
~3
=
in
S
.
So,
he e
exis s
7E
( ,
,0)
such
ha
(5)
=
y
.
Then
he
in e als
[y,
]
and
[ ,
~]
o m
a
ho seshoe
o
G
.
This ends
he
p oo
o
he
lemma
in
he
case
when
has
deg ee
one
.
Finally
assume
ha
deg( )
=
-l
.
By
he
de ini ion
o deg ee o
a
map,
has
a ixed
poin
.
Thus,
Pe ( )
:,A
0
.
Le
us
conside
'
.
I
has
SIMPLE
AND
COMPLEX
DYNAMICS
FOR
CIRCLE
MAPS

31
1
deg ee
1
and
FZ
as a
li ing
.
Thus,
in
iew
o
he
deg ee
one
case, ei he
2
has
a
ho seshoe
o
he e
exis
q
E
N,
pE
7L
and I
=
[x,
x+1]
such
ha
((F2)q-p)(I)
C
I
.
Hence,
ei he
has
a
ho seshoe
o
(F
2
q-p)(I)
C
I
.
g
Rema k
2
.2
.
F om
he
p oo
o
he
abo e
lemma
i
ollows
ha
i
is
a
ci cle
map
sa is ying
(b)
o
Lemma
2
.1,
hen
deg(
)
E
{-1,0,1}
.
3
.
P oo
o
Theo em
1
.2
.
We
shall
s a
wi h
some
echnical
lemmas
on
in e al
maps
.
Lemma
3
.1
.
Le
be
a
con inuous
map
o
he
in e al
I
in o
i sel
ha ing a
ho seshoe
.
Then,
has a
ho seshoe
in
In (I)
.
P oo
..
Since
has
a
ho seshoe
he e
exis
n
E
N
and
h,12
C
I,
wo
closed
in e als
wi h
pai wise
disjoin
in e io s,
such
ha
Il
U12
C
n(I1)

and

h
U
12
C
n
(I2)
.
Then,
he e
exis
J,',
J2
C
Il
and
Ji
,
J2
C
12,
closed
in e als
wi h
pai wise
disjoin in e io s
such
ha
Ji
U
J2
U
Ji
U
JZ
C
2n(J~z)
wi h
i,
j
E
{1,
2}
.
Clea ly,
wo
o
hese
in e als a e
con ained
in In (I)
.
Then,
has
a
ho seshoe
in In (I)
.
Lemma
3
.2
.
Le
be
a
con inuous
map
o
he
in e al
I
=
[a,
a
+
1]
in o
i sel
such
ha
(a+1)
=
(a)+d
wi h
dE
{-1,0,
1}
.
Assume
ha
w
x
( )
is
a simple
se
o
any
xE
I
and
ha
a
E
w
x
( )
and
a
+
1
E
w
y
( )
o
some
x,
y
EI
.
Then
he
ollowing
s a emen s hold
:
(a)
I
w
x(
)
7~
w
y
(
)
hen
w
x
(
)
and
w
y
(
)
a e
ixed
poin s
o
.
(b)
I
w
x
( )
=
w
y
( )
hen
w
x
( )
is
a
pe iodic
o bi
o
pe iod
2 o
.
P oo
.
I
will
be
di ided
in o
se e al
cases
.
Case
1
:
d
=
0
.
We
shall
p o e
ha
{a,
a
+
1}
~
w( )
and
hus
he
lemma
holds
.
In
iew
o
Theo em
A
and
he
ac
ha
o
an
in e al
map
he
chain
ecu en
se
is
a
union
o simple
se s
i
and
only
i
each
omega
limi
se
is
a
simple
se ,
i
is
enough
o
show
ha
i
{a,
a
+
1}
C
w(
)
hen
has
a
ho seshoe
.
I
(a+
1)
=
(a)
=
a
hen,
since
a+
1
E
w
y
(
)
and
w
y
(
)
is
in a ian ,
i
ollows
ha
he e
exis s z
E
I
such
ha
(z)
=
a+
1
.
Thus,
[a,
z]
and
[z,
a
+
1]
o m
a
ho seshoe
o
.
312

LL
.
ALSEDÁ,
V
.
FEDORENKO
I
(a
+
1)
=
(a)
=
a
+
1
we
p oceed
in
a
simila
way
o ob ain
a
ho seshoe
o
.
Assume
now
ha
a
<
(a)
=
(a
+
1)
<
a
+
1
.
As
be o e,
he e
exis
z, z'
É
I
such
ha
(z)
=
a and
(z')
=
a
+
1
.
Mo eo e ,
he e
exis s
,
a
ixed
poin
o
be ween
z
and
z'
.
I
z
<
z'
and
(a)
>_
hen
he
in e als
[a,
z]
and
[z,
]
o m
a
ho se-
shoe
o
.
I
z
<
z'
and
(a)
<
hen
he
ho seshoe
o
is
gi en
by
[ ,
z']
and
[z',
a
+
1]
.
Thus,
i
z
<
z'
we
a e
done
.
I z'
<
z
and
(a)
>
we
se
A
=
[a, z'],
B=
[z',
]
and
C=
[ ,
z]
.
Then
we
ha e
(A)
D
C,
(B)
D
C
and
(C)
D
A
U
B
.
Hence,
2
(A)
D
AUB

and

2
(B)
D
AUB
.
Thus
A
and
B
o m a
ho seshoe
o
.
I z'
<
z
bu
(a)
<
hen,
in
a
simila
way
we
can
see
ha
[ ,
z]
and
[z,
a
+
1]
o m
a
ho seshoe
o
.
Case
2
:
d
=
1
.
We
ha e
ha
(a)
=
a
and
(a
+
1)
=
a
+
1
.
Thus,
w
.,
( )
and
w
y
( )
a e
simple
se s
which
con ain
a
ixed
poin
.
In
iew
o
Rema k
1 .1
we
ob ain
ha
w,
:
(
)
=
{a}
and
w
y
(
)
=
{a
+
1}
.
Case
3
:
d
=
-1
.
We
ha e
ha
{a,
a
+
1}
is
now
a
pe iodic
o bi
o
o pe iod
2
.
Thus,
again
by
Rema k
1
.1
we
see
ha
w
.,
( )
=
w
y
( )
_
{a,
a
+
1}
.
Le
now
bea
ci cle
map,
and
le
F
be a
li ing
o
.
We
say
ha
E
A
i
he e
exis
x
E
lié
and
p
E
7L
such
ha
(F
-
p)
([x,
x
+
1])
C
[x,
x
+
1]
.
Tha
is,
i
F
sa is ies (b)
o
Lemma
2
.1
wi h
q
=
1
.
F om
Rema k
2
.2
we
no e ha
deg(
)
E
{-1,0,1}
.
Lemma
3
.3
.
Each
map
E
A
has
oneand
only
one
om
he
ollow-
ing
wo
p ope ies
:
(a)
has
a
ho seshoe
.
(b)
wx( )
is
a
simple
se
o each
xES
1
.
P oo
::
Le
F
be
a
li ing
o
.
Since
E
A
he e
exis
x
E
I[8
and
pE
9L
such
ha
(F
-
p)
([x,
x
+
1])
C
[x,
x
+
1]
.
In
iew
o
Theo em
A,
Lemmms
3
.1
and
3
.2
and
he
ac
ha
he
chain
ecu en
se o
an
in e al
map
is
union
o
simple
se s
i
and
only
i
each
omega
limi
se
is
a
simple
se
we
ge
ha
(F
-
p)
has
one and
only
one
om
he ollowing
wo
p ope ies
.
(i)

(F
-
p)
has
a
ho seshoe
in
(x,
x
+
1)
.
(ii)
w
y
(F-p)
is
a
simple
se
o
each
y
E
[x,
x+1]
and
i
x
Ew
xl
(F-p)
and
x+1
E
W
X2
(F-p)
o
some
X1,
X2
E
[x,
x+1]
hen
w
xl
(F-p)
SIMPLE
AND
COMPLEX
DYNAMICS
FOR
CIRCLE
MAPS

31
3
and
w
.,
Z
(
-
p)
a e ixed
poin s
o
(F
-
p)
when
w
., l
(F
-
p)
=
w~,
2
(F
-
p)
and
w,
:
l
(F
-
p)
is
a
pe iodic
o bi o
pe iod
2
when
w
.,
(F
-
p) :~
w-2
(F
-
p)
.
Since
(F
-
p)
is
also a
li ing
o
and e
:
(x,
x
+
1)
->
S
1
{e
(x)}
is
a
homeomo phism
we
ge he
desi ed
conclusion
.
Now
we
a e
eady
o
p o e
he
main
esul o his
pape
.
P oo
o
Theo em
1
.2
:
I
ollows
s aigh o wa dly
om
Lemmas
2
.1
and 3
.3
and
om
he
ac
ha
i
sa is ies (a)
o
(b) o
Lemma
2
.1
hen
Pe ( )
:?É
0
.
4
.
A
geome ic
iew-poin
o
Condi ion
(b)
o
Theo em
1 .2
.
Th oughou
his
sec ion
we
will
assume
ha
is
a
ci cle
map
sa is ying
Condi ion
(b)
o
Theo em
1
.2
.
We
shall
y
o
explain
his
Condi ion
in
e ms
o
he
g aph
o
he
map
and
he
chain
ecu en
se
o
.
FYom Rema k
2
.2
we
ha e
ha
deg( )
E
{-1,
0,1}
.

I
deg( )
E
{-1,
0},
by
looking
a
he
p oo
o
Theo em
1
.2,
we
can
p o e
ha
he
si ua ion
o
he
chain
ecu en
se
o
is
he
same
as
in
he
case
o in e al
maps
.
Tha
is,
he
chain
ecu en
se
o
is
he
union
o
all
simple
se s
.
Now
assume
ha
deg( )
=
1
.
Also
om
he
p oo
o
Lemma
2
.1
and
om
he
ac
ha
a
ci cle
map
sa is ies
one
and
only
one
om
he
condi ions
o
Theo em
1
.2
we
see
ha
i
F
is
a
li ing
o
hen
LF
=
{p1
q}
wi h
q
E
N
and
p
E
7L
ela i ely
p ime
.
By
using
he
no a ion
om
he
p oo
o
Lemma
2
.1
we
see
ha
he e
exis s
a
pe iodic
(mod
.
1)
poin
0
o
F
wi h
o a ion
numbe
p/q
such
ha
I
we
se
(Fq
_p)([0,0+11)
C
[0
,
0
+
11
.
Q={FZ(~3)+m
:
i=0,1,2,

,q-1,mEZ}=
=
e

1({ z(e(a))
:

i
=
0,
1, 2,

.
.
,
q
-
1})
_
_
{
. . .
~3-2,
0-1,
/30,
Q1,
02
.
. .
}
hen
i is
no
di icul
o
p o e
ha
(Fq
-p)([~i,Qi+1])
C
[~3i,~3i+1] o
all
i
E
7G
and
ha
o
each
i,
j
E
7G
he
diag am