Publicacions
Ma emá iques,
Vol
35
(1991),
543-558
.
A
CLASSIFICATION
OF
BRAID
TYPES
FOR
PERIODIC
ORBITS
OF
DIFFEOMORPHISMS
OF
SURFACES
OF
GENUS
ONE
WITH
TOPOLOGICAL
ENTROPY
ZERO
Abs ac
GUASCHI,
J
.
LLIBRE
AND
R
.S
.
MACKAY
We
classi y
he
b aid ypes
ha
can
occu
o ini e
unions
o
pe iodic
o bi s
o
di eomo phisms
o su aces
o genus one
wi h
ze o
opological
en opy
.
1
.
In oduc ion
In his
pape
we
classi y
he
b aid
ypes
ha
can
occu
o
ini e
unions
o
pe iodic
o bi s o
di eomo phisms
o
su aces o
genus
one
wi h
ze o
opological
en opy
.
This
ex ends
he
analysis
om
he
case o
genus
ze o
[LMI]
.
The
case
o
mos
in e es
o
us
is
di eomo phisms
o
he
o us,
iso opic
o
he
iden i y
.
This
is
ele an
o
he
beha iou
o
h ee coupled
oscilla o s, o
example
.
A
good
pic u e
o hei
dynamics
is
de eloping
[KMG],
[LM2],[MZ],
[H2],
[F],
[BGKM]
.
We
hope
ha
ou
esul s
will
helo
sol e
he
in iguing
p oblem
o
unde s anding
he
bounda y
o
ze o
opological
en opy
in
he
space
o
C
1
di eomo phisms
o
he
o us
.
We
begin
by
es ablishing
some
no a ion
and
ecalling
he
de ini ion
o
b aid
ype
.
Le
:
X
->
X
be
a
di eomo phism
o
an
o ien ed
mani old
X
.
W i e
h( )
o
he
opological
en opy
o
.
We
w i e
o
-
p,
o
-
o
o ien a ion-
p ese ing
and
e e sing,
espec i ely
.
Gi en
wo
di eomo phisms
:
X
-
;
X,
g
:
Y
~
Y
o
o ien ed
mani olds,
we
w i e
-
g
i
he e
exis s
an
conjugacy
be ween
hem
.
Le
M
be
a
sú ace,
i
.e
.
a
compac
connec ed
o ien ed
:
M
->
M
be
a
di eomo phism
o
M
and
le
P
be
a
ini e
o bi s
o
.
Then we
de ine
-p
:
Mp
,
M-p
by
emo ing
P
om
M
and
ecompac i ying
by
eplacing
he
poin s
o
P
by
ci cles
on which
P
is
he
p ojec i e
ac ion
o
D
[B]
.
o-p
2-mani old
.
Le
union
o
pe iodic
54
4

J
.
GUASCHI,
J
.
LLIBRE,
R
.S
.
MACKAY
Gi en
wo
di eomo phisms
,
g
:
X
->
X
o a
su ace
X
and
ini e
unions
P,
Q
o
pe iodic
o bi s
o
,
g
espec i ely,
we
say he
pai s (P,
) and
(2,
g)
ha e
he
same
b aid
ype
i
he e
exis s
an
o-
p
homeomo phism
k
:
XP
-
XQ
such
ha
k
P
k
-1
is
iso opic
o
gQ
.
The
equi alence
class
o
(P,
),
deno ed
[P,
],
is
called
i s
b aid
ype
.
To
speci y
he
b aid
ype
o
a
pai
(P,
)
we
will
use
Nielsen-Thu s on
he-
o y
o
selec
a
simples
ep esen a i e
o
he
equi alence
class
.
We
e e
he
eade
o
[LM1]
o
he necessa y
in o ma ion
abou
Nielsen-Thu s on
heo y
o
classi ica ion
o
su ace
homeomo phisms
up
o
iso opy,
and
he
de ini ions
o
he
classes
o
di eomo phisms
which
we
call
dise
ees,
e e sing
disc
ees
and
e e sing
annulus
ees
.
The
plan
o
he
pape
is
as ollows
.
In
Sec ion
2,
we
classi y
ini e
o de
homeomo phisms
o
he
o us,
which
is
an
impo an
p elimina y
esul
.
Then
in
Sec ion
3,
we
use
Nielsen-Thu s on heo y
o
iso opc
P
o
a
s anda d
o m,
and
analyse
he
possibili ies
.
Ou
main
esul
is
Theo em
4,
bu
because
i
is
a he
long
o s a e
and
equi es
no ions
in oduced
in
Sec ion
2,
we
lea e
i s
s a emen
un il
Sec ion
3
.
In
Sec ion
4
we
ede i e
esul s
o
[H1],
[H2],
and
[LM2]
as
co olla ies
o
hose
o
Sec ion
3
.
We
hank
he
e e ee
o
a
ca e ul
eading
.
The
i s
wo
au ho s
we e
pa -
ially
suppo ed
by
a
SERC
s uden ship
and a
DGICYT
g an
espec i ely
.
2
.
Classi ica ion
o
ini e
o de
homeomo phisms
o
he
o us
In
his
sec ion,
we
gi e
a
classi ica ion
o
ini e
o de
homeomo phisms
o
he
o us
up
o
o
-
p
conjugacy
.
Le
T
x
,
x E
R2,
be
ansla ion
by
x on
R2,
Le
.
T
x
(y)
=
y
+
x,
yE
8
2
.
Le
R
4

w
E
R/27 7L
be
o a ion
abou
0
on R2 by
angle
w
.
Le
be
he
e lec ion
{x
/
=
x
y
,
_
-y
on
IE8
2
.

De ine
o
o
be
he
g oup
gene a ed
by
Ti
l
o
l
and
Tl
o
,
l
l,
Fo
ha
gene a ed
by
T(I,o)
and
T(2,
~)
.
I
:
R2
_
R2
commu es
wi h
he
ac ion
o
a
g oup
I'
on
R2,
we
de ine
/I'
:
R
2
/I'
,
R
2
/I'
by
iden i ying
poin s
o
he
same
o bi
unde
I'
.
Fo
q
E
NN,
le
wbe
he
quo ien
o
Z
unde
he
equi alence
ela ion
gene a ed
by
he
ela ions
p
-
p
+
q,
p
-
-p
.
Theo em
1
.
I
:
T
2
->
T
2 is
a
homeomo phism
o
ini e
o de ,
hen
i
is
o
-
p
conjuga e
o
one
o
he
ollowing
:
BRAID
TYPES
WITH
ZERO
ENTROPY

54
5
(a)
T(. oyq/I'o o
some
q
E
NI,
p
E
íq,
wi h
p,
q ha ing no
common
ac o
(o de
q)
.
(b)
R
u
,/Fo,
w=
o /2,
w, o
R
W
/Fá,
w=
7 /3,
27 /3
(o de s
4,,2,6,,1
espec i ely)
(see
Figu e
1)
.
(c)
o
T(
p
o)1
q
/Fo,
o
some
q
E
N1, p
E

wi h
p,
qha ing
no
common
ac o
(o de
q
i
q
is
e en,
o de
2q
i
q
is
odd)
.
(d)
o
R,
/2/Fo
(o de
2)
(see
Figu e
2)
.
No
wo
o
he
aboye
a e
o
-
p
conjuga e
.
The
p oo
o
Theo em
1
educes
o
he
classi ica ion
o
isome ies
o
o i,
a
well-s udied
p oblem
(e .g
.
see
[NS])
.
Howe e ,
we
ha e
no
ound
a
compac
ea men
in
he
li e a u e,
no one
which
conside s
he
ques ion
o
which
isome ies
a e
equi alen
up
o
o
-
p
conjugacy
.
So
we
gi e
a
de i a ion
in
he
Appendix
.
Fo
case
(b)
o
Theo em
1,
he
numbe s
o
pe iodie
o bi s o
smalle
pe iod
han
he
o de
a e
gi en
by
he
ollowing
heo em
.
Theo em
2
([E])
.
The
o a ion
o
o de
2 has 4
ixed
poin s
.
The
o a ions
o
o de 3 ha e 3
ixed
poin s
.
The
o a ions
o
o de
4
ha e
2
ixed
poin s
and
one
o bi
o
pe iod
2
.
The
o a ions
o
o de 6 ha e
one
ixed
poin ,
one
o bi
o
pe iod
2,
and
one
o bi
o
pe iod
3
.
(See
Figu e
1)
.
3
.
Resul s
We
i s ly
ecall
some
de ini ions
and
heo ems
ha
we
will
need
.
We
hen
go
on
o s a e
and
p o e
ou
esul s
.
Le
:
M
->
M
be
a
homeomo phism
o
a
su ace
o
genus
one,
hen
i s
comple ion
g
:
T
2
_
T
2
,
is
de ined
by
conside ing
M
o
be
T
2
minus
a
disjoin
union
o
equal
size
dises
D
i
,
and
ex ending
glaD
;
adially
in o
D
i
[E]
.
I
h( )
=
0,
hen
h(g)
=
0
.
We
say
a
simple
closed
cu e
on
M
is
o a ional
i
i is
homo opically
non- i ial
on
T
2
a e
illing
in
he
holes
.
We
ecall
he
ollowing
wo
esul s
.
Theo em
A
([LMl])
.
Le
:
X
->
X
be
a
homeomo phism
o
a
su ace
o
genus
ze o
.
I
is
o
-
p,
i
has
ei he
a
ixed
poin
o
an
in a ian
bounda y
componen
.
I
is
o
-
i
has
ei he
a
ixed
poin
o an
in a ian
bounda y
componen ,
o
an
o bi
o
pe iod
2
o
a
bounda y
componen
o
pe iod
2
.
So
in
bo h
he
o-p
and
o-
cases,
i
P
is
a
ini e
union
o
pe iodic
o bi s
o
,
and P
does
no
ha e
a
bounda y
componen
o
pe iod
1
(o
-
p)
o
pe iod
1
o
2
(o
-
),
we
can
always
append
a ixed
poin
o
pe iod
2
o bi o
P
in
o de
o
achie e
his
.
54
6

J
.
GUASCHI,
J
.
LLIBRE,
R
.S
.
MACKAY
Theo em
B
([LMl])
.
Le
:
X
-->
X
be
a di eomo phism
o
a
su ace
o
genus
ze o,
wi h h( )
=
0,
and
le
P
be
a
ini e
union
o
pe iodic
o bi s
.
(1)
I
P
con ains
a
ixed
poin
o
has
an
in a ian
bounda y
componen ,
hen
(a)
I
is
o
-
p,
[P,
]
has
a
ep esen a i e
which
is
a
dise
ee
.
(b)
I
is
o
-
,
[P,
]
has a
ep esen a i e
which
is
a
e e sing
dise
ee
.
(2)
I
is
o
-
,
and
P
con ains
no
ixed
poin
and
has
no
in a ian
bounda y
componen ,
bu
ei he
P
has a
poin
o
pe iod
,2
o
has a
bounda y
componen
o
pe iod
2,
hen
[P,
]
has
a
ep esen a i e
which
is
a
e e sing
annulus
C ee
.
Le
:
M
,
M
be a
di eomo phism
o a
su ace
o
genus
one
wi h
h( )
=
0,
and
le
P
be
a
ini e
union
o
pe iodic
o bi s
o
.
Le
F
be
a
Thu s on
canonical
o m
o
P
.
Then
om
Thu s on's
classi ica ion
o
su ace
homeo-
mo phisms,
F
is
ei he educible, o o
ini e
o de
.
Fi s
we
conside
he
case
ha
F
has
a
o a ional
educing
cu e
.
Theo em
3
.
Suppose
:
M
-->
M
is
a
di emo phism
o a
su ace
o
genus
one
wi h
h( )
=
0,
and
P
is
a
ini e
union
o
pe iodic
o bi s
o
.
Le
F
be
a
Thu s on
canonical
o m
o
,
such
ha
F
has
a
o a ional
educing
cu e F,
o
pe iod
p
.
Remo e
he
ubula
neighbou hood
o
F
and
i s
images,
o
ob ain
a
disjoin
union
o
punc u ed
annuli
A
i
.
Then
(i)
I
all
Aá
ha e
pe iod
p,
hen
[P,
]
has
a
ep esen a i e
which
is
p
annuli,
joined
by
gene alised
wis s,
pe mu ed
like
a
o a ion,
and
i
is
o
-
p
o
p
is
e en
he e
exis s
a
dise ee
d
:
X'
~
X'
such
ha
FPI
A
j
-
d
o
i
is
o
-
and
p
is
odd
he e
exis s
a
e e sing
dise
ee d'
:
X'
-+
X'
such
ha
FPIAj
-
d'
.
(ii)
Suppose
some
Ai
has
pe iod
q
:~
p
(so
q
=
p/2),
hen
p
=
2,
and
he e
a e
wo
such
annuli
Al,
A
2
.
I
is
o
-
p,
[P,
]
has
a
ep esen a i e
which
is
wo
annuli,
joined
by
gene alised
wis s,
and
he e
exis
dise
ees
Di
:
X'
-+
X',
i
=
1,
2 such
ha
FI
Ai,
-
D
i
.

I
is
o-
,
[P,
]
has
a
ep esen a i e
which
is
wo
annuli,
joined
by
gene alised
wis s,
and
o
each
o
Al,
A
2
he e
exis
e e sing
dise
ees
D'
:
X'
,
X'
such
ha
FIAj
-
D'
o
e e sing
annulus
ees
S
i
:
X
-->
X,
such
ha
FIAj
-
Si,
acco ding
as
FIAj
has an
in a ian
bounda y
componen
o
ixed
poin ,
o
no
.
P oo
.
Suppose
F
has pe iod
p
.
De ine
A
=
{Fk
:
0
<
k
<
p}
.
Then
F
pe mu es
he
elemen s
o
A
.
I
we
emo e
he
ubula
neighbou hood
o
F
and
i s
images,
we
ob ain
a
disjoin
union
o
annuli
A
i
wi h
holes,
which
a e
BRAID
TYPES
WITH
ZERO
ENTROPY

547
pe mu ed
.
Then
he
Ai
ei he
ha e pe iod
p,
o
i
p
is
e en,
some
o
he
A
i
may
ha e
pe iod
p/2
(Le
.
he
wo
bounda ies
o
A
i
in
A
a e
in e changed
by
Fp/
2 )
.
This
gi es
wo
cases
:
(i)
I
he
A
i
ha e
pe iod
p,
F
pe mu es
he
Ai,
and
he e
a e
wo
subcases
:
(a)
I
is
o
-
p,
o
i
is
o
-
and p
is
e en,
hen
FP
I
A
¡
is
o
-
p,
so
by
Theo em
B,
he e
exis s
a disc ee
d
:
X'
,
X'
such ha
FP
I
A
j
-
d
.
(b)
I
is
o
-
and p
is
odd,
hen
FPI
Aj
is
o
-
,
so
by
Theo em
B,
he e
exis s a
e e sing
disc
ee
d'
:
X'
-
X'
such
ha
FPIAj
-
d'
.
(ii)
Suppose
p
is
e en,
and
he e
exis s
some
A
i
(wi hou
loss
o
gene ali y
i
=
0)
such
ha
Ai
has
pe iod
p/2
.
Pu q
=
p/2
.
Then
we
claim
ha
q=1
.
To
p o e
he
claim,
suppose
Ao
has pe iod
q
>
1
.
W i e
Ak
=
Fk(Ao),
1
<
k
<
q
.
Conside
he
si ua ion
on
he
o us
;
hen
each
Ak
has
wo bounda y
componen s
k
ll
,
k
2
)
E
A,
say,
o
0
<_
k
<
q
.
Then
o
j
=~
k,
I'~')
~
kl,
i l
=
1, 2,
Le
.
n o
wo
dis inc
Ak
ha e
a bounda y componen
in
common
.
Fo
suppose
FjZI
.
=
k
¿
l
o
some
i,
l
E
{1, 2},
j
=,A
k
(see
Figu e
3(a))
.
Then
since
F
9
in e changes
k
l l
and
k
2
)
o
each
k,
hen
F
9
( ~
l
l)
=
Fui--1)
=
F9(Fkil)
Fk +I)
.
hence
he e
a e
only
wo
dis inc
elemen s o
A,
so
q
=
1,
a
con adic ion
.
Since
he
Ak
ha e no
bounda y
componen s
in
common,
he e
exis
p ecisely
q
ubula
egions
Bl,
.
. .
,
B
9
,
such
ha
each
Bi
lies
be ween
wo
Ak,
Le
.
each
Bi
has
wo
dis inc
elemen s
o
A
as
i s
( o a ional)
bounda y componen s
.
Conside
one
such
elemen
o
{Bi}9
1
,
B,
say
.
Then
wi hou
loss
o
gene -
ali y,
i
lies
be ween
Ao
and
Ak,
o
some
k,
and
has
bounda y componen s
óll,
1'kl)
E
A
(see
Figu e
3(b))
.
B
is
in a ian
unde
F
2q
.
Since
has pe iod
2q,
B
mus
ha e
pe iod
ei he
q
o
2q
.
Suppose
i
has pe iod
q,
hen
F
9
in e -
changes
i s
bounda y
componen s,
soFk
l)
=
F
9
( o
l
»
.
Bu
F
9
in e changes
he
bounda y componen s
o
Ao
so
F
9
(Fó
')
)
= o
e
) ,
he e o e
Ao
and
Ak
ha e
a
bounda y componen
in
common,
which
we
ha e
shown
does
no
occu
.
Hence
B
has
pe iod
2q
.
Howe e
F
k
(Ao)
=
Ak,
ó
k
<
q,
and
conside ing
F
k
+q(Ao)
=
Ak,
i
nec-
essa y,
hen
F
k
( (
1
»
=
Fkl),
and
F
k
( o
21
)
=
k
21
.
Bu
he
elemen s o
A
a e
pe mu ed
like
a
o a ion
o
o a ion
wi h
e lec ion
by
F,
since
is
a
di eo-
mo phism,
hence
F
k
(Fk
l
» =
ól),
he e o e
F
k
(B)
=
B,
a
con adic ion,
as
B
has
pe iod
2q
.
This
p o es he
claim
.
So q
=
1,
and
has
pe iod
2,
wi h
A
=
{F,
F(F)}
.
I
we
emo e
he
ubula
neighbou hoods
o
and
F(I'),
we
ob ain
wo
disjoin
annuli
A
o
and
A1
.
Then
he e
a e
wo
subcases
o
conside
:

54
8

J
.
GUASCHI,
J
.
LLIBRE,
R
.S
.
MACKAY
(a)
I
is
o
-
p,
hen
FIA
j
(i
=
0, 1)
is
o
-
p,
so
by
Theo em
B,
[P,
]
has
a
ep esen a i e
which
is
wo
annuli
joined
by
gene alised
wis s,
such
ha
FI
A
j
is
a
dise
ee
.
(b)
I
is
-
o,
hen
FEA
¡
(i
=
0,1)
is
o
-
,
so
by
Theo em
B,
['P,
]
has
a
ep esen a i e
which
is
wo
annuli
joined
by
gene alised
wis s,
such
ha
FI
A
j
is
a
e e sing
dise
ee o a
e e sing
annulus
ee,
acco ding
as
has
an
in a ian
bounda y componen
o
P
con ains
a
ixed
poin
in
Ai,
o
no
.
This
comple es
he
p oo
o
Theo em
3
.
Nex
we
conside
he
case
whe e
he e
is
no
o a ional
educing
cu e
.
Ei he
F
is
o
ini e
o de , o
has
a
non- o a ional
educing
cu e
C,
in
which
case
we
may
emo e
he
decomposi ion
componen e
(o
genus
ze o) o
C
and
i s
images
om
M
.
So
in
bo h
cases,
we
can
ind
a
unique
decomposi ion
componen
S
o
genus
one,
such
ha
FAS
is
o
ini e
o de
.
Le
G
:
T2
->
T
2
be
he
comple ion
o
FI
S
.
Then
G
is
o
ini e
o de ,
so
is
o
-
p
conjuga e
o
one
o
he
cases
o
Theo em
1
.
This
leads o
he
ollowing
heo em,
which
is
ou
main
esul
:
Theo em
4
.
Suppose
F
:
M
-
M
is
a
Thu s on
canonical
o m
o a
homeomo phism
o
a
su ace
o
genus
one,
wi h
unique decomposi ion
compo-
nen
S
o
genus
one
.
Le
G
:
T
2
->
T
2
be
he
comple ion
o
FAS
.
Then
one
o
he
ollowing
is
ue
:
(a)
G-
T(P
.o)/q/ o,
and
all
bounda y
componen e
o
S
ha e
pe iod
q
.
F~a -
he ,
o each
o bi
a
l
,. ..,
á
q
o
bounda y
componen e
o
S
no
in
aM,
he e
exis s
a
dise
ee
d
:
X'
,
X'
such
ha
he
componen
Xi
o
M S
inside
Ói is
homeomo phic
o
X',
and
F
9
IXi
-
d
.
(b)
G
-
R,/I'o,
w
=
7 /2,
7
(o de
k
=
4,
2),
o
G-
R

,/ o,
w
=
o /3,
±27x/3
(o de
k
=
6,
3)
.
I
k
=
2,
hen
all
bounda y
componen s
o
S
ha e
pe iod
1 o
2,
wi h
a
mos
4
o
pe iod
1
.

I
k
=
3,
hen
all
bounda y
componen s
o
S
Na e pe iod 1
o
3,
wi h
a
mos
3
o
pe iod
1
.
I
k
=
4,
hen
all
bounda y
componen s
o
S
ha e
pe iod
1,
2
o
4,
wi h
a
mos
2
o
pe iod
1,
and
1
o bi
o
pe iod
2
.
I
k
=
6,
hen
all
bounda y
componen s
o
S
ha e
pe iod
1, 2,
3
o
6
wi h
a
mos
1
o
pe iod
1,
1
o bi
o
pe iod
2,
and
1
o bi
o
pe iod
3
.
In
case,
o
each
o bi
o
bounda y
componen s
o
pe iod
p,
al,
.
. .
,
O
P
,
o
S, no
in
OM,
he e
exis s
a
dise ee
d
:
X'
-4
X'
such
ha
he
componen
X
i
o
M S
inside
á
i
is
homeomo phic
o
X',
and
FPIXi
-
d
.
(e)
(i)
G-
(
o
TOP
o)/q)/Po
.

I
q
is
e en,
all
bounda y
componen s
ha e
pe iod
q
.
I
q
is
odd,
all
bounda y
componen s
ha e
pe iod
q o 2q
.
(ii)
G-
( oR,/2)/I'o
.
All
bounda y
componen s
o
S
ha e
pe iod
ei he
1
o
2
.
BRAID
TYPES
WITH
ZERO
ENTROPY

54
9
In bo h
pa s
o case
(c),
o each
o bi
o
bounda y
componen s
o
pe iod
p,
91
5
. . .
,
ap,
o
S
no
in
OM,
he e
exis s
a
disc
ee
d
:
X'
-
X'
such
ha
he
componen
Xi
o
M S
inside
Vi
is
homeomo phic
o
X'
and
F
P
I
Xi
-
d
i
p
is
e en,
o
he e
exis s
a
e e sing
disc
ee
d'
:
X'
-
X'
such
ha
he
componen
Xi
o
M S
inside
al is
homeomo phic
o
X'
and
F
P
I
Xi
-d
i
p
is
odd
.
P oo
..
We
apply
Theo em
1 o
he
comple ion
G
:
T2
,
T
2
o
FAS
.
Case
(a)
G
^-
TO P
o)/q/I'o,
so
all
poin s
o
T
2
ha e
pe iod
q
unde
G
.
Hence
all
bounda y
componen s
o
S
ha e
pe iod
q,
and
i
we
conside
he
o bi s
a
l
, . . .
.
0,
o
hose
no
in
aM,
he
co esponding
decomposi ion
componen s
Xi
o
M S
inside
al
a e o
genus
ze o,
and we
may
apply
Theo em
B, so
he e
exis s
a
disc
ee
d
:
X'
->
X'
such
ha
X
i
is
homeomo phic
o
X',
and
F
4
1Xi
-
d
.
Case
(b)
G
-
R

,/I'o,
w=
7 /2,
n
:
k
=
4,
2,
o
G -
Ru,/I'o,
w=
7 /3,
±27 /3
:
k
=
6,
3
.
The
s a emen s
abou
he
o bi s
o
bounda y
componen s
a e
an
immedia e
consequence
o
applying
Theo em
2
o
he
comple ion
G
:
T
2
->
T
2
.
As
in
case
(a),
since
G
is
o-
p,
we
ob ain
disc
ees
in
each
o bi o
bounda y
componen s
o
S
no
in
8M
.
Case
(c)
(i)
G-
(
oTO
P
o)/
a
)/I'o
.
F om
he
p oo
o
Theo em
1,
he
bounda y componen s
o
S
ha e
pe iod
q
i
q
is
e en,
o
hey ha e
pe iod
q
o
2q
i
q
is
odd
.
Case
(c)
(ü)
G
-
(
o
R, /2)/I'o
.
Again om
he
p oo
o
Theo em
1,
he
bounda y componen s
o
S
ha e
pe iod
1
o
2
.
In
bo h
pa s
o
case
(c),
each
componen
o
T
2
S
inside a
bounda y
com-
ponen
no
in
8M
o
e en
pe iod
con ains
a
disc
ee since
F
is
o
-
p,
whils
hose
o
odd
pe iod
'
con ain
a
e e sing
disc ee since
F '
is
o- ,
by
Theo em
B
.
55
0

J
.
GUASCHI,
J
.
LLIBRE,
R
.S
.
MACKAY
4
.
Two
Co olla ies
As
a
co olla y
o
he
esul s
o
Sec ion
3
we
ob ain
he
genus
one
case
o
[H1]
:
Theo em
5
.
Le
:
M
-M
be
an
o-
di eomo phism
o
a
su ace
M
o
genus
one
.
I
has
pe iodic
o bi s,
o
o bi s
o
bounda y
componen s,
wi h
3
dis inc
odd
pe os,
hen h( )
>
0
.
To
de i e
his
om
he
abo e,
we
equi e
he
genus
ze o
esul
o
[BF],
[H1],
also
de i ed
by
[LM1]
:
Theo em
C
.
Le
:
X-X
be
an
o
-
di eomo phism o
a
su ace
o
genus
ze o
.
I
has
pe iodic
o bi s
o
o bi s
o
bounda y
componen s
wi h
wo
dis inc
odd
pe os,
hen
h( )
>
0
.
P oo
o
Theo em
5
:
Le
:
M
-~
M
be an
o-
di eomo phism
o
a
su ace
M
o
genus
one
.
Le
P
be
he
union
o
h ee
o bi s
o
dis inc
odd
pe iod
.
Le
F
be a
Thu s on
canonical
o m
o
P
.
Suppose
h( )
--
0
.
Then
he e
a e
wo
cases
:
Case
(a)
:
Suppose
he e
is
a
o a ional
educing
cu e,
I',
say,
wi h
pe iod
p
.
Remo e
i s
annula
neighbou hood
and
i s
images,
hen
we
ob ain
a
disjoin
union
o
annuli
A
i
.
F om
Theo em
3,
he e
a e
wo
possibili ies
.
In
he
i s
case,
he
A
i
ha e
pe iod
p,
so
he
Ai
a e
pe mu ed
by
F, and
hus
p
di ides
he
o de
o
each
pe iodic
o bi ,
hence
p
is
odd
.
So
i
we
conside
any
Ai,
F
P
¡A¡
is
o
-
,
since
is
o
-
.
Ai mus
con ain
bounda y
componen s
co esponding
o
he
h ee
o bi s o
dis inc
odd
pe iod
.
So
by
Theo em
C,
h( )
>
0,
a
con adic ion
.
The
o he
possibili y
is
when
p
=
2,
and
he e
a e
wo
in a ian
annuli
Ao,
Al
.
Then
one
o
Ao,
Al
mus
con ain
bounda y
componen s
co esponding
o a
leas
wo
o
he
h ee
o bi s,
and
since
FIA
jj
(i
=
0,1)
is
o
-
,
Theo em
C
implies
ha
h( )
>
0,
a
con adic ion
.
Case
(b)
:
Suppose
he e
is
no
o a ional
educing
cu e
.
Then
he e
exis s
a
unique
decomposi ion
componen
S
o
genus
one
.
Le
G
:
T2
-> T2
be
he
com-
ple ion
o
FAS
.
Thenwe
a e in
Case
(c)
o
Theo em
4
.
I
G-
( -T(P
o)M/I'o,
all
bounda y
componen s
o
S
lla e
pe iod
q o
2q
.
I
G-
(
o
R,/2)/I'o,
hen
all
bounda y
componen s
o
S
lla e
pe iod
1 o 2
.
In pa icula ,
all
bounda y
componen s
o
S
o
odd
pe iod
lla e
he
same
pe iod
p,
and
he
emaining
bounda y
componen s
co esponding
o
P
mus
lie
wi hin
decomposi ion
com-
ponen s
X
i
o
genus
ze o
whose
ou e
bounda ies
ha e
pe iod
p
.
Since
F
P
I
Xi
is
o- ,
and
a
leas
one
o
he
X
i
con ains
bounda y componen s
o
odd
o de ,
BRAID
TYPES
WITH
ZERO
ENTROPY

55
1
no
p,
co esponding
o
he
o bi s o
P,
hen
Theo em
C
implies ha
h
(
)
>
0,
a
con adic ion
.
As
a
second
co olla y
we
will
de i e
a
esul
o
[LM2],
[H2]
o
di eomo -
phisms
o
he
o us
iso opic o
he
iden i y
.
We
e e
he
eade
o
[LM2]
o
de ini ions
o
li s,
o a ion
ec o s,
e c
.
We
ecall
ha
o
a
con inuous
map
:
T2
,
T2
wi h
li
:
R2
,
I 8
2
,
i
I
is
he
g oup
o in ege
ansla ions
y
,,
:
x
1-->
x
+
m,
xE
R2,
m
E
Z2, and
i
is
homo opic
o
he
iden i y,
hen
y
=
-y
o
all
"
y
EF
.
Theo em
6
.
Le
:
T
2
-
3
T
2
be
a
homeomo phism
o
he
o us
iso opic
o
he
iden i y,
an
.d
suppose
h( )
=
0
.
Then
all
o a ion
ec o s associa ed
wi h
he
pe iodic
o bi s
o
a e
collinea
.
P oo
..
Suppose
ha
has
a
ini e
union
o
pe iodic
o bi s,
wi h
associa ed
dis inc
o a ion
ec o s
pi/gi,
i
=
1,
. . . ,
N
.
Then
om
[LM2],
since
is
homo opic
o
he
iden i y,
o
each
i
E
{1,
. . . ,
N}
he e
exis s
a
pe iodic
o bi
Qá
o
p imi i e
o a ion
ype
(pi,
qi)
.
Le
P
=UN
1
Qá,
and
le
F
be
a
Thu s on
canonical
o m
o
P
.
Suppose
F
is
o
ini e
o de ,
hen
using
(*)
and
Theo em
1,
we
see
ha
F
-
T(P
o)1e/I'o,
o
some q
E
N,
pE
w,
so
all
poin s
o
T
2
a e
pe iodic
wi h
pe iod
qand
o a ion
ec o
(p,
0)/q
.
The
emaining
possibili y
is
ha
F
is
educible (wi h
ini e
o de
componen s,
hough
we
shall
no
need
his)
.
The e
a e
no
non- o a ional
educing
cu es
o
F
.
Fo
suppose
F
we e
such
a cu e,
hen
i
mus
su ound
a
leas
wo
holes,
bu
hey
mus come
om
he
same
o bi ,
so
he
o a ion
ype
o
ha
o bi
canno
be
p imi i e,
which
is
a
con adic ion
.
Suppose
he e
is
a
o a ional
educing
cu e
1'
o
F
.
Le
G
:
T
2
->
T
2
be
he
comple ion
o
F,
and
le
G
:
R2
-->
R2
be
a
li
o
G
.
Le
m
E
71
2
{0} be
he
homo opy
ype
o
1',
and
q
be
i s
pe iod
.
Then
F
li s
o
an
in ini e
se
o
cu es
F,
each
in a ian
unde
T

,,,
which
pa i ion
he
plane
hi o
in ini e
s ips
S
each wi hin
a
bounded
dis ance
o
some
s aigh
line
o
di ec ion
m
.
Fu he mo e,
he e
exis s
pE
71
2
such
ha
FqT
F
=
1',
and
since
F
is
in e ible
he
same
holds
o
he
s ips
S
.
Hence
he
o a ion
se
o
F, and
in
pa icula
he
o a ion ec o s
o
he
chosen
pe iodic
o bi s,
a e
con ained
in
he
s aigh
line
{p/q
+
m
:
E
IR}
.
This
comple es
he
p oo
.
Appendix
:
P oo
o
Theo em
1
To
p o e
Theo em
1,
we
equi e
he
ollowing
:
558

J
.
GUASCHI,
J
.
LLIBRE,
R
.S
.
MACKAY
[NS]
V
.V
.
NIKULIN
AND
I
.R
.
SHAFAREVICH,
"Geome ies
and
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