Pub
.
Ma
.
UAB
N°
22
No
.
1980
Ac es
VII
JMHL
STABILITY
OF
PARABOLIC
POINTSOF AREA
PRESERVING
ANALYTIC
DIF-
FEOMORPHISMS
Ca lesSimó
Facul a
de
Ma emi iques
Uni e si a
.d
e
Ba celona
Abs ac
.
Theo ems
cha ac e izing
s able
pa abolic
poin sa e p o ed
.
Essen-
ially,
s abili y
is
equi alen
o
he
ac
ha
he
gene a ing
unc ion
o
he
di e omo phism,
akingou he
pa
which
gene a es he
iden i y,
has
a
s ic
ex emum
a
he
ixedpoin
.
Wi h
hese
esul s,
he
s udy
o
he
s a
bili y
o
ixed
poin s
o
analy ic
a ea
p ese ing
mappings
(APM) is
ended
.
Some
examples
a e
included,
specially' he
case o
ellip ic
poin s
whose
el--
gen aluesa e
cubic
o ou h
oo s
o
uni y
.
51
.
In oduc ion
and
esul s
.
Le
T an
analy ic
APM
.
The
(Lyapuno )
s abili-
y
o
ixed
poin s
o T is a
me hod
usually
employed
_o
he
s udy
o
he
qua
li a i e
p ope ies
o
pe iodic
o bi s
(P
.
O
.)
in
hamil onian
sys ems
wi h
wo
deg ees
o
eedom,
ia he
Poinca é
mapping
wi h
espec
o_a
su ace
ans
e sal
o
he
P .O
.
in
he
ene gy
le el
H
=h
.
I
he
ixed
poin ,
ha we ake
as
he
o igin,
is
hype bolic,
he ines abili y
o
he
linea
pa
emains
when
nonlinea
e ms
a e
aken
in o
accoun
.
I
he
ixed
poin
is
ellip ic
he
s abili y
o
he linea
pa
is
p ese ed
p o ided
ha
he
éigen alues
a e no
hi d
o
ou h
oo s
o
uni y
and
ha
sui ablecoe icien s
o
he
Bi kho
No mal
Fo m
(B.N .F
.)
a e
no
ze o
.
When
he
ixed
poin
is
deg_e
ne a ed
o
pa abolic,
i
.e
.,
Spec
DT(0)C{±1},
he s abili y
is a
mo e
sub le
ques ion
.
I is
no alwaysenough
o
conside
only
he
lowe
deg eenonli-
-
nea
e ms
o
decide
abóu
s abili y
.
Besides
he
cases
A
3
=1
and
14
=
1,
di--
icul ies
can appea
o
e e y
A,
k- h
oo
o
uni y
i
all
he
de e mined
coe icien s
( he
i s
[
k
2
2
]
ones)
in
he
B
.N
.F
.
a e
.ze o
.
Example
s
o
ines
abili y
exis
o all
k
[91
.
The
case o
A
being
a
k- oo
o
uni y
is
edu-
ceo
o
ne pa abolicone axlng
'1'
ins eaa
o=
'l .
wl nou -
loss
OT
:j-
:
.c alll`
we
can
suppose
ha
in
he pa abolic
case
he
eigen alues
a e
equal
o
one
.
(Take
T2 i
necessa y
.
This
accoun s
also
o
T
o ien a ion
e e sing)
.
In
[111
he ollowing
esul s
a e
p o en
o
he,pa abolic
casewhen
DT(0)
0
can no
be
educed
o
diagonal
o m,
i
.e
.,
DT(0)
=(
;)
in a
sui able
basis
:
1
.1
.
Lemma
.
Le
T(x,y)
= (x +
(x,y),
x+y+g(x,y))
be an
analy íe
APM
uwí h
,g
beg
.ínn,¿ng
wi h
. enme
oj
degnee
a
.Leas
wo
.
Then
heAe
ex,í,a 6
a neah
.
he
.íden, c y
polynomía
.2
change
o6
a~u
:ables
c such
. ha
xhe
ms6onmed
mapp
.íng
T*
=c
-1
Tc

.íl6
g
.í en
by
T*
(x,
y)
=
(x+F
n
(x,y)+on+1
,
x+y+O
n+1
)whexe
Fn
.c,s
a

de-
gnee
n
polynom
.í
.a
e
.
w hou
P¿nean
. enme
and
os
e and6
6oA
a
&eA,íeh
wí h
zeAme
o6
.1'
oweA
degAee
a
keas
s
.
1
.2
.
Theonem
.
In
he
hypo hee
.í s
o6
1
.1
l e
P
n
(z)
=
amzm+0m+1'
a
,~
o
.
Then
he
o ígín
.íes
e~e
undeA
T*
(and
heAelo e
unde
T)
í66
m
íz
odd
and
a
m
<O
.
The objec
o
he
communica ion
is
o
gi e
a
heo em
cha ac e izing

he
s able
pa abolic
poin s o he emaining
case,
i
.e
.,
when
DT(0)
can
be
pu
in
diagonal o m
.
(Then
T
is
nea
he
iden i y
in
a
neighbou nhood
U o
he
ixed
poin )
.
Le
(x',y')
=
T(x,y)
a
canonical
mapping
.

I
Dy y'
is
egula

(as
happens
in
ou
case)
we
can de ine
an
analy ic
gene a ing
unc ion
(see
[11)
G(x,y')
such ha
G(x,y')
=
xy'
+G
(x,y')

and
x'=
D

y=
D
C
.
Fo he
nondianonal
yx
case
o

1
.2
.
we
ge
G(x,y')=-x
2
/2
+
Fn
(u)
du
+0
n+2(x,y')
.
Theo em
1
.2
can
0
be
e o mula ed
as
:
s abili y
is
equi alen
o
G(x,y')
ha ing
a
s ic
ex e
mum
a
he
o igin
.
Tha
his
cha ac e iza ion
is
applicable
o
he
diagonal
case
is
s a ed
in
he
main
esul
:
1
.3
.
Theonem
.
Le
Pbe a
paAabolc
:c
6
.íxed
po,íw
o6
an
analy í
.c
APM,
T,

and
G(x,y')
=xy'
+G(x,y')
a
geneAa í,ng
6unc c
:an
60A T
.
Then
P
Zb
Lyapuno
6.2e
i« Ghay
a
a h,íc
ex xemum
a
P
.
ones
o
McGehee
[71
bu
only
o
he
conse a i e
case
.
§2
.
Ske ch
o
he
p oo
.
Only
he
case
DT(0)
=

1
0
~0
1/
o
1
.3
.
emains
o be
p o ed
.
Ins ead
o
using
he
ac
ha
G
has
a
s ic
ex emum
a
he o igin
we
can equi alen ly
conside
ha ,
nea
he
o igin,
he
se s
G=g
wi h
Ig1
small
and
sui able
sign,
a e closedcu esa ound he o igin
.
The
algo i hm
o
decide
whe he
o
no
G
has
a
s ic
ex emum
a
he
o igin
using
he
New on
polygon
is
de e ed
o
he
nex
sec ion
.
68
As
a
as
ins abili y
is
conce ned
he
esul s
ob ainedhe e
ex end
he
Le
(D
1
be
he
ime uni
low
associa ed
o
he
hamil onian
sys em
wi h ha
mil onian
G
:
m
1
(x,y)
=
(x,y)
.
We
in end
o
use
m1
as
an
app oxima ion
o
T
in
U
.
By
he
way,
i
G,
iis
he
pa ialde i a i e
o
Gw.
.
.
he
i- h
a gumen ,
G
i
,
Gi
,
k
,
. .
.
he
second,
hi d,
.
.
.
pa ial
de i a i es,
be e
app oxima-
) 7
ions
o T
can
be
ob ained
wi h
modi ied
hamil onians
:
H = G
-
2
G
1
G
2
+
112
(G11G2
+ 4G
12
G
1
G2 +
G
22
G~
)
-
6
(G
112G
1
G2 +
G
122
G
2
G2
+
G
11
G
12
G2 +
+
G
22G12
G
2+
G11G22G1G2
+3G
2 2
G
1
G
2
)

+
. .
.
We
ge
inc easingapp oxima ion
aking
e ms
o
inc easingo de
.
Howe e
H=G
is
enough
o
he
p oo
.
In U -
(01

we
de ine
=G(x,y),
a=2n /T( )whe e
T( )
is
he pe iod
o
he
low o
hamil onian
G
along
he closed
cu e
Y=1G(x,y)= I
.
He e
s ands
he
ime
in e al
in
going
om
(x
0
,0)
o
(x,y)
along
Y,
wi h
x0
>0

(one
shows
ha
Yis
s a -shaped
w.
.
.
he
o igin
i
I l
is
small
enough)
.
he
( ,a)
a iables
one
has
m
1
( ,a)
=
( ,a+2n/T( ))
.
A
d
T( )/d
=
0( p),
/3<0
.
The e o e
p essed
in
he
( ,a)
a iables
as
T( ,a)
=
( +¿á ,
a
+2n/T( )+,áa),
begin
wi h
e ms
o
ela i e
high
o de
.
Hence
T
can
be seen
as
wis
[81
and
his
gua an ees
he
exis en e
o
in a ian
cu es om
he
s abili y
ollows
.
0
1
is
a
wis
.
The
In
compu a ion
gi es
ini ial
map
can
be
whe e
A ,Ja
a
pe u bed
whe e
I
G=g,
Ig1
small, does
no de ineclosed
cu es in
U
bu G=0 has
al
b anches
h ough
he o igin
hen we
ge
ins abili y
unde
01
[61
he e o e,unde
T
.
Comple e
p oo sappea
in
[121
.
o
ex
se e--
and,
§3
.
An
algo i hm
o
decide
abou
s abili y
.
Fi s
we
plo
he
New onpoly--
gon
associa ed
o
G
.
A
necessa ycondi ion
o
s abili y
is ha
all he
e icesha e
e en
coo dina es
.
Le
m+ka,
n-k/3,
a,PEZ
+
,
g
.c .d
.
(a,P)=1,
m+ka
k=0
:
be
poin s
in
one
side o
he
polygon
.
The e
we
ge
G=
.
.
.+Y- a
k
x
,y
n-k~3
+
. .
.
.
Le
(P=

akzk

0
.
An
addi ional
necessa y
condi ion
is ha
all
0
he
eal
ze os
o
(p
be o e en
mul iplici y
.
I
he
mul iplici y
is
ze o
his
is
enough
o
s abili y
.
I
i
is
no
ze o, h ee cases
a e
posible,
associa ed
o
each
o such
ze os
:

y=0
(x)
;

x
=
0
(y
a
)
,
a
>
1
;

y = 0
(x
a
)
,
a>
1
.
The
i s
and second
cases
can
be
educed
o
he
hi d
one
h ough
a
o a ion
o
a
elabelling
o
he
axes,
espec i ely
.
The e o e,
we
can
suppose
y
=
mx
p/q
+
.
.
.
,

p/q
>l
.

In oducing
x
=u
q
,
y =
mu
p
+z,
we
ge
a
new
New on
polygon
and
we
p oceed
o
he
analysis
o
e ms
o
he
o m
z
=
0(Up),P>q
.
§4
.
Some
examples
.
a)
We
conside
he
case
,1
a
ou h oo o
uni y
.
A
simple
map
is T(x,y)
=
_
(-y,x+ (y))
(a
de
Jonquié e
map,
no mal
o m
i
Tis a
C emona
map
o
p i
me
deg ee
[31
) .
Taking
T
4 we
can
apply
1
.3
and
§3
.
I
begins
wi h
e ms
o
deg ee
k
and
k is
odd,
he
o igin
is s able
.
I k is
e en
and
has
only
one
e m
(yk
a e
scaling)
he
o igin
is
s able
.
This
is
he
case
o he
classical
Hénon
map
[5,101
wi h
o a ion
angle
a
=7 /2
(k=2)
.
The in a ian
cu es
a e
la in
c oss
shaped
.
Howe e ,
highe
o de
e ms
can
p oduce
ins-
abili y
.
Fo
ins ance,
T(x,y)
=
(-y,
x+y
2
+ay
3
)
is
uns able
o
a
E[-1,0)
.See
[121
.
b) I
11
is
a
cubic
oo o
uni y
and
we
es ic
ou sel es
o T(x,y)
_
=
R2,c/3c)
(x,y
-x
k
)
,
whe e
Rá
is
a
o a ion
o
anggle
S
a ound
he
o igin,
we
ge
s abili y
(uns abili y)
i
k is
odd
(e en)
.
c)
Conce ning
he
es ic ed h ee-body
p oblem,
he s abili y
o
4
o
ma
sses
,a
equal
o
he
c i ical
alues
o
Rou h
(see
[21)
only
he alues
~¿2,u3
emain
o
se le he
ques ion
.
Wi h
he
help
o some
leng hly
compu a
ions
he
esul s
will
appea
elsewhe e
[131
.
O he
applica ions
o
he
s a
bili y
o
bi u ca ion
o bi scan
be
ound
in
[41
.
Re e ences
[11

A nold,
V
.
I
.,
A ez,
A
.
:"E godic
p oblems
o
classical
mechanics",
[21
[31
[41
[51
[61
[71
[91
min, 1968
.
Dép i ,
A
.,
Dép i ,
A
.,
As on,
J
.
72
(1967),
173-179
.
Engel,
W
.,
Ma h
.
Ann
.
136
(1958),
319-325
.
Gómez,
G
.,
Doc o al
Disse a ion,
Uni
.
Au ónoma
Ba celona,
1980
.
Hénon,
M
.,
Qua .Appl
.
Ma h
.
27
(1969),
291-312
.
.
Le sche z,
S
.
:
"Di e en ial
Equa ions
:
Geome ic
heo y",Wiley,
McGehee,
R
.,
J
.
Di e en ial
Eq
.
14
(1973),
70-88
.
Mose ,
J
. :
"S able
and andommo ion
in
dynamical
sys ems",
P ince on
Siegel,
C
.L
.,
Mose ,
J .K
. :
"Lec u es
in
Celes ial
Mechanics",
Sp inge ,
1971
.
[101
Simó,
C
.,
(111
Simó,
c
.,
[121
Simó,
C
.
,
[131
Simó,
C
.,
Uni
.
P ess,
1973
.
Ac as
V
Reunión
Ma emá icos
Exp esión
La ina,
Palma
1978,
361-369
.
P oceed
.
In e n
.
Con
.
GlobalTheo y
o
Dynamical
Sys ems,Ch_i
cago
1979,
o
appea
.
o
appea
.
o
appea
.
Benja
1963
.