Exponentially small estimates for KAM theorem near an elliptic equilibrium point
Abstract
We give a precise statement of KAM theorem for a Hamiltonian system in a neighborhood of an elliptic equilibrium point. If the frequencies of the elliptic point satisfy a Diophantine condition, with exponent $\tau$, and a nondegeneracy condition is fulfilled, we show that in a neighborhood of radius $r$ the measure of the complement of the KAM tori is exponentially small in $(1/r)^{1/(\tau+1)}$. This result is obtained by putting the system in Birkhoff normal form up to an appropriate order, and the key point relies on giving accurate estimates for its terms.
Full text
EXPONENTIALLY SMALL ESTIMATES FOR KAM THEOREM
NEAR AN ELLIPTIC EQUILIBRIUM POINT
AMADEU DELSHAMS
Depa amen de Ma ema icaAplicada I, Uni e si a Poli ecnica de Ca alunya
Diagonal 647, E-08028 Ba celona
AND PERE GUTI
ERREZ
Depa amen de Ma ema ica Aplicada II, Uni e si a Poli ecnica de Ca alunya
Pau Ga gal lo 5, E-08028 Ba celona
Abs ac .
We gi e a p ecise s a emen o KAM heo em o a Hamil onian sys em in a
neighb o ho o d o an ellip ic equilib ium p oin . I he equencies o he ellip ic p oin sa is y
a Diophan ine condi ion, wi h exp onen
, and a nondegene acy condi ion is ullled, we
show ha in a neighb o ho o d o adius
he measu e o he complemen o he KAM
o i is exp onen ially small in (1
=
)
1
=
(
+1)
. This esul is ob ained by pu ing he sys em
in Bi kho no mal o m up o an app op ia e o de , and he key p oin elies on gi ing
accu a e es ima es o i s e ms.
1. In o duc ion
We conside a Hamil onian, wi h
n
deg ees o eedom, ha ing he o igin as an ellip ic
equilib ium p oin . In sui able canonical co o dina es, his sys em can b e pu in he o m
H
(
q p
)=
X
s
2
H
s
(
q p
)
(1)
whe e
H
s
is a homogeneous p olynomial o deg ee
s
in (
q p
) o e e y
s
2, and
H
2
(
q p
)=
1
2
n
X
j
=1
j
q
2
j
+
p
2
j
:
(2)
We a e conce ned wi h he exis ence o
n
-dimensional in a ian o i
in a neighbo hood
o he ellip ic p oin . We s see ha he sys em (1{2) is nea ly-in eg able by pu ing i
in
Bi kho no mal o m
up o an app op ia e deg ee
K
4, p o ided he equency
ec o
=(
1
:::
n
) is non esonan up oo de
K
. A quan i a i e e sion o Bi kho
heo em, implici ly con ained in 4], allows us o ob ain es ima es o he no mal o m.
Like in 9], we conside ac ion{angle a iables in a neighb o ho o d o adius
. Assuming
a sui able nondegene acy condi ion, we apply he known KAM heo em and show ha
mos a jec o ies in a neighb o ho o d o adius
lie in in a ian o i: we ge o he ela i e
measu e o hei complemen an es ima e o he yp e
O
(
K
;
3)
=
2
.
We assume ha
sa ises a Diophan ine condi ion: wi h gi en
>n
;
1and
>
0,
j
k
j
j
k
j
8
k
2
Z
n
n
0
g
(3)
403
whe e wew i e
j
k
j
=
P
n
j
=1
j
k
j
j
.Wesay
o b e
-Diophan ine.
Ou main con ibu ion
is o show ha in his case he es ima es o he Bi kho no mal o m allowus ocho ose
he deg ee
K
as a unc ion o
, gi ing ise o an exp onen ially small es ima e o he yp e
exp
(
;
1
1
=
(
+1)
)
o he measu e o he complemen o hein a ian se .We ema k ha he es ima es
gi en in 4] do no allow o ob ain he exp onen 1
=
(
+ 1), bu a wo se one. Ne e heless,
we shall see ha an imp o emen o ha es ima es leads o he announced exp onen .
2. Es ima es o he Bi kho no mal o m
Gi en
K
4, assume ha he equency ec o
is non esonan up oo de
K
:
k
6
=0
o
k
2
Z
n
0
<
j
k
j
K
. The well-known
Bi kho heo em
1, 7] s a es ha , in some
neighb o ho o d o he o igin, he e exis s a canonical ans o ma ion
(
K
)
, nea o he
iden i y map, such ha
H
(
K
)
=
H
(
K
)
is in Bi kho no mal o m up o deg ee
K
:
H
(
K
)
(
q p
)=
I
+
Z
(
K
)
(
I
)+
R
(
K
)
(
q p
)=
h
(
K
)
(
I
)+
R
(
K
)
(
q p
)
(4)
wi h
Z
(
K
)
(
I
)=
X
4
s
K
s
e en
Z
s
(
I
)
R
(
K
)
(
q p
)=
X
s
K
+1
R
(
K
)
s
(
q p
)
(5)
whe e e e y
Z
s
(
I
) (uniquely de e mined) is a homogeneous p olynomial o deg ee
s=
2in
he ac ion a iables
I
j
=
1
2
q
2
j
+
p
2
j
j
=1
:::n
and e e y
R
(
K
)
s
(
q p
) is a homogeneous p olynomial o deg ee
s
in (
q p
). Since
h
(
K
)
(
I
)is
in eg able, and in a neighb o ho o d o adius
weha e
R
(
K
)
=
O
K
+1
, i u ns ou
ha
H
(
K
)
is a nea ly-in eg able Hamil onian nea he o igin. Howe e , o apply KAM
heo em o
H
(
K
)
we need quan i a i e es ima es o i s e ms.
As in 4], wein o duce he linea change o complex canonical co o dina es
x
j
=
1
p
2
(
q
j
;
ip
j
)
y
j
=
;
i
p
2
(
q
j
+
ip
j
)
j
=1
:::n:
No e ha
q
,
p
a e eal i
y
=
ix
.We dene
j
(
x y
)
j
:= max
j
=1
:::n
q
j
x
j
j
2
+
j
y
j
j
2
.Gi en
>
0,
he eal and complex p olydisks o adius
cen e ed a he o igin will b e deno ed
B
and
b
B
, esp ec i ely.
Fo a gi en homogeneous p olynomial
s
(
x y
)=
X
j
l
+
m
j
=
s
lm
x
l
y
m
,we dene he no m
k
s
k
:=
X
j
l
+
m
j
=
s
j
lm
j
(we use he no a ion
x
l
=
x
l
1
1
x
l
n
n
,
y
m
=
y
m
1
1
y
m
n
n
).
404
P op osi ion 1
Le
H
(
x y
)=
P
s
2
H
s
bea eal Hamil onian wi h
H
2
=
I
, and
assume ha
k
H
s
k
c
s
;
2
d
o
s
3
. Gi en
K
4
, assume ha
j
k
j
K
8
k
2
Z
n
0
<
j
k
j
K
(6)
wi h
0
<
K
1
. Then, he e exis s a eal canonical ans o ma ion
(
K
)
,nea o he
iden i y map, such ha
H
(
K
)
=
H
(
K
)
is in he Bi kho no mal o m
(4{5)
up o
deg ee
K
. Wi h some cons an s
c
1
,
c
2
, one has:
a)
kZ
s
k
c
2
c
s
;
2
1
(
s
;
2)!
3
s
;
1
o
4
s
K
and
s
e en.
b)
R
(
K
)
s
c
2
c
s
;
2
1
(
K
;
3)!(
K
;
2)
s
;
K
+1
3
K
;
1
s
;
K
+1
K
o
s
K
+1
.
c)
The ans o ma ion
(
K
)
is analy ic on
b
B
K
whe e we dene
K
:=
K
c
1
K
:
These es ima es ely on he esul s ob ained in 4], al hough a di ec applica ion would
gi ewo se es ima es, wi h
s
;
3
K
ins ead o
3
s
;
1
in he denomina o s. The imp o e-
men comes om he ac ha , in he cons uc ion o he no mal o m, he only small
di iso s which app ea up o he ob ainmen o
Z
s
co esp ond o he o de s 3
:::s
;
1.
This is c ucial in o de o ge he igh exponen in he es ima es gi en in he las sec ion.
3. Applying KAM heo em
We s ecall a usual s a emen o KAM heo em. Le us conside a nea ly-in eg able
Hamil onian w i en in ac ion{angle a iables
H
(
I
)=
h
(
I
)+
(
I
)
wi h
2
T
n
and
I
2G
R
n
.To show ha mos o he a jec o ies o
H
lie in
n
-dimensional in a ian o i, one usually imp oses one o he ollowing nondegene acy con-
di ions on he equency map
!
=
h
:
de
@!
@I
(
I
)
6
=0 o de
@!
@I
(
I
)
!
(
I
)
!
(
I
)
>
0
!
6
=0
o e e y
I
2G
.We call hese condi ions
Kolmogo o nondegene acy
and
isoene ge ic
nondegene acy,
esp ec i ely.
We deno e
V
(
G
) a complex neighb o ho o d o adius
a ound
G
.
Theo em 2 (KAM heo em)
Conside he Hamil onian
H
=
h
(
I
)+
(
I
)
, analy ic
o
2
T
n
and
I
2V
(
G
)
, wi h
o size
"
. Assume ha
!
=
h
is Kolmogo o o
isoene ge ical ly nondegene a e on
G
.Le
>
0
gi en. Fo some cons an s
C
1
,
C
2
,
C
3
,i
"
C
1
2
C
2
hen he e exis s
I
T
n
G
l led wi h
n
-dimensional in a ian o i o
H
, sa is ying
mes (
T
n
G
)
nI
]
C
3
(diam
G
)
n
;
1
:
(7)
405
Fo mo e de ailed s a emen s and p o o s, see 8, 9, 2, 3, 5]. Gi en
, i u ns ou ha he
in a ian o i o
H
come om in a ian o i o he unp e u b ed sys em
h
wi h equencies
sa is ying a Diophan ine condi ion o he yp e (3), wi h a xed
. Cho osing
p
"
, he
measu e o he complemen in (7) b ecomes
O
(
p
"
).
Now, ou aim is o apply KAM heo em o he Hamil onian
H
(
K
)
=
h
(
K
)
+
R
(
K
)
in o duced in (4{5). We pu his Hamil onian in ac ion{angle a iables byin o ducing he
known canonical change
q
j
=
q
2
I
j
cos
j
p
j
=
q
2
I
j
sin
j
j
=1
:::n:
Howe e , KAM heo em canno b e applied in a di ec way b ecause he change o ac ion{
angle a iables is no analy ic a he hyp e planes
I
j
= 0. Like in 9], his ac o ces
us o emo e a neighbo hood o hese hyp e planes. Thus, o ob ain in a ian o i in he
neighbo hood
B
,we conside o he ac ion a iables he domain
G
:=
(
I
2
R
n
:
I
2
j
I
j
1
2
2
)
wi h
>
0(we use he no a ion
I
a
o mean ha
I
j
a
o
j
=1
:::n
). Wi h a
sui able choice o
, his educ ion o he domain do es no aec essen ially he measu e
es ima es gi en in he nex p op osi ion.
To apply KAM heo em o
H
(
K
)
,we also ha e o equi e ha he equency map
!
(
K
)
=
h
(
K
)
is Kolmogo o o iso ene ge ically nondegene a e. In ac we only assume he
nondegene acy a he o igin i sel , since his suces o ensu e i in a small neighbo hood.
The condi ion we imp ose in ol es he ec o
and he ma ix
A
:=
@
2
Z
4
@I
2
:
Ne e heless, wepoin ou ha highe o de condi ions a e also p ossible.
P op osi ion 3
In he same si ua ion o p oposi ion 1, assume also
de
A
6
=0
o
de
A
>
0
!
6
=0
:
(8)
Le
K
denedasinpa (c) o p oposi ion 1. Fo some cons an s
c
3
,
c
4
,i
0
<
c
3
K
(9)
hen he e exis s
T
(
K
)
B
l led wi h in a ian o i o
H
(
K
)
, sa is ying
mes
h
B
nT
(
K
)
i
c
4
7
K
!
(
K
;
3)
=
2
mes
B
:
(10)
This esul is ob ained applying KAM heo em on he domain
V
(
G
), wi h
"
K
+1
and
(
K
+1)
=
2
. In ac , his is a mo e elab o a ed e sion o a esul gi en in 9], whe e
a measu e es ima e like (10) is ob ained, also wi h he exp onen (
K
;
3)
=
2.
406
4. The Diophan ine case
Finally,we assume ha he equency ec o
sa ises he Diophan ine condi ion (3) wi h
gi en
and
.We hen ake
K
=
K
in (6) and hence condi ion (9) is ullled i we
cho ose
K
(
=
)
1
=
(
+1)
, leading o an exp onen ially small es ima e.
Theo em 4
Le
H
(
x y
)=
P
s
2
H
s
bea eal Hamil onian wi h
H
2
=
I
, and assume
k
H
s
k
c
s
;
2
d
o
s
3
. Assume ha
is
-Diophan ine, wi h
>n
;
1
and
>
0
.
Assume also one o he nondegene acy condi ions
(8)
.Fo some cons an s
c
5
,
c
6
,
c
7
,i
0
<
c
5
hen he e exis s
T
B
l led wi h in a ian o i o
H
, sa is ying
mes
B
n
T
]
c
6
exp
(
;
c
7
1
=
(
+1)
)
mes
B
:
A ela ed esul has b een announced in 6] whe e, o a xed KAM o us o a nea ly-
in eg able Hamil onian, i is shown ha in a neighb o ho o d o adius
he e exis many
in a ian o i, and he measu e o hei complemen is exp onen ially small in 1
=
.
Acknowledgemen s
This wo k has b een pa ially supp o ed by he EC g an ERBCHRXCT940460. Resea ch
by Amadeu Delshams is also pa ially supp o ed by he spanish g an DGICYT PB94{
0215 and he ca alan g an CIRIT GRQ93{1135, and esea chbyPe e Gu ie ez is also
pa ially supp o ed by he U.P.C. g an PR9409.
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.
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one".
J. Di. Eq.
90
, 52{60 .
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p oblem".
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77
, 167{198 .
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.
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J. S a .
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81
, 1{60 .
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