KAM-tori near an analytic elliptic fixed point

0.32 MB·

by  H. Eliasson

KAM-TORI NEAR AN ANALYTIC ELLIPTIC FIXED
POINT
4
L. H. ELIASSON, B. FAYAD, R. KRIKORIAN
1
0
2

n
a
J Dedicated to our friend Alain Chenciner

2
2

]
S
D
.
h
t Abstract. We study the accumulationofanelliptic fixedpoint
a
m of a real analytic Hamiltonian by quasi-periodic invariant tori.
We show that a fixed point with Diophantine frequency vector
[
ω0 isalwaysaccumulatedbyinvariantcomplexanalyticKAM-tori.
1 Indeed,thefollowingalternativeholds: IftheBirkhoffnormalform
v
of the Hamiltonian at the invariant point satisfies a Ru¨ssmann
8
transversality condition, the fixed point is accumulated by real
4
7 analytic KAM-tori which cover positive Lebesgue measure in the
5 phasespace(inthispartitsufficestoassumethatω0hasrationally
. independent coordinates). If the Birkhoff normal form is degener-
1
0 ate, there exists an analytic subvariety of complex dimension at
4 least d+1 passing through 0 that is foliated by complex analytic
1 KAM-tori with frequency ω .
0
:
v Thisisanextensionofpreviousresultsobtainedin[EFK]tothe
i case of an elliptic fixed point.
X
r
a
Contents
1. Introduction 2
2. Power series expansion 8
3. Formal Normal Forms 12
4. A KAM counter term theorem and the frequency map 20
5. Proof of the main results 23
6. The (local) Normal Form Theorem 27
7. Appendix 37
References 42
Date: January 23, 2014.
Supported by ANR-10-BLAN 0102 and ANR-11-BS01-0004.
2 L. H. ELIASSON,B. FAYAD,R. KRIKORIAN
1. Introduction
Let ω Rd and let
0
∈
H(x,y) = ω ,r + 3(x,y)
( ) h 0 i O
∗ r = (r ,...,r ), r = 1(x2 +y2)
(cid:26) 1 d j 2 j j
be a real analytic function defined in a neighborhood of (0,0). The
Hamiltonian system associated to H is given by the vector field X =
H
(∂ H, ∂ H), namely
y x
−
x˙ = ∂ H(x,y)
y
y˙ = ∂ H(x,y).
x
(cid:26) −
The flow of X has a fixed point = (0,0) . We are interested
H 0
P { }
in the study of whether this Hamiltonian system admits, besides ,
0
P
other invariant sets. More precisely, we shall try to find real analytic
KAM-tori for X in a neighborhood of , that is, real analytic La-
H 0
P
grangiantoriinvariant under X oneachoneofwhich theflowofX is
H H
conjugated to a translation flow ϕ ϕ+tω; it is usually required (and
we shall follow this requirement) t7→hat ω Td, the frequency vector, is
∈
in some Diophantine set DC(κ,τ) (κ,τ > 0) defined by the property
κ
(1.1) k,ω k Zd r 0 .
|h i| ≥ k τ ∀ ∈ { }
| |
Wewillsaythatω Rd isirrationalwhenitscoordinatesarerationally
0
∈
independent.
We call the complexification of a real analytic KAM-torus a complex
analytic KAM-torus for X , that is, a complex analytic Lagrangian
H
toric manifold invariant under (the complexification of) X on which
H
the flow is conjugated to a translation flow ϕ ϕ + tω. Note that
7→
there are complex analytic KAM-tori that are not the complexification
ofanyrealKAM-torus. Invariantcomplexanalytictoricmanifoldswere
studied in different settings related to KAM theory (see for example
[Sto]).
Notice that the problem of finding real or complex analytic KAM-
tori in a neighborhood of the invariant fixed point can be given
0
P
various more or less strong forms. For example, one can ask for finding
a set of KAM tori whose Lebesgue density in the phase space tend to
one in the neighborhood of . We shall call this KAM stability.
0
P
In classical KAM theory, an elliptic fixed point is shown to be
0
P
KAM-stable under the hypothesis that ω is irrational (or just suffi-
0
ciently non resonant) and that H satisfies a Kolmogorov non degen-
eracy condition of its Hessian matrix at . Further development of
0
P
KAM-TORI NEAR AN ANALYTIC ELLIPTIC FIXED POINT 3
the theory allowed to relax the non degeneracy condition. In this pa-
per we prove KAM stability of under the Ru¨ssmann transversality
0
P
condition on the Birkhoff normal form of H at .
0
P
We note that for non singular perturbative theory of analytic Hamil-
tonians it is known that the Ru¨ssmann condition is necessary and suf-
ficient for KAM stability – survival after perturbation of a positive
measure set of KAM-tori – of analytic integrable Hamiltonian systems
(see [R] and [Sev]). We stress however that the study of the dynamics
in a neighborhood of an elliptic fixed point, or near a given invariant
torus, is a singular perturbation problem and that, therefore, the latter
results do not apply per se.
The problem is more tricky if no nondegeneracy conditions are im-
posed on the Hamiltonian. In the analytic setting, no examples are
known of an elliptic fixed point with ω irrational that is not KAM
0 0
P
stable. It was conjectured by M. Herman in his ICM98-lecture [H] that
for analytic Hamiltonians, KAM stability holds in the neighborhood of
a KAM torus or of an elliptic fixed point if their frequency is
0 0
T P
assumed to be Diophantine. The conjecture is known to be true in two
degrees of freedom d = 2 [R], but remains open in general.
In this paper, we show that a fixed point with Diophantine frequency
vector ω of an analytic Hamiltonian is always accumulated by com-
0
plex analytic KAM-tori. We previously obtained a similar result in
the neighborhood of an invariant analytic torus with Diophantine fre-
quency vector [EFK]. In the latter setting, the tori obtained were real
analytic, but in the context of elliptic fixed points our method does not
necessarily yield real analytic tori.
The current paper follows the same strategy as in [EFK]and provides
thenecessary modificationsrequiredbytheabsenceofniceaction-angle
coordinates in the neighborhood of the fixed point.
An advantage however of the elliptic fixed point case, compared to
that of an invariant torus, is that the Birkhoff normal form can be
defined and conjugations up to any order can be performed under the
sole condition that ω is irrational. This is why we obtain the KAM
0
stability of any irrational fixed point under the Ru¨ssmann transversal-
ity condition, a result that we could not obtain for an invariant torus
with irrational frequency, except in 2 degrees of freedom (see [EFK],
Sections 2 and 9).
1.1. Statement of the result. Our main theorem is the following.
4 L. H. ELIASSON,B. FAYAD,R. KRIKORIAN
Theorem 1.1. Let H : (R2d,0) R be a real analytic function of
→
the form ( ) and assume that ω is Diophantine. Then, the origin is
0
∗
accumulated by infinitely many complex analytic KAM-tori for X .
H
Let N be the Birkhoff Normal Form of H – for the Birkhoff Nor-
H
mal Form at a Diophantine, and more generally an irrational elliptic
equilibrium, one can consult for example [SM].
We say that N is j-degenerate if there exist j orthonormal vectors
H
γ ,...,γ such that for every r 0 Rd
1 j
∼ ∈
∂ N (r),γ = 0 1 i j,
r H i
h i ∀ ≤ ≤
but no j + 1 orthonormal vectors with this property. Since ω = 0
0
6
clearly j d 1. A 0-degenerate N is also saidto be non-degenerate.
H
≤ −
Our Main Theorem is the consequence of Theorems 1.2 and 1.4 be-
low.
Theorem 1.2. Let H : (R2d,0) R be a real analytic function of the
→
form ( ) and assume that ω is Diophantine. If N is j-degenerate,
0 H
∗
then there exists an analytic subvariety containing 0 of complex dimen-
sion d+j foliated by invariant complex analytic KAM-tori for X with
H
translation vector ω .
0
A stronger result is known when N is (d 1)-degenerate. Indeed
H
−
Ru¨ssmann [R] (in a different setting) proved
Theorem 1.3. If ω is Diophantine and N is (d 1)-degenerate, then
0 H
a full neighborhood of 0 R2d is foliated by real an−alytic KAM-tori for
X with translation vect∈or in Rω .
H 0
OurproofofTheorem1.2(seeSection5.3)willalsoyieldRu¨ssmann’s
result.
In the nondegenrate case we will prove the following.
Theorem 1.4. Let H : (R2d,0) R be a real analytic function of the
→
form ( ) and assume that ω is irrational. If N is non-degenerate,
0 H
then in∗any neighborhood of 0 R2d the set of real analytic KAM-tori
∈
for X is of positive Lebesgue measure and density one at 0.
H
1.2. Strategy of the proof. We adopt a similar strategy to the one
of [EFK] that was inspired by previous works of Herman and Moser.
The basis is a counter term KAM-theorem in which a Hamiltonian as
in ( ) is conjugated, for any action value c 0 Rd and any ω in
∗ ∼ ∈
some fixed Diophantine class, to a Hamiltonian that has an invariant
torus at the action equal to c with frequency ω up to a correction term
Λ(c,ω). Furthermore, for every value c Rd in the neighborhood of 0
∈
KAM-TORI NEAR AN ANALYTIC ELLIPTIC FIXED POINT 5
of the action variable there exists a unique frequency Ω(c) that cancels
the counter term : Λ(c,Ω(c)) = 0. We call the map c Ω(c) the
7→
frequency map. If Ω(c) is Diophantine this yields an invariant KAM-
torus with frequency Ω(c). One can show that the jets of the function
Ω(c) are given by those of the gradient of the Birkhoff normal form
when the latter is well defined (which is the case if ω is irrational since
0
wearedealing withfixed points). Thefollowing alternativethenholds:
either the BNF isnondegenerate andthe functionΩ takes Diophantine
valuesonasetofLebesgue density 1at , which yieldsKAMstability;
0
P
or the BNF is degenerate and we can use the analytic dependance of
the counter term on the action variable to show the existence of a
direction (after a coordinate change in the action variable) that spans
a complex analytic subvariety foliated by complex analytic KAM-tori
with translation vector ω .
0
Let us briefly explain why we do not necessarily obtain real analytic
tori by our method in this context of elliptic fixed point. In the normal
formexpressionwelookforachangeofvariableZ definedinaneighbor-
hood of the origin containing the torus := x2+y2 = c2,j = 1,...d
T { j j j }
such that
d d
ω
(1.2) H Z(x,y) = Γ+ j(x2+y2 c2)+ (x2+y2 c2)F (x,y)
◦ 2 j j − j j j − j j
i=1 j=1
X X
where F = (F ,...,F ) is null on x2 + y2 c2 = 0 (Γ is a constant
1 d
−
which is unimportant). The torus is then invariant by the flow of
T
H Z (hence Z−1 is invariant by H).
◦ T
Since H is real analytic it has a holomorphic extension to a poly-
disk D2d C2d. Notice that the extension of H to D2d thus sat-
ρ ⊂ ρ
isfies H(x,y) = H(x¯,y¯). It will be convenient to make the follow-
ing change of variables: for x,y Cd, define z = 1(x + √ 1y ),
∈ j 2 j − j
w = 1(x √ 1y ) so that r := 1(x2+y2) = z w , j = 1,...,d. One
j 2 j− − j j 2 j j j j
has dz dw = √ 1dx dy. Notice that a function (x,y) f(x,y) is
∧ − − ∧ 7→
˜
real analytic (hence satisfies f(x,y) = f(x¯,y¯)) if and only if f(z,w) :=
˜ ˜ ˜
f(x,y) satisfies the symmetry f(z,w) = f(w¯,z¯). We then say that f is
σ-symmetric (where σ is the involution σ(z,w) = (w¯,z¯)). If f depends
real holomorphically on an extra complex parameter c and smoothly
on an extra real parameter ω, then f˜ : Dd Dd Dd B(0,1) C
ρ × ρ × δ × →
˜ ˜
satisfies f(z,w,c,ω) = f(w¯,z¯,c¯,ω), i.e. it is symmetric with respect
to the involution σ(z,w,c) = (w¯,z¯,c¯). By a slight abuse of notation
we shall call also this property σ-symmetry. One can define a similar
notion of σ-symmetry for diffeomorphisms (see Section 1.3).
6 L. H. ELIASSON,B. FAYAD,R. KRIKORIAN
Equation (1.2) is then equivalent to finding ω R2d, c Rd, an
∈ ∈
˜ ˜
exact symplectic change of coordinates Z for dz dw and maps F null
j
∧
on zw c2 = 0 such that Z˜ and the F˜ are σ-symmetric and
j
−
d d
H˜ Z˜(z,w) = Γ+ ω (z w c2)+ (z w c2)F˜ (z,w).
◦ j j j − j j j − j
i=1 j=1
X X
The searched for torus then corresponds in the (z,w)-coordinates to
z w = c2,j = 1,...,dT (z,w) C2d : σ(z,w) = (z,w) .
{ j j j }∩{ ∈ }
The strategy of the proof is then to find for some values of c Rd
and ω Rd such a normal form. ∈
∈
However, in the j-degenerate case (j = 0), it will only be possible to
do so for some c2 := (c2,...,c2) (but no6t necessarily c itself) in Rd and
1 d
consequently it will not be possible to ensure that the searched for tori
are real. We obtain instead complex analytic KAM-tori for X .
H
T
1.3. Notations. We denote by Dd the polydisk in Cd with radius δ.
δ
More generally if d = (d ,...,d ) and δ = (δ ,...,δ ), then
1 n 1 n
Dd = Dd1 Ddn.
δ δ1 ×···× δn
Let f : De C be a holomorphic function. We denote by ∂ f the
δ → zi
partial derivate of f with respect to z and we use the usual multi-index
i
notation like ∂αf. If z = (z′,z′′) we say that
z
f j(z′)
∈ O
if and only if ∂α′f(ϕ,0,z′′) = 0 for all α′ < j. We shall also use
the same notatioz′ns for Cn-valued functio|ns|f = (f ,...,f ) with the
1 n
absolute value replaced by f = max f (or some other norm on Cn).
i i
| | | |
σ-symmetry. Let σ be the involution (z,w,c) (w¯,z¯,c¯) on Cd
Cd Cd. A holomorphic function f : Dd Dd Dd7→ C is σ-symmetr×ic
× δ× δ× δ →
if, and only if, f σ(z,w,c) = f(z,w,c). This means precisely that
◦
it takes real values on the subspace (z,w,c) = σ(z,w,c) . A (local)
{ }
mapping F preserves this subspace if and only if
σ−1 F σ = F
◦ ◦
– we say then that F is σ-symmetric.
We let ω,σ(Dd Dd Dd) denote the space of σ-symmetric holomor-
phic functCions fδ:×Ddδ×Ddδ Dd C provided with the norm
δ × δ × δ →
f = sup f(z) .
δ
| | z∈De| |
δ
KAM-TORI NEAR AN ANALYTIC ELLIPTIC FIXED POINT 7
Formal power series. Let z = (z ,...,z ). An element
1 n
f C[[z]]
∈
is a formal power series
f = f(z) = a zα
α
α∈Nn
X
whose coefficients a C (possibly vector valued). The notion of σ-
α
∈
symmetry carries over to this more general framework. We denote by
[f] (z) = a zα,
j α
|α|=j
X
the homogenous component of degre j, and
[f]j = [f] .
i
i≤j
X
Parameters. Let B be an open subset of some euclidean space. De-
fine
ω,∞(Dd Dd Dd,B)
C δ × δ × δ
(orforshort ω,∞)tobethesetof ∞ functions(possiblyvectorvalued)
Cδ C
f : Dd Dd Dd B (z,ω) f(z,ω)
δ × δ × δ × ∋ 7→
such that for all ω B 1
∈
f : Dd Dd Dd (ω) f(z,ω)
ω δ × δ × δ ∋ 7→
is a holomorphic function. If in addition, this map is σ-symmetric, we
shall write f ω,σ,∞(Dd Dd Dd,B). We define
∈ C δ × δ × δ
f = sup ∂αf .
|| ||δ,s | ω ω|δ
|α|≤s
(κ,τ)-flat functions. A ∞ function f : Dd B C, (z,ω) f(z,ω)
C × → 7→
is (κ,τ)-flat if, for any set of indices α,β,
∂α∂βf(z,ω) = 0
z ω
whenever ω DC(κ,τ).
∈
Tensorialnotations. When(v ,...,v B(v , ,v )isam-multilinear
1 m) 1 m
7→ ···
form on a vector space V, we shall often see it as a linear form on the
m-th tensorial product V⊗m, and use the corresponding tensorial no-
tations. Also, we denote by the symmetrized tensor product
sym
⊗
1 we apologize for the double use of ω
8 L. H. ELIASSON,B. FAYAD,R. KRIKORIAN
v v = v v where the sum is on all the
1 ⊗ ··· ⊗ m σ σ(1) ⊗ ··· ⊗ σ(m)
permutations of 1,...,d .
{P }
2. Power series expansion
2.1. Expansion with Non Resonant functions. Let
f(z,w) = f zαwβ
α,β
α,β∈N
X
be some holomorphic function defined on a polydisk of (C2,0) – or
more generally a formal power series. We have
∞
f(z,w) = (zw)n f zαwβ
α,β
n=0 αβ=0
X X
and since in the last sum in the previous expression α = 0 or β = 0 we
can find analytic g and h , n Z such that
n n
∈
∞
f(z,w) = (zw)n(g (z)+h (w)).
n n
n=0
X
A similar procedure or a simple induction argument show that if f is
now analytic in some polydisk D2d of (C2d,0) then
ρ
f(z,w) = (z w )n1 (z w )nd f zαwβ
1 1 d d n,α,β
···
n∈Nd (α,β)∈NR
X X
where is the set of (α,β) (Nd)2 such that for all i = 1,...,d,
NR ∈
α β = 0. A power series of the form h(z,w) = h zαwβ
i i (α,β)∈NR α,β
will be called non-resonant and we denote by the vector space of
P
NR
all non-resonant functions. Notice that we allow for the existence of
constant terms in this definition. We can also sday that
f(z,w) = (z w )n1 (z w )nd f (rε1,...,rεd)
1 1 ··· d d n,ε 1 d
n∈Nd ε∈{0,1}d
X X
where f are holomorphic in D2d and where we have used the notation
n,ε ρ
rε = z if ε = 0 and rε = w if ε = 1.
i i i i
The following fact will be useful:
Lemma 2.1. For any (α,β) (Nd)2 there is a unique decomposition
of the form (α,β) = (n,n)+(∈α′,β′) where n Nd and (α′,β′)
∈ ∈ NR
(this means that for any i = 1,...,d, α = n +α′ and β = n +β′).
i i i i i i
KAM-TORI NEAR AN ANALYTIC ELLIPTIC FIXED POINT 9
Proof.
Toprovetheexistenceofsuchadecompositionjusttaken = min(α ,β ),
i i i
i = 1,...,d. To prove uniqueness we observe that if for some i n = n˜ ,
i 6 i
for example n > n˜ , then α′ > α˜ and β′ > β˜ a contradiction with
i i i i i i
(α′,β′) .
∈ NR
(cid:3)
The preceding discussion provides the following decomposition
Lemma 2.2. If f(z,w) is holomorphic on some polydisk Dd Dd there
δ× δ
exists a unique decomposition
(2.3) f(z,w) = (z w )n1 (z w )ndf˜(z,w)
1 1 d d n
···
nX∈Nd
where f˜ are holomorphic on Dd Dd – the series converges
n ∈ NR δ × δ
uniformly on compact sub domains of Dd Dd. Furthermore f is σ-
δ × δ
symmetric ifdand only if all the f˜ are.
n
Proof. To prove uniqueness, one just have to prove that if f is null,
the same is true of all the series f . This is done by looking at the
n
coefficients of the right hand side of (2.3) and by using Lemma 2.1.
The σ-symmetry of the f comes from the uniqueness. (cid:3)
n
If now f depends (or not) on a parameter c = (c ,...,c ), by writing
1 d
z w = c +(z w c )in(2.3)wegetanexpansionuniformlyconverging
j j j j j j
−
on small compact neighborhoods of 0:
(2.4) f(z,w,c) = (z w c )n1 (z w c )ndf (z,w,c).
1 1 1 d d d n
− ··· −
nX∈Nd
We again notice that each f ( , ,c) is non-resonant (for any fixed c)
n
· ·
and
k
(2.5) f (z,w,c) = c(k−n)f˜(z,w,c).
n k
n
k≥n(cid:18) (cid:19)
X
We shall still denote by the set of functions f(z,w,c) which are
NR
non-resonant for each fixed c.
d
Lemma 2.3. If f is σ-symmetric, there exists a unique decomposition
of the form (2.4) where each f is non-resonant and σ-symmetric.
n
Proof. We have to prove that if in (2.4) f is equal to 0 then all the
fn are null. If fn(z,w,c) = α′,β′,γ′fn,α′,β′,γ′zα′wβ′cγ′, the coefficient of
P
10 L. H. ELIASSON,B. FAYAD,R. KRIKORIAN
zαwβ in (2.4) is the sum
n
( 1)|n−k|cγ′+(n−k)fn,α′,β′,γ′
k −
k,α′,β′,γ′ n≥k(cid:18) (cid:19)
X X
(k,k)+(α′,β′)=(α,β)
Sinceinthelastsum(α′,β′) ,thedecomposition(α,β) = (k,k)+
∈ NR
(α′,β′) is unique by Lemma 2.1, and thus the last sum is just (the
summation is in n)
n
( 1)|n−k|cγ′+(n−k)fn,α′,β′,γ′.
k −
n≥k (cid:18) (cid:19)
X
γ′
By assumption, for any k, any (α′,β′) , this has to be equal to
∈ NR
zero for any c in a neighborhood of zero. Multiplying the last sum by
(d+c)k and making the summation on all k 0 one gets
≥
cγ′dnfn,α′,β′,γ′ = 0.
n
Xγ′
This being true for all c,d in a neighborhood of 0 one has f = 0.
n
The σ-symmetry of the f comes from the uniqueness.
n
(cid:3)
Remark Lemmas 2.2 and 2.3 hold in the case of formal series in
C[[z,w,c]].
If p N, we shall denote
∈
(2.6) f (z,w,c) = (zw c)nf (z,w,c)
p n
−
|n|=p
X
We shall use the following notations. We have seen that f(z,w,c)
can be written under the form
(2.7) f(z,w,c) = f(0)(z,w,c)+ f(1)(z,w,c),(zw c) +
h − i
(zw c),f[2](z,w,c)(zw c)
h − − i
or
(2.8) f(z,w,c) = f(0)(z,w,c)+ f(1)(z,w,c),(zw c) +
h − i
(zw c),f(2)(z,w,c)(zw c) +f[3](z,w,c)(zw c)⊗3
h − − i −
where f(0)(z,w,c) and f(1)(z,w,c) (j = 1,...,d), f(2)(z,w,c) (1
j i,j ≤
(2) [2]
i,j d)arein andwherethenotationsf (z,w,c)andf (z,w,c)
≤ NR ij ij
(1 i,j d) denotes respectively the sums f (z,w,c) and
≤ ≤ n=λi+λj n
d
P