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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 324378, 18 pages
doi:10.1155/2010/324378
Research Article
Gevrey Regularity of Invariant Curves of
Analytic Reversible Mappings
Dongfeng Zhang
1
and Rong Cheng
2
1
Department of Mathematics, Southeast University, Nanjing 210096, China
2
College of Mathematics and Physics, Nanjing University of Information Science and Technology,
Nanjing 210044, China
Correspondence should be addressed to Dongfeng Zhang,
Received 19 April 2010; Revised 2 9 October 2010; Accepted 25 December 2010
Academic Editor: Roderick Melnik
Copyright q 2010 D. Zhang and R. Cheng. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings
under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could
be any number μ>τ 2, where τ>m− 1 is the exponent in the small divisors condition and m is
the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of
the reversible mappings in a neighborhood of the union of the invariant curves.
1. Introduction and Main Results
In this paper we consider the following reversible mapping A:
x
1
x h
y
f
x, y
,
y
1
y g
x, y
,
1.1
where the rotation hy is real analytic and satisfies the weaker non-degeneracy condition
h
j
0
0, 0 <j<m, h
m
0
/
0,
1.2
where fx, y and gx, y are real analytic and 2π periodic in x,thevariabley ranges in an
open interval of the real line
. We suppose that the mapping A is reversible with respect to
the involution R : x, y → −x, y,thatis,ARA R. When some nonresonance and non-
degeneracy conditions are satisfied and f, g are sufficiently small, the existence of invariant
2AdvancesinDifference Equations
curve of reversible mapping 1.1 has been proved in 1–3. For related works, we refer the
readers to 4– 6 and the references there.
It is well known that reversible mappings have many similarities as Hamiltonian
systems. Since many KAM theorems are proved for Hamiltonian systems, some math-
ematicians turn to study the regular property of KAM tori with respect to parameters.
One of the earliest results is due to P
¨
oschel 7, who proved that the KAM tori of nearly
integrable analytic Hamiltonian systems form a Cantor family depending on parameters
only in C
∞
-way. Because the notorious small divisors can result in loss of smoothness
with respect to parameters involving in small divisors in KAM steps, we can only expect
Gevrey smoothness of KAM tori even for analytic systems. Gevrey smoothness is a notion
intermediate between C
∞
-smoothness and analyticity see definition below. Popov 8
obtained Gevrey smoothness of invariant tori for analytic Hamiltonian systems. In 9,
Wagener used the inverse approximation lemma to prove a more general conclusion.
Recently, the preceding result has been generalized to R
¨
ussmann’s non-degeneracy condition
10–12. Gevrey smoothness of the family of KAM tori is important for constructing Gevrey
normal form near KAM tori, which can lead to the effective stability 8, 13.
For reversible mappings, if h
y
/
0, the existence of a C
∞
-family of invariant curves
has been proved in 1, 2. But in the case of weaker non-degeneracy condition 1.2,thereis
no result about Gevrey smoothness. In this paper, we are concerned with Gevrey smoothness
of invariant curve of reversible mapping 1.1. T he Gevrey smoothness is expressed by
Gevrey index. In the following, we specifically obtain the Gevrey index of invariant curve
which is related to smoothness of reversible mapping 1.1 and the exponent of the small
divisors condition. Moreover, we obtain a Gevrey normal form of the reversible mappings in
a neighborhood of the union of the invariant curves.
As in 7, 14, 15, we introduce some parameters, so that the existence of invariant curve
of reversible mapping 1.1 can be reduced to that of a family of reversible mappings with
some parameters. We write y p z, and expand hy around p,sothathyhp
1
0
h
y
t
zdt,wherey
t
p tz,0≤ t ≤ 1, z varies in a neighborhood of origin of the real line
.Weputωphp, fx, z; p
1
0
h
y
t
zdt fx, p z, gx, z; pgx, p z and obtain
the family of reversible mappings
x
1
x ω
p
f
x, z; p
,
z
1
z g
x, z; p
.
1.3
Now, we turn to consider this family of reversible mappings with parameters p ∈ Π,where
Π ⊂
is a bounded interval.
Before stating our theorem, we first give some definitions and notations. Usually,
denote by
and
the set of integers and positive integers.
Definition 1 .1. Let D be a domain of
n
.AfunctionF : D → is said to belong to the Gevr ey-
class G
μ
D of index μμ ≥ 1 if F is C
∞
D-smooth and there exists a constant M such that
for all p ∈ D,
∂
β
p
F
p
≤ cM
|β|1
β!
μ
, 1.4
where |β| β
1
··· β
n
and β! β
1
! ···β
n
!forβ β
1
, ,β
n
∈
n
.
Advances in Difference Equations 3
Remark 1.2. By definition, it is easy to see that the Gevrey-smooth functions class G
1
coincides
with the class of analytic functions. Moreover, we have
G
1
⊂ G
μ
1
⊂ G
μ
2
⊂ C
∞
, 1.5
for 1 <μ
1
<μ
2
< ∞.
In this paper, we will prove Gevrey smoothness of function in a closed set, so we give
the following definition.
Definition 1.3. AfunctionF is Gevrey of index μ on a compact set Π
∗
if it can be extended as
a Gevrey function of the same index in a neighborhood of Π
∗
.
Define
D
s, r
{
x, z
∈
/2π × |
|
Im x
|
≤ s,
|
z
|
≤ r
}
, 1.6
and denote a complex neighborhood of Π by
Π
h
p ∈ | dist
p, Π
≤ h
. 1.7
Now the function fx, z; p is real analytic on Ds, r × Π
h
.Weexpandfx, z; p as
Fourier series with respect to x
f
x, z; p
k∈
f
k
z; p
e
ikx
,
1.8
then define
f
Ds,r×Π
h
k∈
f
k
r,h
e
s|k|
,
1.9
where
f
k
r,h
sup
|
z
|
≤r,p∈Π
h
f
k
z; p
.
1.10
We write Fx, z; p ∈ G
1,μ
Ds, r × Π
∗
if Fx, z; p is analytic with respect to x, z on
Ds, r and G
μ
-smooth in p on Π
∗
.
Denote T max
p∈Π
h
|ω
p|. Fix δ ∈ 0, 1 and τ>m− 1, and let μ τ 2 δ and
σ 2/3
δ/τ1δ
.LetW
0
diagρ
−1
0
,r
−1
0
.
4AdvancesinDifference Equations
Theorem 1.4. We consider the mapping A defined in 1.3, which is reversible with respect to the
involution R : x, z → −x, z,thatis,ARA R. Suppose that ωp satisfies the non-degeneracy
condition: ω
j
00, 0 <j<m, ω
m
0
/
0. Suppose that fx, z; p and gx, z; p are real analytic
on Ds, r × Π
h
. Then, there exists γ>0 such that for any 0 <α<1,if
f
Ds,r×Π
h
1
r
g
Ds,r×Π
h
≤ γαs
τ2
,
1.11
there is a nonempty Cantor set Π
∗
⊂ Π, and a family of transformations V
∗
·, ·; p : Ds/2,r/2 →
Ds, r, ∀p ∈ Π
∗
,
x ξ p
∗
ξ; p
,
z η q
∗
ξ, η; p
,
1.12
satisfying V
∗
x, z; p ∈ G
1,μ
Ds/2,r/2 × Π
∗
, and for any β ∈
,
W
0
∂
β
p
V
∗
− id
Ds/2,r/2×Π
∗
≤ cM
β
β!
τ2δ
γ
1/2
,
1.13
where M 2
τ2δ
T 1τ 1 δ
τ1δ
/πα, the constant c depends on n, τ,andδ. Under these
transformations, the mapping 1.3 is transformed to
ξ
1
ξ ω
∗
p
f
∗
ξ, η; p
,
η
1
η g
∗
ξ, η; p
,
1.14
where f
∗
Oη, g
∗
Oη
2
at η 0.Thus,foranyp ∈ Π
∗
, the mapping 1.3 has an invariant
curve Γ such that the induced mapping on this curve is the translation ξ
1
ξ ω
∗
p, whose frequency
ω
∗
p satisfies that
∂
β
p
ω
∗
p
− ω
p
≤ cαM
β
β!
τ2δ
γ
1/2
s
τ2
, ∀β ∈
, 1.15
kω
∗
p
2π
− l
≥
α
2
|
k
|
τ
, ∀
k, l
∈ × \
{
0, 0
}
. 1.16
Moreover, one has meas Π \Π
∗
≤ cα
1/m
.
Remark 1.5. From Theorem 1.4, we can see that for any μ>τ 2, if is sufficiently small, the
family of invariant curves is G
μ
-smoothintheparameters.TheGevreyindexμ τ 2 δ
should be optimal.
Remark 1.6. The derivatives in 1.13 and 1.15 should be understood in the sense of Whitney
16. In fact, the estimates 1.13 and 1.15 also hold in a neighborhood of Π
∗
with the same
Gevrey index.
Advances in Difference Equations 5
2. Proof of the Main Results
In this section, we will prove our Theorem 1.4. But in the case of weaker non-degeneracy
condition, the previous methods in 1, 2 are not valid and the difficulty is how to control the
parameters in small divisors. We use an improved KAM iteration carrying some parameters
to obtain the existence and Gevrey regularity of invariant curves of analytic reversible
mappings. This method is outlined in the paper 7 by P
¨
oschel and adapted to Gevrey classes
in 13 by Popov. We also extend the method of Liu 1, 2.
KAM step
The KAM step can be summarized in the following lemma.
Lemma 2.1. Consider the following real analytic mapping A:
x
1
x ω
p
f
x, z; p
,
z
1
z g
x, z; p
,
2.1
on Ds, r × Π
h
. Suppose the mapping is reversible with respect to the involution R : x, z →
−x, z,thatis,ARA R.Let0 <E<1, 0 <ρ<s/5,andK>0 such that e
−Kρ
E. Suppose
∀p ∈ Π, the following small divisors condition holds:
kω
p
2π
− l
≥
α
|
k
|
τ
, ∀
k, l
∈ × \
{
0, 0
}
, 0 <
|
k
|
≤ K. 2.2
Let
max
p∈Π
h
ω
p
≤ T, h
πα
TK
τ1
.
2.3
Suppose that
f
s,r;h
1
r
g
s,r;h
≤ αρ
τ2
E,
2.4
where the norm ·
s,r;h
indicates ·
Ds,r×Π
h
for simplicity. Then, for any p ∈ Π
h
,thereexistsa
transformation U:
x ξ u
ξ; p
,
z η v
ξ, η; p
,
2.5
which is affine in η, such that the mapping A is transformed to A
U
−1
AU:
ξ
1
ξ ω
p
f
ξ, η; p
,
η
1
η g
ξ, η; p
,
2.6
6AdvancesinDifference Equations
where the new perturbation satisfies
f
s
,r
;h
1
r
g
s
,r
;h
≤
α
ρ
τ2
E
,
2.7
with
s
s − 5ρ, ρ
σρ, μ
√
E, r
μr, E
cE
3/2
,
α
2
≤ α
≤ α,
2.8
where σ is defined in Theorem 1.4. Moreover, one has
ω
p
− ω
p
≤ , ∀p ∈ Π
h
. 2.9
Let α
α − /2πK
τ1
, and denote
R
k
p ∈ Π |
kω
p
2π
− l
<
α
|
k
|
τ
, ∀K<
|
k
|
≤ K
2.10
and Π
Π\ R
k
.Then,∀p ∈ Π
, it follows that
kω
p
2π
− l
≥
α
|
k
|
τ
, ∀
k, l
∈ × \
{
0, 0
}
, 0 <
|
k
|
≤ K
, 2.11
where K
> 0 such that e
−K
ρ
E
.Let
T
T
3
h
,h
πα
T
K
τ1
.
2.12
If h
≤ 2h/3,thenmax
p∈Π
h
|ω
p|≤T
. Moreover, one has
f
s
,r
;h
1
r
g
s
,r
;h
≤
.
2.13
Thus, the above result also holds for A
in place of A.
Proof of Lemma 2.1. The above lemma is actually one KAM step. We divide the KAM step into
several pats.
(A) Truncation
Let Q
f
fx, 0; p, Q
g
gx, 0; pg
z
x, 0; pz. It follows that Q
f
s,r;h
≤ , Q
g
s,r;h
≤ 2r.
Write Q
f
k∈
Q
fk
pe
ikx
, Q
g
k∈
Q
gk
z; pe
ikx
,andlet
R
f
|
k
|
≤K
Q
fk
p
e
ikx
,R
g
|
k
|
≤K
Q
gk
z; p
e
ikx
.
2.14
Advances in Difference Equations 7
By the definition of norm, we have
Q
f
− R
f
s−ρ,r;h
≤ ce
−Kρ
,
Q
g
− R
g
s−ρ,r;h
≤ ce
−Kρ
r.
2.15
(B) Construction of the Transformation
As in 1–3, for a reversible mapping, if the change of variables commutes with the involution
R, then the transformed mapping is also reversible with respect to the same involution R.If
the change of variables U : ξ, η → x, z is of the form
x ξ u
ξ
,
z η v
ξ, η
,
2.16
then from the equality RU UR, it follows that
u
−ξ
−u
ξ
,
v
−ξ, η
v
ξ, η
.
2.17
In this case, the transformed mapping U
−1
AU of A is also reversible with respect to the
involution R : ξ, η → −ξ, η.
In the following, we will determine the unknown functions u and v to satisfy
the condition 2.17 in order to guarantee that the transformed mapping U
−1
AU is also
reversible.
We may solve u and v from the following equations:
u
ξ ω
p
− u
ξ
R
f
ξ
−
R
f
ξ
,
v
ξ ω
p
,η
− v
ξ, η
R
g
ξ, η
−
R
g
ξ, η
,
2.18
where · denotes the mean value of a function over the angular variable ξ. Indeed, we can
solve these functions from the above equations. But the problem is that such functions u and v
do not, in general, satisfy the condition 2.17, that is, the transformed mapping U
−1
AU is no
longer a reversible mapping with respect to R. Therefore, we cannot use the above equations
to determine the functions u and v.
Instead of solving the above equations 2.18, we may find these functions u and v
from the following modified equations:
u
ξ ω
p
− u
ξ
f
ξ
,
v
ξ ω
p
,η
− v
ξ, η
g
ξ, η
,
2.19
8AdvancesinDifference Equations
with
f
ξ
1
2
R
f
ξ
−
R
f
ξ
R
f
−ξ − ω
p
−
R
f
−ξ − ω
p
,
g
ξ, η
1
2
R
g
ξ, η
− R
g
−ξ − ω
p
,η
,
2.20
where · denotes the mean value of a function over the angular variable ξ.
It is easy to verify that
f−ξ − ωp
fξ and g−ξ − ωp,η−gξ, η.So,by
Lemma A.1, the functions u and v meet the condition 2.17. In this case, the transformed
mapping U
−1
AU is also reversible with respect to the involution R : ξ, η → −ξ, η.
Because the right hand sides of 2.19 have the mean value zero, we can solve u, v
from 2.19.Butthedifference equations introduce small divisors. By the definition of Π
h
,it
follows that ∀p ∈ Π
h
,
kω
p
2π
− l
≥
α
2
|
k
|
τ
, ∀
k, l
∈ × \
{
0, 0
}
, 0 <
|
k
|
≤ K. 2.21
Let
f
k
, g
k
be Fourier coefficients of
f and g. Then, we have
u
k
f
k
e
ikωp
− 1
,v
k
g
k
e
ikωp
− 1
, 0 <
|
k
|
≤ K,
2.22
and u
k
0, v
k
0fork 0or|k| >K.Moreover,v is affine in η, u is independent of η.
(C) Estimates of the Transformation
By the definition of norm, we have
f
s−ρ,r;h
≤ c,
g
s−ρ,r;h
≤ cr.
2.23
By Lemma A.1, it follows that
u
s−2ρ,r;h
≤
c
αρ
τ1
,
v
s−2ρ,r;h
≤
cr
αρ
τ1
.
2.24
Using Cauchy’s estimate on the derivatives of u, v,weobtain
u
ξ
s−3ρ,r/2;h
≤
c
αρ
τ2
,
v
ξ
s−3ρ,r/2;h
<
cr
αρ
τ2
,
v
η
s−3ρ,r/2;h
<
c
αρ
τ1
.
2.25
Advances in Difference Equations 9
In the same way as in 1, 2, 4, we can verify that U
−1
AU is well defined in Ds − 5ρ, μr,
0 <μ≤ 1/8. Moreover, according to 2.24–2.25,wehave
W
0
U −id
D
s−5ρ,μr
×Π
h
≤
c
αρ
τ2
,
W
0
DU − Id
W
−1
0
D
s−5ρ,μr
×Π
h
≤
c
αρ
τ2
,
2.26
where ·denotes the maximum of the absolute value of the elements of a matrix, W
0
diagρ
−1
0
,r
−1
0
, DU denotes the Jacobian matrix with respect to ξ, η.
(D) Estimates of the New Perturbation
Let α
α − /2πK
τ1
.Wehave|kω
p/2π − l|≥α
/|k|
τ
, ∀p ∈ Π, ∀0 < |k|≤K. Then,
by the definition of R
k
, it follows that 2.11 holds. Thus, the small divisors condition for the
next step holds.
Let R
f
pR
f
ξ; p,thenwehaveR
f
p≤ αρ
τ2
E.DuetoU
−1
AU A
,we
have
f
ξ, η
u
ξ
− u
ξ
1
− R
f
p
f
ξ u, η v
. 2.27
By the first difference equation of 2.19,wehave
f
u
ξ ω
p
− u
ξ
1
f
ξ u, η v
−
f
ξ
− R
f
p
. 2.28
From the reversibility of A, it follows that
f
−x − ω
p
− f,z g
− f
x, z
0,
g
−x − ω
p
− f,z g
g
x, z
0.
2.29
Hence, we have
f
ξ, η
−
f
ξ
− R
f
p
1
2
f
ξ, η
− R
f
ξ
f
ξ, η
− R
f
−ξ − ω
p
1
2
f
ξ, η
− R
f
ξ
f
−ξ − ω
p
,η
− R
f
−ξ − ω
p
−f
−ξ − ω
p
,η
f
−ξ − ω
p
− f,η g
,
2.30
which yields that
f
ξ, η
−
f
ξ
− R
f
p
≤ cμ ce
−Kρ
2
2
ρ
.
2.31
10 Advances in Difference Equations
By 2.15 and 2.24–2.25, the following estimate of f
holds:
f
≤
u
ξ
·
R
f
p
f
f
ξ
·
u
f
η
·
v
cμ ce
−Kρ
2
2
ρ
≤
c
αρ
τ2
f
c
2
αρ
τ2
cμ ce
−Kρ
.
2.32
Similarly, for g
,weget
g
v
ξ ω
p
,η
− v
ξ
1
,η
1
g
ξ u, η v
− g
ξ, η
, 2.33
1
r
g
≤
c
αμρ
τ2
f
c
αρ
τ1
g
r
c
2
αμρ
τ2
cμ
ce
−Kρ
μ
.
2.34
If is sufficiently small such that
c
αμρ
τ2
<
1
2
,
2.35
then combing with 2.32 and 2.34,wehave
f
s
,r
;h
1
r
g
s
,r
;h
≤
c
2
αμρ
τ2
cμ
ce
−Kρ
μ
.
2.36
Suppose h
≤ 2/3h. Then, by Cauchy’s estimates, we have
ω
p
− ω
p
≤
3
h
, ∀p ∈ Π
h
.
2.37
Let T
T 3/h. Then, max
p∈Π
h
|ω
p|≤T
.
Moreover, by the definition of ρ
and E
,wehave
f
s
,r
;h
1
r
g
s
,r
;h
≤
c
2
αμρ
τ2
≤ cα
ρ
τ2
E
3/2
α
ρ
τ2
E
.
2.38
Thus, this ends the proof of Lemma 2.1.
Setting the Parameters and Iteration
Now, we choose some suitable parameters so that the above iteration can go on infinitely.
At the initial step, let ρ
0
1 − σs/10, r
0
r,
0
α
0
ρ
τ2
0
E
0
.LetK
0
satisfy e
−K
0
ρ
0
E
0
,
α
0
α>0, ω
0
ω, T
0
T max
p∈Π
h
|ω
p|.Denote
Π
0
p ∈ Π |
kω
p
2π
− l
≥
α
|
k
|
τ
, ∀0 <
|
k
|
≤ K
0
. 2.39
Advances in Difference Equations 11
Choose h α
1/m
. Note that this choice for h is only for measure estimate for parameters
and has no conflict with the assumption in Theorem 1.4, since we can use a smaller h.
Let h
0
πα
0
/T
0
K
τ1
0
≤ h and μ
0
E
1/2
0
. Assume the above parameters are all well
defined for j. Then, we define ρ
j1
σρ
j
, r
j1
μ
j
r
j
and E
j1
cE
3/2
j
, α
j1
α
j
−
j
/2πK
τ1
j
.
Define
j1
, μ
j1
, K
j1
,andh
j1
in the same way as the previous step.
Let
Π
j
p ∈ Π
j−1
|
kω
j
p
2π
− l
≥
α
j
|
k
|
τ
, ∀K
j−1
<
|
k
|
≤ K
j
. 2.40
Denote Π
h
j
{ξ ∈ | distξ, Π
j
≤ h
j
} and D
j
Ds
j
,r
j
for simplicity. By the iteration
lemma, we have a sequence of transformations U
j
:
x ξ u
j
ξ; p
,
z η v
j
ξ, η; p
,
2.41
such that for any p ∈ Π
h
j
, U
j
: D
j
→ D
j−1
, satisfying
W
j
U
j
− id
D
j
×Π
h
j
≤
c
j
α
j
ρ
τ2
j
,
W
j
DU
j
− Id
W
−1
j
D
j
×Π
h
j
≤
c
j
α
j
ρ
τ2
j
,
2.42
where W
j
diagρ
−1
j
,r
−1
j
, DU
j
denotes the Jacobian matrix with respect to ξ, η.
Thus, the transformation V
j
U
0
◦ U
1
···U
j
is well defined in D
j
× Π
h
j
and is seen to
take A
0
into
A
j
V
−1
j
A
0
V
j
.
2.43
More precisely, if we write A
0
as
x
1
x ω
p
f
x, z; p
,
z
1
z g
x, z; p
2.44
and express V
j
in the form
x ξ p
j
ξ; p
,
z η q
j
ξ, η; p
,
2.45
12 Advances in Difference Equations
then A
0
is transformed into A
j
:
ξ
1
ξ ω
j
p
f
j
ξ, η; p
,
η
1
η g
j
ξ, η; p
,
2.46
satisfying
f
j
D
j
×Π
h
j
1
r
j
g
j
D
j
×Π
h
j
≤
j
α
j
ρ
τ2
j
E
j
,
ω
j1
p
− ω
j
p
D
j
×Π
h
j
≤ c
j
.
2.47
In the following, we will check the assumptions in the iteration lemma to ensure that
KAM step is valid for all j ≥ 0.
Since E
j1
cE
3/2
j
and x
j
K
j
ρ
j
−ln E
j
,ifE
0
is sufficiently small such that
−ln c/ ln E
j
≤ 31 −σ/2, it follows that 3/2 ≤ K
j1
/K
j
≤ 3/2σ.Thush
j1
≤ 2/3h
j
.
By the definition of α
j
,wehave
α
j1
α
j
−
j
2π
K
τ1
j
α
j
1 −
1
2π
x
τ2
j
e
−x
j
. 2.48
If E
0
is sufficiently small such that
∞
j0
1 −
1
2π
x
τ2
j
e
−x
j
1 − O
1
x
0
≥
1
2
,
2.49
then we obtain α
0
/2 ≤ α
j
≤ α
0
and so α
j
/2 ≤ α
j1
≤ α
j
, ∀j ≥ 0.
Obviously, if E
0
is sufficiently small, the assumption 2.35 holds.
By σ 2/3
δ/τ1δ
, it is easy to see that K
δ
j1
ρ
τ1δ
j1
≥ K
δ
j
ρ
τ1δ
j
.IfE
0
is sufficiently
small and so x
0
is sufficiently large such that K
δ
0
ρ
τ1δ
0
x
δ
0
ρ
τ1
0
≥ 1, then we have K
δ
j
ρ
τ1δ
j
≥
1, ∀j ≥ 0.
Suppose max
p∈Π
h
j
|ω
j
p|≤T
j
.LetT
j1
T
j
3
j
/h
j
. Then, we have
max
p∈Π
h
j1
|ω
j1
p|≤T
j1
.
By iteration, T
j1
T
0
j
i0
3
i
/h
i
≤ T
0
∞
j0
3T
j
/πx
τ2
j
e
−x
j
. Suppose T
j
≤ T 1,
then we have
∞
j0
3T
j
/πx
τ2
j
e
−x
j
≤ 3T 1/π
∞
j0
x
τ2
j
e
−x
j
.IfE
0
is sufficiently small such
that
∞
j0
x
τ2
j
e
−x
j
≤ π/3T 1,thenT
0
≤ T
j1
≤ T
0
1.
Advances in Difference Equations 13
Convergence of Iteration in Gevrey Space G
1,τ2δ
Ds/2,r/2 × Π
∗
Now, we prove convergence of KAM iteration. Let V
j
U
0
◦U
1
···◦U
j
: D
j
×Π
h
j
→ D
0
×Π
h
0
,
and write V
j
in the form
x ξ p
j
ξ; p
,
z η q
j
ξ, η; p
.
2.50
In the same way as in 4, 7,wehave
W
0
V
j
− V
j−1
D
j
×Π
h
j
≤
c
j
α
j
ρ
τ2
j
,
W
0
D
V
j
− V
j−1
D
j
×Π
h
j
≤
c
j
α
j
ρ
τ2
j
,
2.51
where ·denotes the maximum of the absolute value of the elements of a matrix.
By Cauchy’s estimate we have
W
0
∂
β
p
V
j
− V
j−1
D
j
×Π
j
≤
cE
j
β!
h
β
j
,
2.52
W
0
∂
β
p
D
V
j
− V
j−1
D
j
×Π
j
≤
cE
j
β!
h
β
j
,
∂
β
p
ω
j1
p
− ω
j
p
D
j
×Π
j
≤
c
j
β!
h
β
j
.
2.53
Let P
j,β
cE
j
β!/h
β
j
and Q
j,β
c
j
β!/h
β
j
.ByK
δ
j
ρ
τ1δ
j
≥ 1 and the definition of h
j
,we
have
P
j,β
≤ c
2
T
0
1
πα
β
x
τ1δβ
j
β!e
−x
j
,
≤ c
2
T
0
1
πα
β
x
β
j
e
−κx
j
τ1δ
β!e
−x
j
/2
,
2.54
where κ 1/2τ 1 δ. It is easy to see that
x
β
j
e
−κx
j
≤ β!κ
−β
.
2.55
14 Advances in Difference Equations
Thus, we have
P
j,β
≤ cM
β
β!
τ2δ
E
1/2
j
,
2.56
where M 2
τ2δ
T
0
1τ 1 δ
τ1δ
/πα, c depends on n, τ,andδ.
In the same way, we have
Q
j,β
≤ cαM
β
β!
τ2δ
E
1/2
j
ρ
τ2
j
.
2.57
Note that s
j
→ s/2, r
j
→ 0, h
j
→ 0, as j →∞.LetD
∗
Ds/2, 0, Π
∗
∩
j≥0
Π
j
and V
∗
lim
j →∞
V
j
.SinceV
j
is affine in η, these estimates 2.52–2.56 imply that ∂
β
p
V
j
is
uniformly convergent to ∂
β
p
V
∗
on Ds/2,r/2 and satisfies
W
0
∂
β
p
V
∗
− id
Ds/2,r/2×Π
∗
≤ cM
β
β!
τ2δ
E
1/2
0
.
2.58
Since E
0
10/1 − σ
τ1
γ,thisproves1.13.
Let ω
∗
lim
j →∞
ω
j
. It follows that
∂
β
p
ω
∗
p
− ω
p
Π
∗
≤ cαM
β
β!
τ2δ
E
1/2
0
ρ
τ2
0
.
2.59
Moreover, we have |kω
∗
p/2π − l|≥α
∗
/|k|
τ
, ∀k, l ∈ × \{0, 0}, ∀p ∈ Π
∗
,where
α
∗
lim
j →∞
α
j
with α/2 ≤ α
∗
≤ α.Thus1.15 and 1.16 hold.
Whitney Extension in Gevrey Classes
In this section, we apply the Whitney extension theorem in Gevrey classes 13, 17, 18 to
extend V
∗
as a Gevrey function of the same Gevrey index in a neighborhood of Π
∗
.
Denote S
j
V
j
− V
j−1
, then for any positive integers β, γ,andm ∈
with β ≤ m,we
denote
R
m
p
∂
β
p
S
j
ξ, η, p
: ∂
β
p
S
j
ξ, η, p
−
βγ≤m
p −p
γ
∂
βγ
p
S
j
ξ, η, p
γ!
.
2.60
In order to apply the Whitney extension theorem in Gevrey classes for function V
∗
,weare
going to estimate R
m
p
∂
β
p
S
j
ξ, 0,p. First, we suppose that |p−p
|≤h
j
/8, p, p
∈ Π
∗
.Expanding
in Π
j
the analytic function p → ∂
β
p
S
j
ξ, 0,p, ξ ∈
1
, in Taylor series with respect to p at p
,
and using the Cauchy estimate, we evaluate
L
m
j,β
:
W
0
R
m
p
∂
β
p
S
j
ξ, 0,p
,ξ∈
1
,p,p
∈ Π
∗
. 2.61
Advances in Difference Equations 15
For β ≤ m 1, we have
β γ
!
γ!
≤ 2
βγ
β! ≤ 2
βγ
m 1
!
m −β 1
!
. 2.62
Then we obtain as above
L
m
j,β
≤
βγ≥m1
p −p
γ
W
0
∂
βγ
p
S
j
ξ, 0,p
γ!
≤
βγ≥m1
c
p − p
γ
E
j
β γ
!2
βγ
γ!h
βγ
j
≤ c
m 1
!
p −p
m−β1
m −β 1
!
4
m1
E
j
h
m1
j
βγ≥m1
4
p −p
h
−1
j
βγ−m−1
2.63
and we get
L
m
j,β
≤ c
m 1
!
p −p
m−β1
m − β 1
!
4
m1
E
j
h
m1
j
≤ c
4M
m1
p −p
m−β1
m − β 1
!
m 1
!
τ2δ
E
1/2
j
,
2.64
where M 2
τ2δ
T
0
1τ 1 δ
τ1δ
/πα, c depends on n, τ,andδ. Similarly, for |p −p
|≥
h
j
/8, we obtain the same inequality.
Let Π
∗
∩
j≥0
Π
j
and V
∗
lim
j →∞
V
j
. According to 2.64, the limit R
m
p
∂
β
p
S
j
ξ, 0,p
satisfies that
W
0
R
m
p
∂
β
p
V
∗
− id
Ds/2,r/2×Π
∗
≤ c
4M
m1
p −p
m−β1
m −β 1
!
m 1
!
τ2δ
E
1/2
0
.
2.65
Since V
∗
satisfies 2.58 and 2.65, by Theorem 3.7 and Theorem 3.8 in 13,wecanextend
V
∗
as a Gevrey function of the same Gevrey index in a neighborhood of Π
∗
.Thus,bythe
definition of Gevrey function in a closed set, V
∗
x, z; p ∈ G
1,μ
Ds/2,r/2 × Π
∗
,satisfiesthe
estimate 1.13 and 1.15 in a neighborhood of Π
∗
.
16 Advances in Difference Equations
Note that one can also use the inverse approximation lemma in 19 to prove the
preceding Whitney extension for V
∗
.
Estimates of Measure for Parameters
Now we estimate the Lebesgue measure of the set Π
∗
, on which the small divisors condition
holds in the KAM iteration. By the analyticity of ωp and ω
m
0
/
0, m>1, for almost all
points in Π, ω
m
p
/
0. Without loss of generality, we suppose ω
m
p
/
0, ∀p ∈ Π. Then, by
the KAM step, we have
Π \ Π
∗
j≥0
R
j
k
,
2.66
where
R
j
k
p ∈ Π
j−1
|
kω
j
p
2π
− l
<
α
j
|
k
|
τ
, ∀K
j−1
<
|
k
|
≤ K
j
2.67
with K
−1
0.
By Lemma A.2,wehave
meas
R
j
k
≤ c
K
j−1
<
|
k
|
≤K
j
α
j
|
k
|
τ1
1/m
,
≤ cα
1/m
K
j−1
<
|
k
|
≤K
j
1
|
k
|
τ1/m
.
2.68
Since τ>m− 1, we have
meas
Π \Π
∗
≤ cα
1/m
0
/
k∈
1
|
k
|
τ1/m
≤ cα
1/m
.
2.69
Appendix
A. Some Results on Difference Equation and Measure Estimate
In this section, we formulate some lemmas which have been used in the previous section. For
detailed proofs, we refer to 1, 20.
In the construction of the transformation in Lemma 2.1, we will meet the following
difference equation:
l
x ω
− l
x
g
x
. A.1
Advances in Difference Equations 17
Lemma A.1. Suppose that l(x), g(x) are real analytic on Ds,whereDs{x ∈
/2π ||Imx|≤
s}. Suppose ω satisfies the Diophantine condition |kω/2π − l|≥α/|k|
τ
, ∀k, l ∈ × \{0, 0},
then for any 0 <s
<s, the difference equation A.1 has the unique solution lx ∈ Ds
satisfying
l
x
s
≤
c
α
s − s
τ1
g
x
s
.
A.2
Moreover, if g−x − ωgx,thenlx is odd in x if g−x − ω−gx, lx is even in x.
Lemma A.2. Suppose gx is mth differentiable function on the closure
I of I,whereI ⊂ is an
interval. Let I
h
{x ||gx| <h,x∈ I}, h>0.If|g
m
x|≥d>0 for all x ∈ I,whered is a
constant, then
meas
I
h
≤ ch
1/m
,
A.3
where c 22 3 ··· m d
−1
.
Acknowledgments
We would like to thank the referees very much for their valuable comments and suggestions.
D. Zhang was supported by the National Natural Science Foundation of China Grants nos.
1082603511001048 and the Specialized Research Fund for the Doctoral Program of Higher
Education for New Teachers Grant no. 200802861043. R. Cheng was supported by the
National Natural Science Foundation of China Grant no. 11026212.
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