Jiang Journal of Inequalities and Applications (2017) 2017:39 DOI 10.1186/s13660-017-1307-1
RESEARCH
Open Access
Gevrey-smoothness of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings Shunjun Jiang* *
Correspondence: [email protected] College of Sciences, Nanjing Tech University, Nanjing, 210009, China
Abstract This paper provides a normal form for a class of lower dimensional hyperbolic invariant tori of nearly integrable symplectic mappings with generating functions. We prove the persistence and the Gevrey-smoothness of the invariant tori under some conditions. Keywords: symplectic mappings; KAM iteration; Gevrey-smoothness; normal form
1 Introduction and main results Area-preserving mappings have some dynamical properties similar to Hamiltonian systems, and hence become an important test ground of all kinds of theories for studying Hamiltonian systems, such as Poincaré [, ] on three body problem, Moser [] on the differentiable form of KAM theory, Aubry and Mather [–] on Aubry-Mather theorem, Conley and Zehnder [, ] on symplectic topology. So area-preserving mappings have attracted many scholars’ interest. We refer to [–]. Among all the mappings, symplectic mappings are special for their symplectic structures; we refer to [–] for more results on symplectic structures. On the other hand, many mathematicians turn to the study of the connection between the KAM tori and the parameter. The first work is due to Pöschel [] who proved that the persisting invariant tori are C ∞ -smooth in the frequency parameter. Popov [] obtained the Gevrey-smoothness, a notion intermediate between C ∞ -smoothness and analyticity, of invariant tori in the frequencies under the Kolmogorov non-degeneracy condition. Xu and You [] obtained a similar result under the Rüssmann non-degeneracy condition by an improved KAM method. For more results, we refer to [, ]. Motivated by [, ], we consider the persistence and the Gevrey-smoothness of lower dimensional hyperbolic invariant tori for symplectic mappings determined by generating functions under Rüssmann’s non-degeneracy condition. We consider the following parameterized symplectic mapping: ˆ yˆ , vˆ ) ∈ Tn × Rm × Rn × Rm , (·; ξ ) : (x, u, y, v) ∈ Tn × W × O × W → (ˆx, u, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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where ξ ∈ ⊂ O is a parameter and O ⊂ Rn is a bounded closed connected domain. Suppose (·; ξ ) is implicitly defined by ⎧ ⎪ ⎪xˆ = ∂yˆ H(x, u, yˆ , vˆ ; ξ ), ⎪ ⎪ ⎪ ⎨y = ∂ H(x, u, yˆ , vˆ ; ξ ), x
(.)
⎪ ⎪ uˆ = ∂vˆ H(x, u, yˆ , vˆ ; ξ ), ⎪ ⎪ ⎪ ⎩ v = ∂u H(x, u, yˆ , vˆ ; ξ ), where H(x, u, yˆ , vˆ ; ξ ) = N + P,
(.)
N(x, u, yˆ , vˆ ; ξ ) = x + ω(ξ ), yˆ + Au, vˆ + Bu, u + C vˆ , vˆ .
(.)
Suppose A is a constant matrix and B, C are symmetric. If P = , can be expressed explicitly as xˆ = x + ω(y), yˆ = y, – – uˆ = A – C AT B u + C AT v, We define
A – C(AT )– B = –(AT )– B
C(AT )– (AT )–
– – vˆ = – AT Bu + AT v.
(.)
. m×m
Denote the eigenvalues of by (λ , λ , . . . , λm ). We call the lower dimensional invariant torus elliptic if |λi | = , λi = , ∀i = , , . . . , m and hyperbolic if |λi | = , ∀i = , , . . . , m. We note that, although some results on symplectic mappings can be anticipated by Hamiltonian systems, there are still many differences for lower dimensional invariant tori. The first one is concerned with the relations of variables. In symplectic mappings, some variables determined by generating functions take on an implicit form and hence lead to more difficulties than in a Hamiltonian system. The second one is the non-degeneracy condition, which will result in a more complicated proof for estimate of measure. Before presenting the main result, we give some assumptions and definitions. Assumption (Rüssmann’s non-degeneracy condition) There exists an integer n¯ > such that β
rank ∂ξ ω(ξ ) : ≤ |β| ≤ n¯ = n,
∀ξ ∈ ,
where β T β β β ∂ξ ω(ξ ) = ∂ξ ω (ξ ), ∂ξ ω (ξ ), . . . , ∂ξ ωn (ξ ) with |β|
∂ξ ωi (ξ )
β
∂ξ ωi (ξ ) =
β
β
β
∂ξ ∂ξ · · · ∂ξn n
,
(.)
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and β = (β , β , . . . , βn ). Assumption (Hyperbolic condition) Let A = diag(a , a , . . . , am ), B = diag(b , b , . . . , bm ), C = diag(c , c , . . . , cm ). Define i =
ai –bi ci + , i = , , . . . , m. ai
Suppose i > , i = , , . . . , m.
Remark . By direct calculation, we have the eigenvalues of ,
λi =
i ±
( i ) – ,
i = , , . . . , m.
If i > , we have |λi | = , i = , , . . . , m, so the lower dimensional invariant torus is hyperbolic. If otherwise i < , we have |λi | = , and hence the lower dimensional invariant torus is elliptic. Definition . Let O ⊂ Rn be a bounded closed connected domain. A function F : O → R is said to belong to Gevrey-class Gμ (O) of index μ (μ ≥ ) if F is C ∞ (O)-smooth and there exists a constant J such that for all p ∈ O , β ∂ F(p) ≤ cJ |β|+ β!μ , p where |β| = β + β + · · · + βn and β!μ = β !β ! · · · βn ! for β = (β , β , . . . , βn ) ∈ Zn+ . Remark . By definition, it is easy to see that the Gevrey-smooth function class G coincides with the analytic function class. Moreover, we have G ⊂ Gμ ⊂ Gμ ⊂ C ∞ for < μ < μ < ∞. Set
Ts = x ∈ Cn /πZn : | Im x|∞ ≤ s ,
Br = y ∈ Cn : |y| ≤ r , and
Wr = w ∈ Cm : |w| ≤ r . Denote by D(s, r) = Ts × Wr × Br × Wr . Here, |x|∞ = max≤j≤n |xj |, |y| = |w| = ( ≤j≤m |wj | ) . Denote
= ξ ∈ O | dist(ξ , ∂ O) ≥ h
≤j≤n |yj |
and
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and
h = ξ ∈ Cn | dist(ξ , ) ≤ h . Definition . f ∈ G,μ (D(s, r) × ) means that f ∈ C ∞ (D(s, r) × ) and f (x, y, u, v; ξ ) is analytic with respect to (x, y, u, v) on D(s, r) and Gμ -smooth in ξ on . ¯ Below we define some norms. If P(x, u, yˆ , vˆ ; ξ ) is analytic in (x, u, yˆ , vˆ ) on D(s, r) and ntimes continuously differentiable in ξ on , we have P(x, u, yˆ , vˆ ; ξ ) =
Pk (u, yˆ , vˆ ; ξ )eik,x ,
k∈Zn
where Pk (u, yˆ , vˆ ; ξ ) =
Pklij (ξ )ˆyl ui vˆ j .
l,i,j
Define P D(s,r)× =
|Pk |r es|k| ,
k∈Zn
where |Pk |r =
sup
(u,ˆy,ˆv)∈Wr ×Br ×Wr i,j,l
Pklij s yˆ l ui vˆ j .
This norm is apparently stronger than the super-norm. Moreover, the Cauchy estimate of analytic functions is also valid under this norm. Let XP = (–∂yˆ P, –∂vˆ P, ∂x P, ∂u P) and denote a weighted norm by XP r;D(s,r)× = ∂yˆ P D(s,r)× + ∂vˆ P D(s,r)× r + ∂x P D(s,r)× + ∂u P D(s,r)× , r r where ∂xˆ P D(s,r)× =
∂xˆ j P D(s,r)× ,
j
∂yˆ P D(s,r)× = max ∂yˆ j P D(s,r)× j
and ∂u P D(s,r)× =
∂uj P s,r
j
∂vˆ P D(s,r)× is defined similarly.
,
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Now we introduce the main result. Let τ ≥ nn¯ – . For δ ∈ (, ), let μ = τ + δ + and δ σ = ( ) τ ++δ . Theorem . Consider the symplectic mapping (·; ξ ), which is implicitly defined by a | ≤ T. Suppose that Assumptions , generating function H(·; ξ ) in (.). Let maxξ ∈h | ∂ω ∂ξ hold. Then there exists γ > such that for any < α < , if XP r;D(s,r)×d = ≤ γ α ¯ν ρ ν ¯ the following results hold true: with ν¯ = n¯ + , ν = τ (n¯ + ) + n + n, (i) There exist a non-empty Cantor-like subset ∗ ⊂ and, for ξ ∈ ∗ , a symplectic mapping ∗ (·; ξ ), where ∗ ∈ G,μ with β ∂ (∗ – id) s r ≤ cρ ν J |β| β!μ γ (n+) ξ r;D( , )×∗
(.)
for ∀β ∈ Zn+ and J = T+ [ (μ–)(n+) ]μ– . Moreover, ∗ = ∗– ◦ ◦ ∗ is generated by α H∗ = N∗ + P∗ as in (.) satisfying N∗ (x, u, yˆ , vˆ ; ξ ) = x + ω∗ , yˆ + A∗ u, vˆ + B∗ u, u + C∗ vˆ , vˆ , P∗ (x, u, yˆ , vˆ ; ξ ) = Plij (x; ξ )ˆyl ui vˆ j . |i|+|j|+|l|≥
(ii) Hence, for ξ ∈ ∗ , the symplectic mapping (·; ξ ) admits a lower dimensional invariant torus Tξ = ∗ T n , , , ; ξ , whose frequencies ω∗ satisfy β ∂ ω∗ (ξ ) – ω(ξ ) ≤ cρ ν J |β| β!μ γ (n+) ξ
(.)
ω∗ (ξ ), k + πl ≥
(.)
and α ( + |k|)τ
for all ξ ∈ ∗ , = k ∈ Zn . Moreover, we have
meas( \ ∗ ) ≤ cα m .
2 The proof of main results We will use the idea for Hamiltonian systems in [] to prove our results. In Section ., one KAM step iteration is presented. The key lies in solving a homological equation. Then we will show the KAM step can iterate infinitely in Section .. Convergence of the iteration and the estimate of measure will be presented in Sections . and .. 2.1 KAM-step Iteration Lemma Consider a symplectic mapping (·; ξ ) defined in Theorem .. Let < E < , < ρ = ( – σ )s/ < s and < η < . Let K > satisfy η e–Kρ = E. Let ∂ω max ≤ T, ξ ∈h ∂ξ
h=
α . ( + K)τ + T
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Moreover, ω(ξ ) satisfies that: for k ∈ Zn \ {}, l ∈ Z, k, ω + πl ≥
α . ( + |k|)τ
(.)
Suppose Assumptions , hold. Suppose that P satisfies XP r;D(s,r)×d ≤ = η α ¯ν ρ ν E, ¯ Then we have the following results: with < α < , ν¯ = n¯ + , ν = τ (n¯ + ) + n + n. () ∀ξ ∈ h , there exists a symplectic diffeomorphism (·; ξ ) with c , α ν¯ ρ ν c ≤ ν¯ ν+ , α ρ
– id r;D(s–ρ, r )×h ≤ D – id r;D(s–ρ, r )×h
such that the conjugate mapping + (·; ξ ) = – ◦ ◦ is generated by H+ (·; ξ ) = N+ + P+ , where N+ = x + ω+ (ξ ), yˆ + A+ u, vˆ + B+ u, u + C+ vˆ , vˆ and P+ satisfies XP r+ ;D(s+ ,r+ )×d ≤ η+ α+¯ν ρ+ν E+ = + , with s+ = s – ρ,
ρ+ = σρ,
α ≤ α+ ≤ α.
r+ = ηr,
η+ = E+ ,
E+ = E ,
Furthermore, we have ω+ (ξ ) – ω(ξ ) ≤ ,
∀ξ ∈ h .
(.)
() Let α+ = α – (K + )τ + , ¯ = ξ ∈ : k, ω+ (ξ ) <
α+ n , k ∈ Z , K < |k| ≤ K + , ( + |k|)τ
(.)
¯ Then, for ∀ξ ∈ + , ∀k ∈ Zn and < |k| ≤ K+ , we have and + = \ . k, ω+ (ξ ) ≥
α+ , ( + |k|)τ –K+ ρ+ η+
where K+ > such that e
(.)
= E+ .
α+ ∂ω+ () Let T+ = T + and h+ = (K +) τ + T . If h+ ≤ h, we have maxξ ∈h+ | ∂ξ | ≤ T+ , + + where h+ is the complex h+ -neighborhood of + . h
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A. The equivalent form of (.). Let P(p, qˆ ) = P (x) + P (x), yˆ + P (x), u + P (x), vˆ + P (x)u, vˆ + P (x)u, u + P (x)ˆv, vˆ l i j + Plij yˆ u vˆ
(.)
|i|+|j|+|l|≥
with ∂ l+i+j P . Plij = l i j ∂ yˆ ∂u ∂ vˆ u=,ˆy=,ˆv= Let Q(x, u, vˆ ) = Q (x)u, vˆ + Q (x)u, u + Q (x)ˆv, vˆ with Q (x) = P (x), Q (x) = P (x), Q (x) = P (x). Then we rewrite H as H = N + Q + (P – Q), where N + Q is the new main term and P – Q is the new small term. Now we will study the following function which is equivalent to (.): H(x, u, yˆ , vˆ ; ξ ) = N(x, u, yˆ , vˆ ; ξ ) + P(x, u, yˆ , vˆ ; ξ ),
(.)
where N = x + ω(ξ ), yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ) and P = P (x) + P (x), yˆ + P (x), u + P (x), vˆ +
Plij yˆ l ui vˆ j .
|l|+|i|+|j|≥
B. Generating functions of conjugate mappings. For convenience, let p = (x, u) and q = (y, v). pˆ and qˆ have a similar meaning. The symplectic structure becomes dp ∧ dq on Rn+m × Rn+m . Consider a symplectic mapping : (p, q) → (ˆp, qˆ ) generated by pˆ = ∂qˆ H(p, qˆ ),
q = ∂p H(p, qˆ ),
(.)
where H(p, qˆ ) = N(p, qˆ ) + P(p, qˆ ), where N is the main term and P is a small perturbation. We need a symplectic transformation : (p+ , q+ ) → (p, q) generated by q = q+ + F (p, q+ ),
p+ = p + F (p, q+ ),
(.)
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with F (p, q+ ) =
∂F(p, q+ ) , ∂p
F (p, q+ ) =
∂F(p, q+ ) . ∂q+
The generating function is p, q+ + F(p, q+ ) with F being a small function. By (.) and (.), we have a conjugate mapping = – ◦ ◦ : (p+ , q+ ) → (ˆp+ , qˆ + ) implicitly by pˆ + = H (p, qˆ ) + F (ˆp, qˆ + ),
q+ = H (p, qˆ ) – F (p, q+ )
(.)
with H (p, qˆ ) =
∂H(p, qˆ ) , ∂p
H (p, qˆ ) =
∂H(p, qˆ ) . ∂ qˆ
So we have the following lemma. Lemma . ([]) The conjugate symplectic mapping + can be implicitly determined by a generating function H+ (p+ , qˆ + ) through pˆ + = ∂qˆ + H+ (p+ , qˆ + ),
q+ = ∂p+ H+ (p+ , qˆ + ),
(.)
where H+ (p+ , qˆ + ) = H(p, qˆ ) + H (p, qˆ )F (p, q+ ) – H (p, qˆ )F (ˆp, qˆ + ) + F(ˆp, qˆ + ) – F(p, q+ ) – F (p, q+ )F (p, q+ ),
(.)
where p, pˆ , qˆ , q+ depend on (p+ , qˆ + ) as explained above. Set z = (p+ , qˆ + ). We have H+ (z) = H(z) + F N (z), qˆ + – F p+ , N (z) + ϒ(z)
(.)
with N (z) =
∂N(p+ , qˆ + ) , ∂p+
N (z) =
∂N(p+ , qˆ + ) ∂ qˆ +
and ϒ(z) satisfying |||Xϒ |||r;D(s–ρ,r/) ≤
c , α ¯ν ρ ν
where ν¯ = n¯ + and ν = τ (n¯ + ) + n¯ + n.
(.)
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C. Truncation. Let P = R + (P – R),
(.)
where R(p, qˆ ) =
k k k k P (x) + P (x), yˆ + P (x), u + P (x), vˆ
(.)
|k|≤K
= R (x) + R (x), yˆ + R (x), u + R (x), vˆ ,
(.)
and P–R=
Plij yˆ l ui vˆ j +
|l|+|i|+|j|=
Plij yˆ l ui vˆ j .
|l|+|i|+|j|≥
Let F(p, qˆ ) possess the same form as (.). D. Extending the small divisor estimate. ∀ξ ∈ h , there exists ξ ∈ such that |ξ – ξ | < h. So we have, for < |k| ≤ K , k, ω(ξ ) + πl ≥ k, ω(ξ ) + πl – k, ω(ξ ) – ω(ξ ) ≥
α . ( + |k|)τ
E. Homological equation. By (.), it follows that N(p+ , qˆ + ) + P(p+ , qˆ + ) – F p+ , Np (p+ , qˆ + ) + F Nq (p+ , qˆ + ), qˆ + + ϒ(z) ¯ + , qˆ + ) + P(p ¯ + , qˆ + ). = N(p
(.)
For simplicity, below we drop the subscripts ‘+’ in p+ and qˆ + . Similar to the discussion in [], we get the homological equations. –F Nq (p, qˆ ), qˆ + F p, Np (p, qˆ ) = R – [R].
(.)
To solve (.), we need some preparations. Let x + ω = x˜ . Since pˆ = Nqˆ (p, pˆ ) = x˜ , (A + Q )u + (C + Q )v and q = Np (p, qˆ ) = yˆ + Qx , (A + Q )ˆv + (B + Q )u , we have F p, Np (p, qˆ ) = F (x) + F (x), yˆ + Qx + F (x), u + F (x), (A + Q )T vˆ + (B + Q )u
(.)
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and F Nqˆ (p, qˆ ), qˆ = F (˜x) + F (˜x), yˆ + F (˜x), vˆ + F (˜x), (A + Q )T u + (C + Q )ˆv .
(.)
So we get F Nqˆ (p, qˆ ), qˆ – F p, Np (p, qˆ ) = L + L + L – F (x), Qx , where L = F (˜x) – F (x) + F (˜x) – F (x), yˆ , L = AT F (˜x) – F (x) – BF (x), u + QT F (˜x) – QT F (x), u and L = C T F (˜x) + F (x) – AF (x), u + QT F (˜x) – Q F (x), u . After these preparations, we can solve (.) which is equivalent to solving the following: ⎧ ⎨F
x) – F (x) = R (x) – [R ], (˜ ⎩F (˜x) – F (x) = R (x) – [R ],
(.)
and ⎧ ⎨AT F
x) – F (x) – BF (x) + QT F (˜x) – QT F (x) = R (x), (˜
⎩C T F (˜x) + F (x) – AF (x) + QT F (˜x) – Q F (x) = R (x).
(.)
Firstly, we solve (.) for F and F . Expand F (x) and R (x): F (x) =
Fk eik,x ,
k∈Zn
R (x) =
Rk eik,x .
k∈Zn
Then we get Fkj =
Rkj ek –
(.)
with ek = eik,ω , k = . By Assumption , we have the following estimate: Fk s–ρ ≤
c Rk s . αρ τ +n
So we have XF r;D(s–ρ)× ≤
c R s . α ν¯ ρ ν
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Similarly we have XF r;D(s–ρ)× ≤
c R s . α ν¯ ρ ν
m Next we will get F and F from (.). Let F = (F , . . . , F ) and F = (F ,..., l l Expand Fi j (x) and Ri j (x):
m F ).
l Fi j (x) =
l ik,x Fki , j e
k∈Zn
Rli j (x) =
Rlki j eik,x
k∈Zn
with l = , , . . . , m and (i , j ) = (, ), (, ). Let
l F X= , l F
aek – Mk = bek
Rl Y= , Rl –b . ek – a
l To get the estimate of Fi j (x), we rewrite (.) as the following form:
Mk Xk eik,x =
k∈Zn
Yk eik,x +
k∈Zn
Nk Xk eik,x ,
(.)
k∈Zn
where Nk is composed of the components of Qj , j = , , , . We can set |Nk | ≤ . By a direct calculation, we have |Mk | = |ek – λi |ek – λi , where λi =
i +
( i ) – ,
λi =
i –
( i ) – ,
i = , , . . . , m,
are the eigenvalues of . By Assumption , we have |λi | = , |λi | = . Since |ek | = , it follows that |Mk | > c > . We rewrite (.) as X = Y + X. Since |Mk | > , we have the operator is invertible and hence X = – (Y + X). Set X = X – – X, then we have X = L– Y . So X – X = – X – – X ≤ – · · X – X .
Jiang Journal of Inequalities and Applications (2017) 2017:39
Set =
c ,
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then we have
X – X ≤ X – X . By the implicit function theorem, we have X ≤ c Y , with c depending on A, B, C. So Fi j D(s–ρ,r)× ≤
cr Ri j D(s,r)× α ν¯ ρ ν
with (i , j ) = (, ), (, ). From the above discussion, we have XF r;D(s–ρ,r)× ≤
c c XR r;D(s,r)× ≤ ν¯ ν , α ν¯ ρ ν α ρ
(.)
where ν¯ = n¯ + and ν = τ (n¯ + ) + n¯ + n. By (.) and (.), we obtain c , α ν¯ ρ ν c D – id r;D(s–ρ, r )× ≤ ν¯ ν+ . α ρ
– id r;D(s–ρ, r )× ≤
From the above discussion, we get the conjugate mapping + (·; ξ ) = – ◦ ◦ gener¯ where ated by H+ = N¯ + P, N¯ = [R ] + x + ω(ξ ) + R , yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ), and P¯ = ϒ + (P – R) – F (x), Qx . Recalling (.), we find there are second order terms of u, vˆ in P+ , so we will put these terms into the main term. Let Q+ = –F (x), Qx + ϒ , where ϒ contains the second order term on u, vˆ in ϒ. Then we get H+ = N+ + P+ , where N+ = [R ] + x + ω(ξ ) + R , yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ) + Q+ and P+ = (P – R) + ϒ – ϒ . We note that N+ has the same form as N .
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Since
P–R=
Plij yˆ l ui vˆ j +
|l|+|i|+|j|=
Plij yˆ l ui vˆ j ,
|l|+|i|+|j|≥
we have e–Kρ XP–R ηr;D(s–ρ,ηr)× ≤ c · η + . η
(.)
By (.) and (.), we have e–Kρ c XP+ ηr;D(s–ρ,ηr)× ≤ c · η + + ¯ν ν . η η α ρ F. Choice of parameters. We choose < E < and set η = E,
e–Kρ = E, η
= η α ν ρ ν E,
h=
α . (K + )τ + T
Fix σ ∈ (, ). We define ρ+ = σρ,
s+ = s – ρ,
α+ = α – (K + )τ + ,
r+ = ηr, + = cη,
E+ = cE .
By the estimate of P+ , supposing α < α+ , we have c e–Kρ XP+ ηr;D(s–ρ,ηr)×+ ≤ c · η + + ¯ν ν η η α ρ ≤ cη = cα ν ρ ν E ≤ cα+¯ν ρ+τ E+ . Setting + = cα+ ρ+τ E+ , we arrive at XP+ r+ ;D(s+ ,r+ )×+ ≤ + . By Iteration Lemma, we have k, ω+ (ξ ) + πl ≥ k, ω(ξ ) + πl – k, ω+ (ξ ) – ω(ξ ) ≥
α – ( + K)τ + τ ( + |k|)
=
α+ , ( + |k|)τ
where ξ ∈ + and = k ≤ K . So we choose α+ = α – ( + K)τ + . By the choice of α+ , the ¯ it follows, for ∀ξ ∈ + , ¯ in (.) and + = \ , definition of k, ω+ (ξ ) + πl ≥
α+ , ( + |k|)τ
∀k ∈ Zn , < |k| ≤ K+ .
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Now we give the choice of T+ . Suppose h+ ≤ h. By the Cauchy estimate, for ξ ∈ +h+ , we have ∂ ω+ (ξ ) – ω(ξ ) /∂ξ ≤ |ω+ (ξ ) – ω(ξ )|h ≤ . h+ h – h+ h Define T+ = T +
h
and h+ =
α+ , T+ (+K+ )τ +
then we have
max |∂ω+ /∂ξ | ≤ max ∂ ω+ – ω(ξ ) /∂ξ + max |∂ω/∂ξ | ≤ T+ .
ξ ∈h+
ξ ∈h+
ξ ∈h+
Thus all the parameters for H+ are defined, and so Iteration Lemma is proved.
2.2 Iteration Define inductive sequences ρj+ = σρj ,
sj+ = sj – ρ,
αj+ = αj – ( + Kj )τ + j , ηj+ = Ej+ ,
rj+ = ηj rj , Ej+ = cEj ,
Tj+ = Tj +
j , dj
ν j+ = αj¯ν ρj+ Ej+ ηj+ ,
and e–Kj+ ρj+ = Ej+ , ηj+
αj+ . ( + K)τj + Tj+
hj+ =
Define j+ = ξ ∈ j : ωj (ξ ) + πl, k ≥
αj , Kj < |k| ≤ Kj+ (|k| + )τ
and
hj+ = ξ ∈ C n : dist(ξ , j+ ) ≤ hj+ . In the following we give some estimates for Gevrey-smoothness. K Let γj = Kj ρj = – ln Ej . We have Kj+j = lnlnEcj + σ , and hence ≤
α T (+K )τ enough. If < Kj < Kj+ , we have = αj+j Tj+j (+Kj+j )τ ≤ , and ∂ω means h+ ≤ h holds. Suppose maxξ ∈hj | ∂ξj | ≤ Tj . Let Tj+ = hj+ hj
|
∂ωj+ | ∂ξ
=|
∂(ωj+ –ωj +ωj ) | ∂ξ δ
Kj+ Kj
ρ
for E small
hence hj+ ≤ hj , which Tj +
j . dj
Then we have δ
δ
τ + ≤ Tj+ . By the choice of σ , we can easily get that ρj+ γj+ ≥ ρj γj τ + . δ
Since ρ γτ + ≥ , we have ρj γj τ + ≥ for all j > . By the definitions of Tj , hj and j , we have Tj+ = Tj + ing γj = – ln Ej j τ –γi Ti i= (γi ) e
≤
( )j
j dj
= T +
j
i= (γi )
τ –γi
e
Ti . Not-
and Ej ≤ (cE ) , we can choose E to be sufficiently small such that ≤ , then we have T ≤ Tj ≤ T + . Similarly, we have αj ≤ αj+ ≤ αj .
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By Iteration Lemma, there exists a sequence of symplectic mappings {j (·; ξ )}, generated by p, q+ + Fj (p, q+ ), satisfying j – id rj ;D(sj –ρj ,rj )×hj ≤
cj , αjν¯ ρjν
Dj – Id rj ;D(sj –ρj ,rj )×hj ≤
cj . ν¯ ν+ αj ρj
Define j = ◦ ◦· · ·◦j . Then we have a sequence of symplectic mappings {j+ (·; ξ ) = ( j )– ◦ j ◦ j }, generated by Hj+ (·; ξ ) = Nj+ + Pj+ , where Nj+ = x + ωj+ (ξ ), yˆ + Aj+ u, vˆ + Bj+ u, u + Cj+ vˆ , vˆ with |ωj+ – ωj | ≤ cj ,
∀j ≥ ,
and XPj+ rj+ ;D(sj+ ,rj+ )×hj+ ≤ j+ .
2.3 The convergence of the KAM iteration Now we prove the convergence of the KAM iteration. Similar to [], we have j ν¯ ν – j– ≤ cαj– ρj– Ej– , r ;D(s –ρ ,r )× j
j
j j
hj
and j ν¯ ν+ D – j– ≤ αj– ρj– Ej– . r ;D(s –ρ ,r )× j
j
j j
hj
By the Cauchy estimate, we have ν¯ ν cαj– ρj– Ej– β! β j ∂ – j– ≤ , ξ |β!| rj ;D(sj –ρj ,rj )×hj hj ν+ ν¯ cαj– ρj– Ej– β! β j ∂ D – j– ≤ ξ |β!| rj ;D(sj –ρj ,rj )×hj hj
and β j ∂ ω – ωj– ≤ cj– β! . ξ |β!| j hj β
Let Uj =
ν¯ ρ ν E β! cαj– j– j– |β!|
hj δ
β
and Gj =
cj– β! |β!|
hj
β
β
. Now we estimate Uj and Gj for β ∈ Zn+ .
Since ρj γj τ + ≥ for all j > , we have
ρj
δ
≤ γj τ + . Then we have Kj =
means that Kjτ + ≤ γjτ ++δ . Noting that hj =
αj (K+)τj + Tj
γj ρj
+ τ δ+
≤ γj
, which
, Tj < T + , α ≤ αj and Ej– = Ej =
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γj
e– , we have β Uj
≤ cα ν¯ ρjν β! ≤ cρjν
T + α
(T + ) α
|β|
|β|
γjτ ++δ
|β|
β – γj β! γj e
e–
γj
(τ +δ)(n+)
β
j τ +δ – γj – (τ +δ)(n+) e n+
· · · γj n e
γ
≤ cρjν J |β| β!μ Jj(n+) , where J = T+ [ (μ–)(n+) ]μ– , μ = τ + δ, and c only depends on n, α, μ. α In the same way, we have
Gj ≤ cρjν J |β| β!μ Ej(n+) . β
Note that sj → s , rj → , hj → as j → ∞. Let D∗ = D( s , ), ∗ = limj→∞ j . Then we have β ∂ (∗ – id) r ≤ cρν J |β| β!μ E(n+) , ξ ;D(∗)×∗
j j≥
and ∗ =
∀β ∈ Zn+ .
β
Since j is affine in y, j is also affine in y, and hence we have the convergence of ∂ξ j to β ∂ξ ∗ on D( s , r ) and β ν |β| μ (n+) ∂ (∗ – id) r s r ≤ cρ J β! E , ξ ;D( , )×∗
∀β ∈ Zn+ . Thus we proved (.). Let ω∗ = limj→∞ ωj . Similarly, it follows that β ∂ ω∗ (ξ ) – ω (ξ ) ≤ cρ ν J |β| β!μ E (n+) , ξ ∗
∀β ∈ Zn+ .
Moreover we have ω∗ (ξ ), k ≥
α∗ ( + |k|)τ
for all ξ ∈ ∗ and = k ∈ Zn , where α∗ = limj→∞ α j , with (.) and (.).
α
≤ α∗ ≤ α. Thus we proved
2.4 Estimate of measure We note that β ≥ in Assumption for symplectic mappings , while β ≥ in Hamiltonian systems [, ]. So the non-degeneracy condition in symplectic mappings and that in Hamiltonian systems are different. It means that the estimate of measure is different in two cases. But the proof for symplectic mappings is similar to [, ], so we omit the details. Competing interests The author declares that they have no competing interests.
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Author’s contributions The article is a work of the author who contributed to the final version of the paper. The author read and approved the final manuscript. Acknowledgements The paper was completed during the author’s visit to the Department of Mathematics of Pennsylvania State University, which was supported by Nanjing Tech University. The author thanks Professor Mark Levi for his invitation, hospitality and valuable discussions. The work is supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (14KJB110009). The work is partly supported by the Natural Science Foundation of Jiangsu Province (BK20140927) and (BK20150934). The work is also in part supported by the Natural Science Foundation of China (11301263). Received: 25 September 2016 Accepted: 26 January 2017 References 1. Poincaré, H, Magini, R: Les méthodes nouvelles de la mécanique céleste. Il Nuovo Cimento (1895-1900) 10, 128-130 (1899) 2. Poincaré, H: Sur un théoreme de géométrie. Rendiconti del Circolo Matematico di Palermo (1884-1940) 33, 375-407 (1912) 3. Moser, J: On Invariant Curves of Area-Preserving Mappings of an Annulus. Vandenhoeck & Ruprecht, Gottingen (1962) 4. Aubry, S, Le Daeron, PY: The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states. Physica D 8, 381-422 (1983) 5. Aubry, S, Abramovici, G: Chaotic trajectories in the standard map. The concept of anti-integrability. Physica D 43, 199-219 (1990) 6. Mather, J: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457-467 (1982) 7. Mather, J: Non-existence of invariant circles. Ergod. Theory Dyn. Syst. 4, 301-309 (1984) 8. Mather, J: More Denjoy minimal sets for area preserving diffeomorphisms. Comment. Math. Helv. 60, 508-557 (1985) 9. Conley, C, Zehnder, E: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73, 33-50 (1983) 10. Conley, C, Zehnder, E: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37, 207-253 (1984) 11. Cheng, C-Q, Sun, Y-S: Existence of invariant tori in three-dimensional measure-preserving mappings. Celest. Mech. Dyn. Astron. 47, 275-292 (1989) 12. de la Llave, R, James, JM: Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete Contin. Dyn. Syst. 32, 4321-4360 (2012) 13. Dullin, HR, Meiss, JD: Resonances and twist in volume-preserving mappings. SIAM J. Appl. Dyn. Syst. 11, 319-349 (2012) 14. Fox, AM, Meiss, JD: Greene’s residue criterion for the breakup of invariant tori of volume-preserving maps. Physica D 243, 45-63 (2013) 15. Gelfreich, V, Simó, C, Vieiro, A: Dynamics of 4D symplectic maps near a double resonance. Physica D 243, 92-110 (2013) 16. Rüssmann, H: On the existence of invariant curves of twist mappings of an annulus. In: Geometric Dynamics, pp. 677-718. Springer, Berlin (1983) 17. Xia, Z-H: Existence of invariant tori in volume-preserving diffeomorphisms. Ergod. Theory Dyn. Syst. 12, 621-631 (1992) 18. Bi, Q-Y, Xun, J-X: Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings. Qual. Theory Dyn. Syst. 13, 269-288 (2014) 19. Lu, X-Z, Li, J, Xu, J-X: A KAM theorem for a class of nearly integrable symplectic mappings. J. Dyn. Differ. Equ. 27, 1-24 (2015). 20. Shang, Z-J: A note on the KAM theorem for symplectic mappings. J. Dyn. Differ. Equ. 12, 357-383 (2000) 21. Zhu, W, Liu, B, Liu, Z: The hyperbolic invariant tori of symplectic mappings. Nonlinear Anal. 68, 109-126 (2008) 22. Pöschel, J: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35, 653-696 (1982) 23. Popov, G: Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms. Ann. Henri Poincaré 1, 223-248 (2000) 24. Xu, J-X, You, J-G: Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann’s non-degeneracy condition. J. Differ. Equ. 235, 609-622 (2007) 25. Zhang, D-F, Xu, J-X: On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition. Discrete Contin. Dyn. Syst., Ser. A 16, 635-655 (2006) 26. Zhang, D-F, Xu, J-X: Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmanns non-degeneracy condition. J. Math. Anal. Appl. 322, 293-312 (2006) 27. Pöschel, J: On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202, 559-608 (1989)
RESEARCH
Open Access
Gevrey-smoothness of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings Shunjun Jiang* *
Correspondence: [email protected] College of Sciences, Nanjing Tech University, Nanjing, 210009, China
Abstract This paper provides a normal form for a class of lower dimensional hyperbolic invariant tori of nearly integrable symplectic mappings with generating functions. We prove the persistence and the Gevrey-smoothness of the invariant tori under some conditions. Keywords: symplectic mappings; KAM iteration; Gevrey-smoothness; normal form
1 Introduction and main results Area-preserving mappings have some dynamical properties similar to Hamiltonian systems, and hence become an important test ground of all kinds of theories for studying Hamiltonian systems, such as Poincaré [, ] on three body problem, Moser [] on the differentiable form of KAM theory, Aubry and Mather [–] on Aubry-Mather theorem, Conley and Zehnder [, ] on symplectic topology. So area-preserving mappings have attracted many scholars’ interest. We refer to [–]. Among all the mappings, symplectic mappings are special for their symplectic structures; we refer to [–] for more results on symplectic structures. On the other hand, many mathematicians turn to the study of the connection between the KAM tori and the parameter. The first work is due to Pöschel [] who proved that the persisting invariant tori are C ∞ -smooth in the frequency parameter. Popov [] obtained the Gevrey-smoothness, a notion intermediate between C ∞ -smoothness and analyticity, of invariant tori in the frequencies under the Kolmogorov non-degeneracy condition. Xu and You [] obtained a similar result under the Rüssmann non-degeneracy condition by an improved KAM method. For more results, we refer to [, ]. Motivated by [, ], we consider the persistence and the Gevrey-smoothness of lower dimensional hyperbolic invariant tori for symplectic mappings determined by generating functions under Rüssmann’s non-degeneracy condition. We consider the following parameterized symplectic mapping: ˆ yˆ , vˆ ) ∈ Tn × Rm × Rn × Rm , (·; ξ ) : (x, u, y, v) ∈ Tn × W × O × W → (ˆx, u, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Jiang Journal of Inequalities and Applications (2017) 2017:39
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where ξ ∈ ⊂ O is a parameter and O ⊂ Rn is a bounded closed connected domain. Suppose (·; ξ ) is implicitly defined by ⎧ ⎪ ⎪xˆ = ∂yˆ H(x, u, yˆ , vˆ ; ξ ), ⎪ ⎪ ⎪ ⎨y = ∂ H(x, u, yˆ , vˆ ; ξ ), x
(.)
⎪ ⎪ uˆ = ∂vˆ H(x, u, yˆ , vˆ ; ξ ), ⎪ ⎪ ⎪ ⎩ v = ∂u H(x, u, yˆ , vˆ ; ξ ), where H(x, u, yˆ , vˆ ; ξ ) = N + P,
(.)
N(x, u, yˆ , vˆ ; ξ ) = x + ω(ξ ), yˆ + Au, vˆ + Bu, u + C vˆ , vˆ .
(.)
Suppose A is a constant matrix and B, C are symmetric. If P = , can be expressed explicitly as xˆ = x + ω(y), yˆ = y, – – uˆ = A – C AT B u + C AT v, We define
A – C(AT )– B = –(AT )– B
C(AT )– (AT )–
– – vˆ = – AT Bu + AT v.
(.)
. m×m
Denote the eigenvalues of by (λ , λ , . . . , λm ). We call the lower dimensional invariant torus elliptic if |λi | = , λi = , ∀i = , , . . . , m and hyperbolic if |λi | = , ∀i = , , . . . , m. We note that, although some results on symplectic mappings can be anticipated by Hamiltonian systems, there are still many differences for lower dimensional invariant tori. The first one is concerned with the relations of variables. In symplectic mappings, some variables determined by generating functions take on an implicit form and hence lead to more difficulties than in a Hamiltonian system. The second one is the non-degeneracy condition, which will result in a more complicated proof for estimate of measure. Before presenting the main result, we give some assumptions and definitions. Assumption (Rüssmann’s non-degeneracy condition) There exists an integer n¯ > such that β
rank ∂ξ ω(ξ ) : ≤ |β| ≤ n¯ = n,
∀ξ ∈ ,
where β T β β β ∂ξ ω(ξ ) = ∂ξ ω (ξ ), ∂ξ ω (ξ ), . . . , ∂ξ ωn (ξ ) with |β|
∂ξ ωi (ξ )
β
∂ξ ωi (ξ ) =
β
β
β
∂ξ ∂ξ · · · ∂ξn n
,
(.)
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and β = (β , β , . . . , βn ). Assumption (Hyperbolic condition) Let A = diag(a , a , . . . , am ), B = diag(b , b , . . . , bm ), C = diag(c , c , . . . , cm ). Define i =
ai –bi ci + , i = , , . . . , m. ai
Suppose i > , i = , , . . . , m.
Remark . By direct calculation, we have the eigenvalues of ,
λi =
i ±
( i ) – ,
i = , , . . . , m.
If i > , we have |λi | = , i = , , . . . , m, so the lower dimensional invariant torus is hyperbolic. If otherwise i < , we have |λi | = , and hence the lower dimensional invariant torus is elliptic. Definition . Let O ⊂ Rn be a bounded closed connected domain. A function F : O → R is said to belong to Gevrey-class Gμ (O) of index μ (μ ≥ ) if F is C ∞ (O)-smooth and there exists a constant J such that for all p ∈ O , β ∂ F(p) ≤ cJ |β|+ β!μ , p where |β| = β + β + · · · + βn and β!μ = β !β ! · · · βn ! for β = (β , β , . . . , βn ) ∈ Zn+ . Remark . By definition, it is easy to see that the Gevrey-smooth function class G coincides with the analytic function class. Moreover, we have G ⊂ Gμ ⊂ Gμ ⊂ C ∞ for < μ < μ < ∞. Set
Ts = x ∈ Cn /πZn : | Im x|∞ ≤ s ,
Br = y ∈ Cn : |y| ≤ r , and
Wr = w ∈ Cm : |w| ≤ r . Denote by D(s, r) = Ts × Wr × Br × Wr . Here, |x|∞ = max≤j≤n |xj |, |y| = |w| = ( ≤j≤m |wj | ) . Denote
= ξ ∈ O | dist(ξ , ∂ O) ≥ h
≤j≤n |yj |
and
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and
h = ξ ∈ Cn | dist(ξ , ) ≤ h . Definition . f ∈ G,μ (D(s, r) × ) means that f ∈ C ∞ (D(s, r) × ) and f (x, y, u, v; ξ ) is analytic with respect to (x, y, u, v) on D(s, r) and Gμ -smooth in ξ on . ¯ Below we define some norms. If P(x, u, yˆ , vˆ ; ξ ) is analytic in (x, u, yˆ , vˆ ) on D(s, r) and ntimes continuously differentiable in ξ on , we have P(x, u, yˆ , vˆ ; ξ ) =
Pk (u, yˆ , vˆ ; ξ )eik,x ,
k∈Zn
where Pk (u, yˆ , vˆ ; ξ ) =
Pklij (ξ )ˆyl ui vˆ j .
l,i,j
Define P D(s,r)× =
|Pk |r es|k| ,
k∈Zn
where |Pk |r =
sup
(u,ˆy,ˆv)∈Wr ×Br ×Wr i,j,l
Pklij s yˆ l ui vˆ j .
This norm is apparently stronger than the super-norm. Moreover, the Cauchy estimate of analytic functions is also valid under this norm. Let XP = (–∂yˆ P, –∂vˆ P, ∂x P, ∂u P) and denote a weighted norm by XP r;D(s,r)× = ∂yˆ P D(s,r)× + ∂vˆ P D(s,r)× r + ∂x P D(s,r)× + ∂u P D(s,r)× , r r where ∂xˆ P D(s,r)× =
∂xˆ j P D(s,r)× ,
j
∂yˆ P D(s,r)× = max ∂yˆ j P D(s,r)× j
and ∂u P D(s,r)× =
∂uj P s,r
j
∂vˆ P D(s,r)× is defined similarly.
,
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Now we introduce the main result. Let τ ≥ nn¯ – . For δ ∈ (, ), let μ = τ + δ + and δ σ = ( ) τ ++δ . Theorem . Consider the symplectic mapping (·; ξ ), which is implicitly defined by a | ≤ T. Suppose that Assumptions , generating function H(·; ξ ) in (.). Let maxξ ∈h | ∂ω ∂ξ hold. Then there exists γ > such that for any < α < , if XP r;D(s,r)×d = ≤ γ α ¯ν ρ ν ¯ the following results hold true: with ν¯ = n¯ + , ν = τ (n¯ + ) + n + n, (i) There exist a non-empty Cantor-like subset ∗ ⊂ and, for ξ ∈ ∗ , a symplectic mapping ∗ (·; ξ ), where ∗ ∈ G,μ with β ∂ (∗ – id) s r ≤ cρ ν J |β| β!μ γ (n+) ξ r;D( , )×∗
(.)
for ∀β ∈ Zn+ and J = T+ [ (μ–)(n+) ]μ– . Moreover, ∗ = ∗– ◦ ◦ ∗ is generated by α H∗ = N∗ + P∗ as in (.) satisfying N∗ (x, u, yˆ , vˆ ; ξ ) = x + ω∗ , yˆ + A∗ u, vˆ + B∗ u, u + C∗ vˆ , vˆ , P∗ (x, u, yˆ , vˆ ; ξ ) = Plij (x; ξ )ˆyl ui vˆ j . |i|+|j|+|l|≥
(ii) Hence, for ξ ∈ ∗ , the symplectic mapping (·; ξ ) admits a lower dimensional invariant torus Tξ = ∗ T n , , , ; ξ , whose frequencies ω∗ satisfy β ∂ ω∗ (ξ ) – ω(ξ ) ≤ cρ ν J |β| β!μ γ (n+) ξ
(.)
ω∗ (ξ ), k + πl ≥
(.)
and α ( + |k|)τ
for all ξ ∈ ∗ , = k ∈ Zn . Moreover, we have
meas( \ ∗ ) ≤ cα m .
2 The proof of main results We will use the idea for Hamiltonian systems in [] to prove our results. In Section ., one KAM step iteration is presented. The key lies in solving a homological equation. Then we will show the KAM step can iterate infinitely in Section .. Convergence of the iteration and the estimate of measure will be presented in Sections . and .. 2.1 KAM-step Iteration Lemma Consider a symplectic mapping (·; ξ ) defined in Theorem .. Let < E < , < ρ = ( – σ )s/ < s and < η < . Let K > satisfy η e–Kρ = E. Let ∂ω max ≤ T, ξ ∈h ∂ξ
h=
α . ( + K)τ + T
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Moreover, ω(ξ ) satisfies that: for k ∈ Zn \ {}, l ∈ Z, k, ω + πl ≥
α . ( + |k|)τ
(.)
Suppose Assumptions , hold. Suppose that P satisfies XP r;D(s,r)×d ≤ = η α ¯ν ρ ν E, ¯ Then we have the following results: with < α < , ν¯ = n¯ + , ν = τ (n¯ + ) + n + n. () ∀ξ ∈ h , there exists a symplectic diffeomorphism (·; ξ ) with c , α ν¯ ρ ν c ≤ ν¯ ν+ , α ρ
– id r;D(s–ρ, r )×h ≤ D – id r;D(s–ρ, r )×h
such that the conjugate mapping + (·; ξ ) = – ◦ ◦ is generated by H+ (·; ξ ) = N+ + P+ , where N+ = x + ω+ (ξ ), yˆ + A+ u, vˆ + B+ u, u + C+ vˆ , vˆ and P+ satisfies XP r+ ;D(s+ ,r+ )×d ≤ η+ α+¯ν ρ+ν E+ = + , with s+ = s – ρ,
ρ+ = σρ,
α ≤ α+ ≤ α.
r+ = ηr,
η+ = E+ ,
E+ = E ,
Furthermore, we have ω+ (ξ ) – ω(ξ ) ≤ ,
∀ξ ∈ h .
(.)
() Let α+ = α – (K + )τ + , ¯ = ξ ∈ : k, ω+ (ξ ) <
α+ n , k ∈ Z , K < |k| ≤ K + , ( + |k|)τ
(.)
¯ Then, for ∀ξ ∈ + , ∀k ∈ Zn and < |k| ≤ K+ , we have and + = \ . k, ω+ (ξ ) ≥
α+ , ( + |k|)τ –K+ ρ+ η+
where K+ > such that e
(.)
= E+ .
α+ ∂ω+ () Let T+ = T + and h+ = (K +) τ + T . If h+ ≤ h, we have maxξ ∈h+ | ∂ξ | ≤ T+ , + + where h+ is the complex h+ -neighborhood of + . h
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A. The equivalent form of (.). Let P(p, qˆ ) = P (x) + P (x), yˆ + P (x), u + P (x), vˆ + P (x)u, vˆ + P (x)u, u + P (x)ˆv, vˆ l i j + Plij yˆ u vˆ
(.)
|i|+|j|+|l|≥
with ∂ l+i+j P . Plij = l i j ∂ yˆ ∂u ∂ vˆ u=,ˆy=,ˆv= Let Q(x, u, vˆ ) = Q (x)u, vˆ + Q (x)u, u + Q (x)ˆv, vˆ with Q (x) = P (x), Q (x) = P (x), Q (x) = P (x). Then we rewrite H as H = N + Q + (P – Q), where N + Q is the new main term and P – Q is the new small term. Now we will study the following function which is equivalent to (.): H(x, u, yˆ , vˆ ; ξ ) = N(x, u, yˆ , vˆ ; ξ ) + P(x, u, yˆ , vˆ ; ξ ),
(.)
where N = x + ω(ξ ), yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ) and P = P (x) + P (x), yˆ + P (x), u + P (x), vˆ +
Plij yˆ l ui vˆ j .
|l|+|i|+|j|≥
B. Generating functions of conjugate mappings. For convenience, let p = (x, u) and q = (y, v). pˆ and qˆ have a similar meaning. The symplectic structure becomes dp ∧ dq on Rn+m × Rn+m . Consider a symplectic mapping : (p, q) → (ˆp, qˆ ) generated by pˆ = ∂qˆ H(p, qˆ ),
q = ∂p H(p, qˆ ),
(.)
where H(p, qˆ ) = N(p, qˆ ) + P(p, qˆ ), where N is the main term and P is a small perturbation. We need a symplectic transformation : (p+ , q+ ) → (p, q) generated by q = q+ + F (p, q+ ),
p+ = p + F (p, q+ ),
(.)
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with F (p, q+ ) =
∂F(p, q+ ) , ∂p
F (p, q+ ) =
∂F(p, q+ ) . ∂q+
The generating function is p, q+ + F(p, q+ ) with F being a small function. By (.) and (.), we have a conjugate mapping = – ◦ ◦ : (p+ , q+ ) → (ˆp+ , qˆ + ) implicitly by pˆ + = H (p, qˆ ) + F (ˆp, qˆ + ),
q+ = H (p, qˆ ) – F (p, q+ )
(.)
with H (p, qˆ ) =
∂H(p, qˆ ) , ∂p
H (p, qˆ ) =
∂H(p, qˆ ) . ∂ qˆ
So we have the following lemma. Lemma . ([]) The conjugate symplectic mapping + can be implicitly determined by a generating function H+ (p+ , qˆ + ) through pˆ + = ∂qˆ + H+ (p+ , qˆ + ),
q+ = ∂p+ H+ (p+ , qˆ + ),
(.)
where H+ (p+ , qˆ + ) = H(p, qˆ ) + H (p, qˆ )F (p, q+ ) – H (p, qˆ )F (ˆp, qˆ + ) + F(ˆp, qˆ + ) – F(p, q+ ) – F (p, q+ )F (p, q+ ),
(.)
where p, pˆ , qˆ , q+ depend on (p+ , qˆ + ) as explained above. Set z = (p+ , qˆ + ). We have H+ (z) = H(z) + F N (z), qˆ + – F p+ , N (z) + ϒ(z)
(.)
with N (z) =
∂N(p+ , qˆ + ) , ∂p+
N (z) =
∂N(p+ , qˆ + ) ∂ qˆ +
and ϒ(z) satisfying |||Xϒ |||r;D(s–ρ,r/) ≤
c , α ¯ν ρ ν
where ν¯ = n¯ + and ν = τ (n¯ + ) + n¯ + n.
(.)
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C. Truncation. Let P = R + (P – R),
(.)
where R(p, qˆ ) =
k k k k P (x) + P (x), yˆ + P (x), u + P (x), vˆ
(.)
|k|≤K
= R (x) + R (x), yˆ + R (x), u + R (x), vˆ ,
(.)
and P–R=
Plij yˆ l ui vˆ j +
|l|+|i|+|j|=
Plij yˆ l ui vˆ j .
|l|+|i|+|j|≥
Let F(p, qˆ ) possess the same form as (.). D. Extending the small divisor estimate. ∀ξ ∈ h , there exists ξ ∈ such that |ξ – ξ | < h. So we have, for < |k| ≤ K , k, ω(ξ ) + πl ≥ k, ω(ξ ) + πl – k, ω(ξ ) – ω(ξ ) ≥
α . ( + |k|)τ
E. Homological equation. By (.), it follows that N(p+ , qˆ + ) + P(p+ , qˆ + ) – F p+ , Np (p+ , qˆ + ) + F Nq (p+ , qˆ + ), qˆ + + ϒ(z) ¯ + , qˆ + ) + P(p ¯ + , qˆ + ). = N(p
(.)
For simplicity, below we drop the subscripts ‘+’ in p+ and qˆ + . Similar to the discussion in [], we get the homological equations. –F Nq (p, qˆ ), qˆ + F p, Np (p, qˆ ) = R – [R].
(.)
To solve (.), we need some preparations. Let x + ω = x˜ . Since pˆ = Nqˆ (p, pˆ ) = x˜ , (A + Q )u + (C + Q )v and q = Np (p, qˆ ) = yˆ + Qx , (A + Q )ˆv + (B + Q )u , we have F p, Np (p, qˆ ) = F (x) + F (x), yˆ + Qx + F (x), u + F (x), (A + Q )T vˆ + (B + Q )u
(.)
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and F Nqˆ (p, qˆ ), qˆ = F (˜x) + F (˜x), yˆ + F (˜x), vˆ + F (˜x), (A + Q )T u + (C + Q )ˆv .
(.)
So we get F Nqˆ (p, qˆ ), qˆ – F p, Np (p, qˆ ) = L + L + L – F (x), Qx , where L = F (˜x) – F (x) + F (˜x) – F (x), yˆ , L = AT F (˜x) – F (x) – BF (x), u + QT F (˜x) – QT F (x), u and L = C T F (˜x) + F (x) – AF (x), u + QT F (˜x) – Q F (x), u . After these preparations, we can solve (.) which is equivalent to solving the following: ⎧ ⎨F
x) – F (x) = R (x) – [R ], (˜ ⎩F (˜x) – F (x) = R (x) – [R ],
(.)
and ⎧ ⎨AT F
x) – F (x) – BF (x) + QT F (˜x) – QT F (x) = R (x), (˜
⎩C T F (˜x) + F (x) – AF (x) + QT F (˜x) – Q F (x) = R (x).
(.)
Firstly, we solve (.) for F and F . Expand F (x) and R (x): F (x) =
Fk eik,x ,
k∈Zn
R (x) =
Rk eik,x .
k∈Zn
Then we get Fkj =
Rkj ek –
(.)
with ek = eik,ω , k = . By Assumption , we have the following estimate: Fk s–ρ ≤
c Rk s . αρ τ +n
So we have XF r;D(s–ρ)× ≤
c R s . α ν¯ ρ ν
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Similarly we have XF r;D(s–ρ)× ≤
c R s . α ν¯ ρ ν
m Next we will get F and F from (.). Let F = (F , . . . , F ) and F = (F ,..., l l Expand Fi j (x) and Ri j (x):
m F ).
l Fi j (x) =
l ik,x Fki , j e
k∈Zn
Rli j (x) =
Rlki j eik,x
k∈Zn
with l = , , . . . , m and (i , j ) = (, ), (, ). Let
l F X= , l F
aek – Mk = bek
Rl Y= , Rl –b . ek – a
l To get the estimate of Fi j (x), we rewrite (.) as the following form:
Mk Xk eik,x =
k∈Zn
Yk eik,x +
k∈Zn
Nk Xk eik,x ,
(.)
k∈Zn
where Nk is composed of the components of Qj , j = , , , . We can set |Nk | ≤ . By a direct calculation, we have |Mk | = |ek – λi |ek – λi , where λi =
i +
( i ) – ,
λi =
i –
( i ) – ,
i = , , . . . , m,
are the eigenvalues of . By Assumption , we have |λi | = , |λi | = . Since |ek | = , it follows that |Mk | > c > . We rewrite (.) as X = Y + X. Since |Mk | > , we have the operator is invertible and hence X = – (Y + X). Set X = X – – X, then we have X = L– Y . So X – X = – X – – X ≤ – · · X – X .
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Set =
c ,
Page 12 of 17
then we have
X – X ≤ X – X . By the implicit function theorem, we have X ≤ c Y , with c depending on A, B, C. So Fi j D(s–ρ,r)× ≤
cr Ri j D(s,r)× α ν¯ ρ ν
with (i , j ) = (, ), (, ). From the above discussion, we have XF r;D(s–ρ,r)× ≤
c c XR r;D(s,r)× ≤ ν¯ ν , α ν¯ ρ ν α ρ
(.)
where ν¯ = n¯ + and ν = τ (n¯ + ) + n¯ + n. By (.) and (.), we obtain c , α ν¯ ρ ν c D – id r;D(s–ρ, r )× ≤ ν¯ ν+ . α ρ
– id r;D(s–ρ, r )× ≤
From the above discussion, we get the conjugate mapping + (·; ξ ) = – ◦ ◦ gener¯ where ated by H+ = N¯ + P, N¯ = [R ] + x + ω(ξ ) + R , yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ), and P¯ = ϒ + (P – R) – F (x), Qx . Recalling (.), we find there are second order terms of u, vˆ in P+ , so we will put these terms into the main term. Let Q+ = –F (x), Qx + ϒ , where ϒ contains the second order term on u, vˆ in ϒ. Then we get H+ = N+ + P+ , where N+ = [R ] + x + ω(ξ ) + R , yˆ + Au, vˆ + Bu, u + C vˆ , vˆ + Q(x, u, vˆ ) + Q+ and P+ = (P – R) + ϒ – ϒ . We note that N+ has the same form as N .
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Since
P–R=
Plij yˆ l ui vˆ j +
|l|+|i|+|j|=
Plij yˆ l ui vˆ j ,
|l|+|i|+|j|≥
we have e–Kρ XP–R ηr;D(s–ρ,ηr)× ≤ c · η + . η
(.)
By (.) and (.), we have e–Kρ c XP+ ηr;D(s–ρ,ηr)× ≤ c · η + + ¯ν ν . η η α ρ F. Choice of parameters. We choose < E < and set η = E,
e–Kρ = E, η
= η α ν ρ ν E,
h=
α . (K + )τ + T
Fix σ ∈ (, ). We define ρ+ = σρ,
s+ = s – ρ,
α+ = α – (K + )τ + ,
r+ = ηr, + = cη,
E+ = cE .
By the estimate of P+ , supposing α < α+ , we have c e–Kρ XP+ ηr;D(s–ρ,ηr)×+ ≤ c · η + + ¯ν ν η η α ρ ≤ cη = cα ν ρ ν E ≤ cα+¯ν ρ+τ E+ . Setting + = cα+ ρ+τ E+ , we arrive at XP+ r+ ;D(s+ ,r+ )×+ ≤ + . By Iteration Lemma, we have k, ω+ (ξ ) + πl ≥ k, ω(ξ ) + πl – k, ω+ (ξ ) – ω(ξ ) ≥
α – ( + K)τ + τ ( + |k|)
=
α+ , ( + |k|)τ
where ξ ∈ + and = k ≤ K . So we choose α+ = α – ( + K)τ + . By the choice of α+ , the ¯ it follows, for ∀ξ ∈ + , ¯ in (.) and + = \ , definition of k, ω+ (ξ ) + πl ≥
α+ , ( + |k|)τ
∀k ∈ Zn , < |k| ≤ K+ .
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Now we give the choice of T+ . Suppose h+ ≤ h. By the Cauchy estimate, for ξ ∈ +h+ , we have ∂ ω+ (ξ ) – ω(ξ ) /∂ξ ≤ |ω+ (ξ ) – ω(ξ )|h ≤ . h+ h – h+ h Define T+ = T +
h
and h+ =
α+ , T+ (+K+ )τ +
then we have
max |∂ω+ /∂ξ | ≤ max ∂ ω+ – ω(ξ ) /∂ξ + max |∂ω/∂ξ | ≤ T+ .
ξ ∈h+
ξ ∈h+
ξ ∈h+
Thus all the parameters for H+ are defined, and so Iteration Lemma is proved.
2.2 Iteration Define inductive sequences ρj+ = σρj ,
sj+ = sj – ρ,
αj+ = αj – ( + Kj )τ + j , ηj+ = Ej+ ,
rj+ = ηj rj , Ej+ = cEj ,
Tj+ = Tj +
j , dj
ν j+ = αj¯ν ρj+ Ej+ ηj+ ,
and e–Kj+ ρj+ = Ej+ , ηj+
αj+ . ( + K)τj + Tj+
hj+ =
Define j+ = ξ ∈ j : ωj (ξ ) + πl, k ≥
αj , Kj < |k| ≤ Kj+ (|k| + )τ
and
hj+ = ξ ∈ C n : dist(ξ , j+ ) ≤ hj+ . In the following we give some estimates for Gevrey-smoothness. K Let γj = Kj ρj = – ln Ej . We have Kj+j = lnlnEcj + σ , and hence ≤
α T (+K )τ enough. If < Kj < Kj+ , we have = αj+j Tj+j (+Kj+j )τ ≤ , and ∂ω means h+ ≤ h holds. Suppose maxξ ∈hj | ∂ξj | ≤ Tj . Let Tj+ = hj+ hj
|
∂ωj+ | ∂ξ
=|
∂(ωj+ –ωj +ωj ) | ∂ξ δ
Kj+ Kj
ρ
for E small
hence hj+ ≤ hj , which Tj +
j . dj
Then we have δ
δ
τ + ≤ Tj+ . By the choice of σ , we can easily get that ρj+ γj+ ≥ ρj γj τ + . δ
Since ρ γτ + ≥ , we have ρj γj τ + ≥ for all j > . By the definitions of Tj , hj and j , we have Tj+ = Tj + ing γj = – ln Ej j τ –γi Ti i= (γi ) e
≤
( )j
j dj
= T +
j
i= (γi )
τ –γi
e
Ti . Not-
and Ej ≤ (cE ) , we can choose E to be sufficiently small such that ≤ , then we have T ≤ Tj ≤ T + . Similarly, we have αj ≤ αj+ ≤ αj .
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By Iteration Lemma, there exists a sequence of symplectic mappings {j (·; ξ )}, generated by p, q+ + Fj (p, q+ ), satisfying j – id rj ;D(sj –ρj ,rj )×hj ≤
cj , αjν¯ ρjν
Dj – Id rj ;D(sj –ρj ,rj )×hj ≤
cj . ν¯ ν+ αj ρj
Define j = ◦ ◦· · ·◦j . Then we have a sequence of symplectic mappings {j+ (·; ξ ) = ( j )– ◦ j ◦ j }, generated by Hj+ (·; ξ ) = Nj+ + Pj+ , where Nj+ = x + ωj+ (ξ ), yˆ + Aj+ u, vˆ + Bj+ u, u + Cj+ vˆ , vˆ with |ωj+ – ωj | ≤ cj ,
∀j ≥ ,
and XPj+ rj+ ;D(sj+ ,rj+ )×hj+ ≤ j+ .
2.3 The convergence of the KAM iteration Now we prove the convergence of the KAM iteration. Similar to [], we have j ν¯ ν – j– ≤ cαj– ρj– Ej– , r ;D(s –ρ ,r )× j
j
j j
hj
and j ν¯ ν+ D – j– ≤ αj– ρj– Ej– . r ;D(s –ρ ,r )× j
j
j j
hj
By the Cauchy estimate, we have ν¯ ν cαj– ρj– Ej– β! β j ∂ – j– ≤ , ξ |β!| rj ;D(sj –ρj ,rj )×hj hj ν+ ν¯ cαj– ρj– Ej– β! β j ∂ D – j– ≤ ξ |β!| rj ;D(sj –ρj ,rj )×hj hj
and β j ∂ ω – ωj– ≤ cj– β! . ξ |β!| j hj β
Let Uj =
ν¯ ρ ν E β! cαj– j– j– |β!|
hj δ
β
and Gj =
cj– β! |β!|
hj
β
β
. Now we estimate Uj and Gj for β ∈ Zn+ .
Since ρj γj τ + ≥ for all j > , we have
ρj
δ
≤ γj τ + . Then we have Kj =
means that Kjτ + ≤ γjτ ++δ . Noting that hj =
αj (K+)τj + Tj
γj ρj
+ τ δ+
≤ γj
, which
, Tj < T + , α ≤ αj and Ej– = Ej =
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γj
e– , we have β Uj
≤ cα ν¯ ρjν β! ≤ cρjν
T + α
(T + ) α
|β|
|β|
γjτ ++δ
|β|
β – γj β! γj e
e–
γj
(τ +δ)(n+)
β
j τ +δ – γj – (τ +δ)(n+) e n+
· · · γj n e
γ
≤ cρjν J |β| β!μ Jj(n+) , where J = T+ [ (μ–)(n+) ]μ– , μ = τ + δ, and c only depends on n, α, μ. α In the same way, we have
Gj ≤ cρjν J |β| β!μ Ej(n+) . β
Note that sj → s , rj → , hj → as j → ∞. Let D∗ = D( s , ), ∗ = limj→∞ j . Then we have β ∂ (∗ – id) r ≤ cρν J |β| β!μ E(n+) , ξ ;D(∗)×∗
j j≥
and ∗ =
∀β ∈ Zn+ .
β
Since j is affine in y, j is also affine in y, and hence we have the convergence of ∂ξ j to β ∂ξ ∗ on D( s , r ) and β ν |β| μ (n+) ∂ (∗ – id) r s r ≤ cρ J β! E , ξ ;D( , )×∗
∀β ∈ Zn+ . Thus we proved (.). Let ω∗ = limj→∞ ωj . Similarly, it follows that β ∂ ω∗ (ξ ) – ω (ξ ) ≤ cρ ν J |β| β!μ E (n+) , ξ ∗
∀β ∈ Zn+ .
Moreover we have ω∗ (ξ ), k ≥
α∗ ( + |k|)τ
for all ξ ∈ ∗ and = k ∈ Zn , where α∗ = limj→∞ α j , with (.) and (.).
α
≤ α∗ ≤ α. Thus we proved
2.4 Estimate of measure We note that β ≥ in Assumption for symplectic mappings , while β ≥ in Hamiltonian systems [, ]. So the non-degeneracy condition in symplectic mappings and that in Hamiltonian systems are different. It means that the estimate of measure is different in two cases. But the proof for symplectic mappings is similar to [, ], so we omit the details. Competing interests The author declares that they have no competing interests.
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Author’s contributions The article is a work of the author who contributed to the final version of the paper. The author read and approved the final manuscript. Acknowledgements The paper was completed during the author’s visit to the Department of Mathematics of Pennsylvania State University, which was supported by Nanjing Tech University. The author thanks Professor Mark Levi for his invitation, hospitality and valuable discussions. The work is supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (14KJB110009). The work is partly supported by the Natural Science Foundation of Jiangsu Province (BK20140927) and (BK20150934). The work is also in part supported by the Natural Science Foundation of China (11301263). Received: 25 September 2016 Accepted: 26 January 2017 References 1. Poincaré, H, Magini, R: Les méthodes nouvelles de la mécanique céleste. Il Nuovo Cimento (1895-1900) 10, 128-130 (1899) 2. Poincaré, H: Sur un théoreme de géométrie. Rendiconti del Circolo Matematico di Palermo (1884-1940) 33, 375-407 (1912) 3. Moser, J: On Invariant Curves of Area-Preserving Mappings of an Annulus. Vandenhoeck & Ruprecht, Gottingen (1962) 4. Aubry, S, Le Daeron, PY: The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states. Physica D 8, 381-422 (1983) 5. Aubry, S, Abramovici, G: Chaotic trajectories in the standard map. The concept of anti-integrability. Physica D 43, 199-219 (1990) 6. Mather, J: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457-467 (1982) 7. Mather, J: Non-existence of invariant circles. Ergod. Theory Dyn. Syst. 4, 301-309 (1984) 8. Mather, J: More Denjoy minimal sets for area preserving diffeomorphisms. Comment. Math. Helv. 60, 508-557 (1985) 9. Conley, C, Zehnder, E: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math. 73, 33-50 (1983) 10. Conley, C, Zehnder, E: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37, 207-253 (1984) 11. Cheng, C-Q, Sun, Y-S: Existence of invariant tori in three-dimensional measure-preserving mappings. Celest. Mech. Dyn. Astron. 47, 275-292 (1989) 12. de la Llave, R, James, JM: Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete Contin. Dyn. Syst. 32, 4321-4360 (2012) 13. Dullin, HR, Meiss, JD: Resonances and twist in volume-preserving mappings. SIAM J. Appl. Dyn. Syst. 11, 319-349 (2012) 14. Fox, AM, Meiss, JD: Greene’s residue criterion for the breakup of invariant tori of volume-preserving maps. Physica D 243, 45-63 (2013) 15. Gelfreich, V, Simó, C, Vieiro, A: Dynamics of 4D symplectic maps near a double resonance. Physica D 243, 92-110 (2013) 16. Rüssmann, H: On the existence of invariant curves of twist mappings of an annulus. In: Geometric Dynamics, pp. 677-718. Springer, Berlin (1983) 17. Xia, Z-H: Existence of invariant tori in volume-preserving diffeomorphisms. Ergod. Theory Dyn. Syst. 12, 621-631 (1992) 18. Bi, Q-Y, Xun, J-X: Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings. Qual. Theory Dyn. Syst. 13, 269-288 (2014) 19. Lu, X-Z, Li, J, Xu, J-X: A KAM theorem for a class of nearly integrable symplectic mappings. J. Dyn. Differ. Equ. 27, 1-24 (2015). 20. Shang, Z-J: A note on the KAM theorem for symplectic mappings. J. Dyn. Differ. Equ. 12, 357-383 (2000) 21. Zhu, W, Liu, B, Liu, Z: The hyperbolic invariant tori of symplectic mappings. Nonlinear Anal. 68, 109-126 (2008) 22. Pöschel, J: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35, 653-696 (1982) 23. Popov, G: Invariant tori, effective stability, and quasimodes with exponentially small error terms I - Birkhoff normal forms. Ann. Henri Poincaré 1, 223-248 (2000) 24. Xu, J-X, You, J-G: Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann’s non-degeneracy condition. J. Differ. Equ. 235, 609-622 (2007) 25. Zhang, D-F, Xu, J-X: On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition. Discrete Contin. Dyn. Syst., Ser. A 16, 635-655 (2006) 26. Zhang, D-F, Xu, J-X: Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmanns non-degeneracy condition. J. Math. Anal. Appl. 322, 293-312 (2006) 27. Pöschel, J: On elliptic lower dimensional tori in Hamiltonian systems. Math. Z. 202, 559-608 (1989)
