ARNOLD DIFFUSION IN ARBITRARY DEGREES OF FREEDOM
AND 3-DIMENSIONAL NORMALLY HYPERBOLIC INVARIANT
CYLINDERS
P. BERNARD∗ , V. KALOSHIN# , K. ZHANG∗∗
1. Introduction
Let (𝜃, 𝑝) ∈ 𝕋𝑛 × 𝑈 be the phase space of an integrable Hamiltonian system 𝐻0 (𝑝)
with 𝕋𝑛 being the 𝑛-dimensional torus 𝕋𝑛 = ℝ𝑛 /ℤ𝑛 ∋ 𝜃 = (𝜃1 , ⋅ ⋅ ⋅ , 𝜃𝑛 ) and 𝑈 being
an open set in ℝ𝑛 , 𝑝 = (𝑝1 , ⋅ ⋅ ⋅ , 𝑝𝑛 ) ∈ 𝐵 𝑛 . Assume that 𝐻0 is strictly convex, i.e.
Hessian ∂𝑝2𝑖 𝑝𝑗 𝐻0 is strictly positive definite.
Consider a smooth time periodic perturbation
𝐻𝜀 (𝜃, 𝑝, 𝑡) = 𝐻0 (𝑝) + 𝜀𝐻1 (𝜃, 𝑝, 𝑡),
𝑡 ∈ 𝕋 = ℝ/𝕋.
We study Arnold diffusion for this system, namely, existence of orbits {(𝜃, 𝑝)(𝑡)}𝑡 such
that
∣𝑝(𝑡) − 𝑝(0)∣ > 𝑂(1) independently of 𝜀.
We say that 𝐻0 has a resonance of order 𝑚 < 𝑛 at a point 𝑝 ∈ 𝐵 𝑛 if there are 𝑚
linearly independent integer vectors 𝑘1 , . . . , 𝑘𝑚 ∈ ℤ𝑛 such that 𝑘𝑗 ⋅ ∇𝐻0 (𝑝) = 0 for
𝑗 = 1, ⋅ ⋅ ⋅ , 𝑚. We say that a resonance is of co-dimension 𝑠 if it is of order 𝑛 − 𝑠. Due
to the theorem on implicit function and convexity of 𝐻0 a resonance of codimension
𝑠 (if non empty) locally defines a surface of dimension 𝑠. We would like to study
dynamics near a resonance of codimension one, i.e. near a segment in 𝐵 𝑛 . For any
resonance of codimension one there is an integer linear symplectic transformation
which brings integer vectors 𝑘1 , . . . , 𝑘𝑛−1 ∈ ℤ𝑛 , defining the resonance, to the form
𝑘𝑗 = (0, ⋅ ⋅ ⋅ , 1𝑗 , 0, ⋅ ⋅ ⋅ , 0). Since we are interested in a local property assume that a
resonance, denoted Γ, of codimension one is of the following form:
(∂𝑝1 𝐻0 (𝑝), ⋅ ⋅ ⋅ , ∂𝑝𝑛−1 𝐻0 (𝑝)) = (𝜃˙1 , ⋅ ⋅ ⋅ , 𝜃˙𝑛−1 ) = 0 for 𝜀 = 0.
∑
In the case 𝐻0 (𝑝) = 21 𝑛𝑗=1 𝑝2𝑗 we have Γ = {(𝑝1 , ⋅ ⋅ ⋅ , 𝑝𝑛−1 ) = 0}. Thus, it is naturally
parametrized by 𝑝𝑛 .
Consider the space of 𝐶 𝑟 perturbations 𝐶 𝑟 (𝕋𝑛 × 𝑈 × 𝕋, ℝ) with a natural 𝐶 𝑟 norm
given maximum of all partial derivatives of order up to 𝑟. Denote by 𝑆 𝑟 the unit
sphere in this space.
1
2
P. Bernard, V. Kaloshin, K. Zhang
Theorem 1.1. For 𝑟 ≥ 4, there is an open and dense set 𝒰 ⊂ 𝑆 𝑟 , a nonnegative
function 𝑙 : 𝑆 𝑟 −→ ℝ+ with 𝑙∣𝒰 > 0 and a positive function 𝜀0 = 𝜀0 (𝐻1 ), we write
𝒱 = {𝜖𝐻1 : 𝐻1 ∈ 𝒰}. We have that, for an open and dense set of 𝜖𝐻1 ∈ 𝒱 the
Hamiltonian system 𝐻𝜖 = 𝐻0 + 𝜖𝐻1 has an orbit {(𝜃, 𝑝)(𝑡)}𝑡 whose action component
∣𝑝(𝑇 ) − 𝑝(0)∣ > 𝑙(𝐻1 ).
Moreover, for all 0 < 𝑡 < 𝑇 the action component 𝑝(𝑡) stays close to the codimension
one resonance Γ.
Remark 1.1. This Theorem provides a form of Arnold diffusion for generic Hamiltonian systems. The type of generic condition in Theorem 1.1 is a version of Mather’s
cusp residue condition introduced in [Ma3].
The present work is in large part inspired by the work of Mather [Ma3, Ma4, Ma5].
In [Ma3], Mather announced a much stronger version of Arnold diffusion for 𝑛 = 2
(the system is time-periodic hence the degree of freedom is 2 12 ). The proof of Mather’s
result is partially written and available [Ma4], and he has given lectures about the
proof [Ma5]. One of the ideas underlying his proof is to construct diffusion along a
segment of a resonance and away from other low order resonances. Conceptually, the
proof of our result is similar to the part of Mather’s proof [Ma4] for single resonances.
The novelty of our approach is the use of normal form theory and normally hyperbolic
cylinders in an a priori stable setting. Application of normal forms to construct
normally 3-dimensional hyperbolic invariant cylinders in apriori stable situation in
3 degrees of freedom is proposed in [KZZ]. Independently in the case of arbitrary
degrees of freedom it is proposed in [Be3]. In the latter it is shown that such cylinders
have length independent of 𝜖.
The proof of this Theorem proceeds in three steps.
Step 1. Build a normal form for 𝐻𝜀 for for 𝑝 near Γ. In section 3 we prove the
existence of a normal form, which takes a particular nice form along subsegments of
Γ, which we will call passage segments, defined in the next section. The length and
choice of the passage segments depends on 𝐻0 and 𝐻1 only.
Step 2. For 𝐻1 ∈ 𝒰, we establish existence of finitely many 3-dimensional normally
hyperbolic cylinder along Γ. This is discussed in Section 4.
Step 3. For a generic perturbation, we show that there exists diffusion orbit along a
passage segment, using the normally hyperbolic cylinders. This steps uses variational
methods of Bernard [Be1] and of Cheng-Yan [CY1, CY2] which are based on ideas of
Mather (see [Ma4]). These constructions are discussed in Section 5 and Section 6.
Arnold diffusion along normally hyperbolic invariant cylinders
3
2. Notations and terminology
We denote 𝜃𝑠 = (𝜃1 , ⋅ ⋅ ⋅ , 𝜃𝑛−1 ), 𝑝𝑠 = (𝑝1 , ⋅ ⋅ ⋅ , 𝑝𝑛−1 ) and 𝜃𝑓 = 𝜃𝑛 , 𝑝𝑓 = 𝑝𝑛 . These are
the slow-fast variables associated to the resonance Γ = {∂𝑝𝑠 𝐻0 (𝑝) = 0}. It is natural
to use 𝑝𝑓 as a parameter for Γ, i.e. we may write Γ ∩ 𝐵 = {𝑝∗ (𝑝𝑓 ) = (𝑝𝑠∗ (𝑝𝑓 ), 𝑝𝑓 ), 𝑝𝑓 ∈
[𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ]}.
The averaged Hamiltonian associated to the resonance Γ is given by
∫∫
𝑠
𝑍(𝜃 , 𝑝) =
𝐻1𝑠 (𝜃𝑠 , 𝑝𝑠 , 𝜃𝑓 , 𝑝𝑓 , 𝑡) 𝑑𝜃𝑓 𝑑𝑡.
We would like to impose the following set of non-degeneracies and notations. Consider
the function 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) as a family of functions on 𝕋𝑛−1 parametrized by 𝑝𝑓 .
Call a value 𝑝𝑓 on Γ regular if 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) has a unique global maximum on
𝑠
𝕋 ∋ 𝜃𝑠 at some 𝜃∗𝑠 = 𝜃𝑠 (𝑝𝑓 ). We say the maximum is non-degenerate if the Hessian
of 𝑍 with respect to 𝜃𝑠 is strictly negative definite.
Call a value 𝑝𝑓 on Γ bifurcation if 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) has exactly two global maxima on
𝑠
𝕋 ∋ 𝜃𝑠 at some 𝜃1𝑠 = 𝜃1𝑠 (𝑝𝑓 ) and 𝜃2𝑠 = 𝜃2𝑠 (𝑝𝑓 ).
Call a regular 𝑝𝑓 on Γ non-degenerate if the unique maximum is non-degenerate.
If 𝑝𝑓 is a bifurcation, it is called non-degenerate if both maxima are non-degenerate,
furthermore, the values at these maxima moves with different speed with respect to
the parameter 𝑝𝑓 Otherwise, it is called degenerate.
The generic condition that defines 𝒰 ⊂ 𝑆𝑟 is a higher dimensional version of the
conditions (C1)-(C3) given by Mather [Ma3]. These conditions may be described as
follows: Each value 𝑝𝑓 ∈ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] is a non-degenerate regular or bifurcation point.
Note that the non-degeneracy condition implies that there are at most finitely many
bifurcation points. Let 𝑎1 < ⋅ ⋅ ⋅ < 𝑎𝑠−1 be the set of bifurcation points in the interval
(𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ), and consider the partition of the interval [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] by {[𝑎𝑗 , 𝑎𝑗+1 ]}𝑠−1
𝑗=0 .
Here we give an explicit quantitative version of the above condition: There exists
𝜆 > 0 such that
[G0] There are smooth functions 𝜃𝑗𝑠 (𝑝𝑓 ) : [𝑎𝑗 − 𝜆, 𝑎𝑗+1 + 𝜆] −→ 𝕋𝑛 , 𝑗 = 0, ⋅ ⋅ ⋅ , 𝑠 − 1,
such that for each 𝑝𝑓 ∈ [𝑎𝑗 − 𝜆, 𝑎𝑗+1 + 𝜆], 𝜃𝑗𝑠 (𝑝𝑓 ) is a local maximum of
𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) satisfying −∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃𝑗𝑠 , 𝑝) ≥ 𝜆𝐼.
[G1] For 𝑝𝑓 ∈ (𝑎𝑗 , 𝑎𝑗+1 ), 𝜃𝑗𝑠 is the unique maximum for 𝑍. for 𝑝𝑓 = 𝑎𝑗+1 , 𝜃𝑗𝑠 and
𝑠
𝜃𝑗+1
are the only maxima.
𝑓
[G2] At 𝑝 = 𝑎𝑗+1 the maximum value of 𝑍 has different derivatives with respect
to 𝑝𝑓 , i.e.
𝑑
𝑑
𝑠
𝑓
𝑠
𝑍(𝜃
(𝑎
),
𝑝
(𝑝
))
=
∕
𝑍(𝜃𝑗+1
(𝑎𝑗+1 ), 𝑝∗ (𝑝𝑓 )).
𝑗+1
∗
𝑗
𝑑𝑝𝑓
𝑑𝑝𝑓
4
P. Bernard, V. Kaloshin, K. Zhang
Theorem 2.1. The set 𝒰 of functions 𝐻1 ∈ 𝑆𝑟 such that the corresponding 𝑍(𝜃𝑠 , 𝑝)
satisfies conditions [G0]-[G2] is open and dense.
The proof of Theorem 2.1 will be given in Appendix A.
Write 𝜔(𝑝) = ∂𝑝 𝐻0 (𝑝) = (∂𝑝𝑠 𝐻0 , ∂𝑝𝑓 𝐻0 ), clearly for any 𝑝 ∈ Γ we have that 𝜔(𝑝) =
(0, ∂𝑝𝑓 𝐻0 )). We say that 𝑝𝑓 has an additional resonance if there exists integers 𝑘𝑛 , 𝑙
such that 𝑘𝑛 ∂𝑝𝑓 𝐻0 (𝑝) + 𝑙 = 0. Given a large integer 𝐾, let
(1)
Σ𝐾 = {𝑝 ∈ Γ ∩ 𝐵;
∃𝑘𝑛 , 𝑙 ∈ ℤ, ∣𝑘𝑛 ∣, ∣𝑙∣ ≤ 𝐾, 𝑘𝑛 ⋅ ∂𝑝𝑓 𝐻0 (𝑝) + 𝑙 = 0}.
Given 𝐻1 ∈ 𝒰, we will define a small 𝛿 = 𝛿(𝐻1 , 𝑛, 𝑟) > 0 and integer 𝐾 = 𝐾(𝛿, 𝑛, 𝑟)
and call the elements of Σ𝐾 punctures. We need to exclude a neighborhood of
1
the punctures from Γ ∩ 𝐵. Let 𝑈3𝜖 16 (Σ𝐾 ) stand for 3𝜖 6 neighborhood of Σ𝐾 , then
Γ ∩ 𝐵 ∖ 𝑈3𝜖 16 (Σ𝐾 ) is a collection of disjoint segments. Each of these segments is
called a passage segment. On a neighborhood of each passage segment there exists a
convenient normal form for the Hamiltonian 𝐻𝜖 .
3. Normal forms
Let Γ0 = {(𝑝𝑠 = 𝑝∗ (𝑝𝑓 )) : 𝑝𝑓 ∈ [𝑎− , 𝑎+ ]} be one of the passage segments. The goal
of this segment is to prove the following theorem:
Theorem 3.1. [Normal Form] For 𝐻𝜖 = 𝐻0 + 𝜖𝐻1 , 𝐻1 ∈ 𝒰 with 𝑟 ≥ 4. Given any
small 𝛿 > 0, there exists 𝑐(𝑛, 𝑟) > 0, 𝜖0 = 𝜖0 (𝛿, 𝑛, 𝑟), such that for
2
𝐾(𝛿, 𝑛, 𝑟) = 𝑐(𝑐(𝑛, 𝑟)−1 𝛿)− 𝑟−3
and define Σ𝐾 and Γ0 accordingly, we have that for 0 < 𝜖 < 𝜖0 there exists a 𝐶 ∞
change of coordinates Φ defined on 𝑈𝜖 16 (Γ0 ) with
(2)
𝑁𝜖 = 𝐻𝜖 ∘ Φ = 𝐻0 (𝑝) + 𝜖𝑍(𝜃𝑠 , 𝑝) + 𝜖𝑅(𝜃, 𝑝, 𝑡)
such that ∥𝑅∥𝐶 2 ≤ 𝑐𝛿 for a constant 𝑐 independent of 𝜆, 𝑛 and 𝑟. Moreover, ∥Φ −
1
1
𝑖𝑑∥𝐶 0 ≤ 𝐷′ 𝜖 2 and ∥Φ − 𝑖𝑑∥𝐶 2 ≤ 𝐷′ 𝜖 6 for some constant 𝐷′ .
Remark 3.1. [Length of passage segment] On the interval, the distance between 2
adjacent rationals with denominator at most 𝐾 is 𝐾12 . It follows that the distance
4
between 𝑝𝑓1 , 𝑝𝑓2 ∈ Σ𝐾 (see (3)) is at least ∥∂ 2 𝐻0−1 ∥ 𝐾12 ≥ 𝑐(𝑐(𝑛, 𝑟)−1 𝛿) 𝑟−3 , assuming
that ∥∂ 2 𝐻0−1 ∥ is bounded by some universal constant.
To prove Theorem 3.1 we proceed in 3 steps. We first show that in general we
may do averaging near a resonance to provide a normal form 𝑁𝜖 . We then show that
this normal form takes our desired form on the set 𝑈𝜖 16 (Γ0 ). However, the averaging
procedure lowers smoothness, in particular, the technique requires the smoothness
Arnold diffusion along normally hyperbolic invariant cylinders
5
𝑟 ≥ 𝑛 + 5. To obtain a result that does not require this relation between 𝑟 and 𝑛, we
use a smooth approximation trick that goes back to Moser.
3.1. Normal form near a resonance. We will prove a general result on the normal
form of an autonomous Hamiltonian system. The time periodic version will come as
a corollary.
Consider the Hamiltonian 𝐻𝜖 (𝜑, 𝐽) = 𝐻0 (𝐽) + 𝜖𝐻1 (𝜑, 𝐽), where (𝜑, 𝐽) ∈ 𝕋𝑚 × ℝ𝑚 .
Let 𝐵 = {∣𝐽∣ ≤ 1} be the unit ball in ℝ𝑚 . Given any integer vector 𝑘 ∈ ℤ𝑚 ∖ {0},
let ∣𝑘∣ = max{𝑘𝑖 }. To avoid zero denominators in some calculations, we make the
unusual convention that ∣(0, ⋅ ⋅ ⋅ , 0)∣ = 1.
Let 𝜌 : ℝ −→ ℝ be a 𝐶 ∞ bump function, i.e.
{
1, ∣𝑥∣ ≤ 1
𝜌(𝑥) =
0, ∣𝑥∣ ≥ 2
and 0 ≤ 𝜌(𝑥) ≤ 1 in between. Choose some 0 < 𝛽 < 1, for each 𝑘 ∈ ℤ𝑚 we define the
𝐽 𝐻0
function 𝜌𝑘 (𝐽) = 𝜌( 𝑘⋅∂
).
𝜖𝛽 ∣𝑘∣
We have the following
Theorem 3.2. Let 𝛿 > 0 be a small parameter. For 𝐶 𝑟 Hamiltonian 𝐻𝜖 = 𝐻0 + 𝜖𝐻1
there exists constants 𝑐, 𝜅𝑚 and 𝑐𝑚 such that if the parameters 𝛽, 𝐾, 𝑟 and 𝜖 satisfy
(1) 𝑟 ≥ 𝑚 + 4.
(2) 𝑐𝜅𝑚 𝐾 −𝑟+𝑚+3 ∥𝐻1 ∥𝐶 𝑟 ≤ 21 𝛿.
(3) 𝑐𝑚 𝜖1−4𝛽 ∥𝐻0 ∥4𝐶 3∑
∥𝐻1 ∥2𝐶 𝑟 ≤ 12 𝛿.
(Here 𝜅𝑚 = 𝑘∈ℤ𝑚 ∣𝑘∣−𝑚−1 , 𝑐 is independent constant and 𝑐𝑚 depends only
on 𝑚).
Then there exists a change of coordinates Φ defined on 𝕋𝑚 × 𝐵, such that
𝐻𝜖′ = 𝐻𝜖 ∘ Φ = 𝐻0 (𝐽) + 𝜖𝑅1 + 𝜖𝑅2 ,
here we abuse notation by still using (𝜑, 𝐽) for the new coordinates. We have that
∑
∙ 𝑅1 = 𝑘∈ℤ𝑚 ,∣𝑘∣≤𝐾 𝜌𝑘 (𝐽)ℎ𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) , here ℎ𝑘 (𝐽) is the 𝑘 𝑡ℎ coefficient for the
Fourier expansion of 𝐻1 .
∙ ∥𝑅2 ∥𝐶 𝑟 ≤ 𝛿. We also have that ∥Φ−𝑖𝑑∥𝐶 2 ≤ 𝐷′ 𝜎, where 𝐷′ is an independent
constant and 𝜎 = 𝑐𝑚 𝜖1−4𝛽 ∥𝐻0 ∥4𝐶 4 ∥𝐻1 ∥𝐶 𝑟 .
To apply it to the time periodic system, we consider the equivalent autonomous
system 𝐻𝜖 (𝜃, 𝑝, 𝑠, 𝐸) = 𝐻0 (𝑝) + 𝐸 + 𝜖𝐻1 (𝜃, 𝑝, 𝑠), here 𝑠 is time and 𝐸 is the conjugate
action variable. Here (𝜃, 𝑝, 𝑠, 𝐸) ∈ 𝕋𝑛 × ℝ𝑛 × 𝕋 × ℝ, hence the previous theorem
applies with 𝑚 = 𝑛 + 1. We have
6
P. Bernard, V. Kaloshin, K. Zhang
Theorem 3.3. Assume that 𝛿, 𝛽, 𝐾, 𝑟 and 𝜖 satisfy the conditions of the previous
theorem, then there exists a change of coordinates Φ under which
𝐻𝜖′ = 𝐻𝜖 ∘ Φ = 𝐻0 (𝑝) + 𝐸 + 𝜖𝑅1 + 𝜖𝑅2
where
∙
𝑅1 = 𝑅1 (𝑝, 𝜃, 𝑠) =
∑
𝜌𝑘,𝑙 (𝑝)ℎ𝑘,𝑙 (𝑝)𝑒2𝜋𝑖(𝑘⋅𝜃+𝑙𝑠) ,
(𝑘,𝑙)∈ℤ𝑛 ×ℤ,∣(𝑘,𝑙)∣≤𝐾
here ℎ𝑘,𝑙 (𝑝) are the coefficients of the Fourier expansion of 𝐻1 and 𝜌𝑘,𝑙 (𝑝) =
𝐽 𝐻0 +𝑙
).
𝜌( 𝑘⋅∂
𝜖𝛽 ∣(𝑘,𝑙)∣
∙ ∥𝑅2 ∥𝐶 2 ≤ 𝛿.
∙ Φ is identity in the 𝑠 (time) component and ∥Φ − 𝐼𝑑∥𝐶 2 ≤ 𝐷′ 𝜎.
We will then try to prove Theorem 3.2. To avoid cumbersome notations, we will
use two generic constants 𝑐 and 𝑐𝑚 whose meaning will vary depending on context.
The constant 𝑐 is independent of all parameters, while the constant 𝑐𝑚 depends only
on the dimension 𝑚. These constants will be fixed at the end of the proof and they
are the constants in the statement of the Theorem.
We have the following basic estimates about the Fourier series of a function 𝑔(𝜑, 𝐽).
Lemma 3.1. For 𝑔(𝜑, 𝐽) ∈ 𝐶 𝑟 (𝕋𝑚 × 𝐵), we have
(1) Given any multi-index 𝛼 = (𝛼1 , ⋅ ⋅ ⋅ , 𝛼𝑚 ), let ∣𝛼∣ = 𝛼1 + ⋅ ⋅ ⋅ 𝛼𝑚 . In what
follows whenever we use the letter 𝛼 it stands for a multi-index. We have
that ∥∂𝐽 𝛼 𝑔𝑘 (𝐽)∥𝐶 0 ≤ ∣𝑘∣−𝑟+∣𝛼∣ ∥𝑔∥𝐶 𝑟 . Moreover, let 𝑙 ≤ 3 be an integer, then
∥𝑔𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) ∥𝐶 𝑙 ≤ 𝑐∣𝑘∣−𝑟+𝑙 ∥𝑔∥𝐶 𝑟 .
Let’s also fix the notation that 𝑙 will always stand for a positive integer less
or equal to 3.
(2) Let ℎ𝑘 (𝐽) be a series of functions
such that ∥∂𝐽 𝛼 ℎ𝑘 (𝐽)∥𝐶 0 ≤ ∣𝑘∣−𝑟+∣𝛼∣ 𝑀 for
∑
some 𝑀 > 0. We have ∥ 𝑘∈ℤ𝑚 ℎ𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) ∥𝐶 𝑙 ≤ 𝑐𝜅𝑚 𝑀 assuming that
𝑟 ≥ 𝑚 + 𝑙 +∑
1.
+
−𝑟+𝑚+3
(3) Let Π𝐾 𝑔 = ∣𝑘∣>𝐾 𝑔𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) . Then ∥Π+
∥𝑔∥𝐶 𝑟 .
𝑘 𝑔∥𝐶 2 ≤ 𝑐𝜅𝑚 𝐾
Proof. 1. Let 𝑘𝑗 = max{𝑘1 , ⋅ ⋅ ⋅ , 𝑘𝑚 } and let 𝑏 = 𝑟 − ∣𝛼∣. We have that
∥∂𝐽 𝛼 ℎ𝑘 ∥𝐶 0 ∣𝑘∣𝑏 = ∥∂𝐽 𝛼 ℎ𝑘 ∥𝐶 0 𝑘𝑗𝑏 ≤ ∥∂𝑝𝛼 𝜑𝑏𝑗 𝑔∥𝐿1 ≤ ∥∂𝑝𝛼 𝜑𝑏𝑗 𝑔∥𝐶 0 ≤ ∥𝑔∥𝐶 𝑟 .
For the second claim we have that
∑
∥𝑔𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) ∥𝐶 𝑙 ≤
(2𝜋)∣𝛽∣ ∥∂𝐽 𝛼 𝑔𝑘 (𝐽)∥𝐶 0 ∣𝑘∣∣𝛽∣ ≤ 𝑐∣𝑘∣−𝑟+𝑙 ∥𝑔∥𝐶 𝑟 .
∣𝛼+𝛽∣=𝑙
Arnold diffusion along normally hyperbolic invariant cylinders
2.
∥
∑
𝑘∈ℤ𝑚
∑
ℎ𝑘 (𝐽)𝑒2𝜋𝑖(𝑘⋅𝜑) ∥𝐶 𝑙 ≤
−𝑚−1
∑
𝑘∈ℤ𝑚
7
𝑐𝑙 ∣𝑘∣−𝑟+𝑙 𝑀 ≤ 𝑐𝑙 𝜅𝑚 𝑀,
recall that 𝜅𝑚 = 𝑘∈ℤ𝑚 ∣𝑘∣
.
3.
∑
∑
∥Π+
∣𝑘∣−𝑟+2 ∥𝑔∥𝐶 𝑟 ≤ 𝑐𝐾 −𝑟+𝑚+3
∣𝑘∣−𝑚−1 ∥𝑔∥𝐶 𝑟
𝐾 𝑔∥𝐶 2 ≤ 𝑐
∣𝑘∣>𝐾
∣𝑘∣>𝐾
≤ 𝑐𝐾
−𝑟+𝑚+3
𝜅𝑚 ∥𝑔∥𝐶 𝑟 = 𝑐𝜅𝑚 𝐾 −𝑟+𝑚+3 ∥𝑔∥𝐶 𝑟 .
□
Proof of Theorem 3.2. Let 𝐺(𝜑, 𝐽) be the function that solves the cohomological
equation
{𝐻0 , 𝐺} + 𝐻1 = 𝑅1 + 𝑅+ ,
+
where 𝑅+ = Π𝐾 𝐻1 . We have the following explicit formula for 𝐺:
∑ (1 − 𝜌𝑘 (𝐽))ℎ𝑘 (𝐽)
𝑒2𝜋𝑖(𝑘⋅𝜑) .
𝐺=
𝑘 ⋅ ∂ 𝐽 𝐻0
∣𝑘∣≤𝐾
𝐺 is well defined thanks to the smoothing terms 1 − 𝜌𝑘 we introduced, as whenever
𝑘 ⋅ ∂𝐽 𝐻0 = 0 we also have 1 − 𝜌𝑘 = 0 and that term is considered non-present.
Let Φ𝑡 be the Hamiltonian flow generated by 𝜖𝐺, and the change of coordinates
Φ = Φ1 . By the Lie method of making coordinate changes, we get that
∫ 1
′
2
𝐻𝜖 = 𝐻0 + 𝜖𝑅1 + 𝜖𝑅+ + 𝜖
{𝐹𝑡 , 𝐺} ∘ Φ𝑡 𝑑𝑡,
0
∫1
where 𝐹𝑡 = 𝑅1 + 𝑅+ + 𝑡(𝐻1 − 𝑅1 − 𝑅+ ). Write 𝑅2 = 𝑅+ + 𝜖 0 {𝐹𝑡 , 𝐺} ∘ Φ𝑡 𝑑𝑡.
It follows from Lemma 3.1 that ∥𝑅+ ∥𝐶 2 ≤ 𝑐𝜅𝑚 𝐾 −𝑟+𝑚+2 ∥𝐻1 ∥𝐶 𝑟 ≤ 12 𝛿, it remains
∫1
to treat 𝜖 0 {𝐹𝑡 , 𝐺} ∘ Φ𝑡 𝑑𝑡. To estimate the norm of 𝐹𝑡 , it is convenient to write
𝐹𝑡 = 𝐹𝑡′ + (1 − 𝑡)𝑅1 , where 𝐹𝑡′ = (1 − 𝑡)𝑅+ + 𝑡𝐻1 . Notice that the coefficients of
the Fourier expansion of 𝐹𝑡′ is simply a constant times that of 𝐻1 , Lemma 3.1 then
implies that
∥𝐹𝑡′ ∥𝐶 3 ≤ 𝑐𝜅𝑚 ∥𝐻1 ∥𝐶 𝑟
as long as 𝑟 ≥ 𝑚 + 4.
We still need to estimate the norm of 𝑅1 and 𝐺. These estimates require, additional
estimates of the smoothing terms 𝜌𝑘 as well as the small denominators 𝑘 ⋅ ∂𝐽 𝐻0 . We
have the following
- For any 𝑘 ∈ ℤ𝑚 and any 𝐽 such that 𝜌𝑘 (𝐽) ∕= 0, we have that ∣(𝑘 ⋅ ∂𝐽 𝐻0 )−1 ∣ ≤
2𝜖−𝛽 ∣𝑘∣−1 .
8
P. Bernard, V. Kaloshin, K. Zhang
∥(𝑘 ⋅ ∂𝐽 𝐻0 )−1 ∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽(𝑙+1) ∥𝐻0 ∥𝑙+1
𝐶4 .
- The derivative of 𝜌𝑘 (𝐽) can be estimated in the following way,
∥𝜌𝑘 (𝐽)∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽𝑙 ∥𝐻0 ∥𝑙𝐶 4 .
Here we are using the following estimates on the derivative of composition
of functions: For 𝑓 : ℝ𝑚 −→ ℝ and 𝑔 : ℝ𝑚 −→ ℝ𝑚 we have ∥𝑓 ∘ 𝑔∥𝐶 𝑙 ≤
𝑐𝑚,𝑙 ∥𝑓 ∥𝐶 𝑙 ∥𝑔∥𝑙𝐶 𝑙 .
- For each multi-index ∣𝛼∣ ≤ 3, we have that
(
)
∥∂𝐽 𝛼 (1 − 𝜌𝑘 (𝐽))ℎ𝑘 (𝐽)(𝑘 ⋅ ∂𝐽 𝐻0 )−1 ∥𝐶 0
∑
≤
∥1 − 𝜌𝑘 (𝐽)∥𝐶 ∣𝛼1 ∣ ∥ℎ𝑘 ∥𝐶 ∣𝛼2 ∣ ∥(𝑘 ⋅ ∂𝐽 𝐻0 )−1 ∥𝐶 ∣𝛼3 ∣
𝛼1 +𝛼2 +𝛼3 =𝛼
≤ 𝑐𝑚
∑
𝛼1 +𝛼2 +𝛼3 =𝛼
∣𝛼 ∣+1
𝜖−𝛽(∣𝛼1 ∣+1) ∥𝐻0 ∥𝐶 41
∣𝛼 ∣
⋅ ∣𝑘∣−𝑟+∣𝛼2 ∣ ∥𝐻1 ∥𝐶 𝑟 ⋅ 𝜖−𝛽∣𝛼3 ∣ ∥𝐻0 ∥𝐶 43
∣𝛼∣+1
≤ 𝑐𝑚 Δ∣𝛼∣+1 ∣𝑘∣−𝑟+∣𝛼∣ ∥𝐻0 ∥𝐶 4 ∥𝐻1 ∥𝐶 𝑟 .
∑
Since 𝐺(𝜑, 𝐽) = 𝑘∈ℤ𝑚 (1−𝜌𝑘 (𝐽))ℎ𝑘 (𝐽)(𝑘⋅∂𝐽 𝐻0 )−1 𝑒2𝜋𝑖(𝑘⋅𝜑) , apply Lemma 3.1,
part 2, we have that
𝑟
∥𝐺∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽(𝑙+1) ∥𝐻0 ∥𝑙+1
𝐶 4 ∥𝐻1 ∥𝐶 .
∑
- Finally since 𝑅1 = ∣𝑘∣≤𝐾 𝜌𝑘 (𝐽)𝑒2𝜋𝑘⋅𝜑 and ∥𝜌𝑘 ℎ𝑘 ∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽𝑙 ∣𝑘∣−𝑟+𝑙 ∥𝐻0 ∥𝑙𝐶 4 ∥𝐻1 ∥𝐶 𝑟 ,
we have that ∥𝑅1 ∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽𝑙 ∥𝐻0 ∥𝑙𝐶 4 ∥𝐻1 ∥𝐶 𝑟 , assuming that 𝑟 ≥ 𝑚 + 𝑙 + 1.
It follows that
∥𝐹𝑡 ∥𝐶 𝑙 ≤ ∥𝑅1 ∥𝐶 𝑙 + ∥𝐹𝑡′ ∥𝐶 𝑙 ≤ 𝑐𝑚 𝜖−𝛽𝑙 ∥𝐻0 ∥𝑙𝐶 4 ∥𝐻1 ∥𝐶 𝑟 .
Using the above estimates, we have
∑
∥{𝐹𝑡 , 𝐺}∥𝐶 2 ≤
∥𝐹𝑡 ∥𝐶 ∣𝛼1 ∣ ∥𝐺∥𝐶 ∣𝛼2 ∣ ≤ 𝑐𝑚 𝜖−4𝛽 ∥𝐻0 ∥4𝐶 4 ∥𝐻1 ∥2𝐶 𝑟 .
∣𝛼1 +𝛼2 ∣≤3, ∣𝛼1 ∣,∣𝛼2 ∣≥1
∫1
It remains to estimate 0 {𝐹, 𝐺} ∘ Φ𝑡 . The following is a paraphrased Lemma 3.15
from [DH]:
Assume that ∥𝜖𝐺∥𝐶 3 < 1 , then there exists constants 𝐷 and 𝐷′ such that
∥Φ𝑡 ∥𝐶 2 ≤ 𝐷,
∥Φ𝑡 − 𝑖𝑑∥𝐶 2 ≤ 𝐷′ 𝜖∥𝐺∥𝐶 3 ,
0 ≤ 𝑡 ≤ 1.
In particular, we get that ∥Φ − 𝑖𝑑∥𝐶 2 ≤ 𝐷′ 𝜎. Using the above estimate, we have that
1
𝜖∥{𝐹𝑡 , 𝐺} ∘ Φ𝑡 ∥𝐶 2 ≤ 𝑐𝑚 𝜖∥{𝐹𝑡 , 𝐺}∥𝐶 2 ∥Φ𝑡 ∥2𝐶 2 ≤ 𝑐𝑚 𝐷2 𝜖𝜖−4𝛽 ∥𝐻0 ∥4𝐶 4 ∥𝐻1 ∥2𝐶 𝑟 ≤ 𝛿,
2
Arnold diffusion along normally hyperbolic invariant cylinders
9
where 𝑐𝑚 𝐷2 in the last formula will be our final choice of 𝑐𝑚 in Condition 2. It follows
that ∥𝑅2 ∥𝐶 2 ≤ ∥𝑅+ ∥𝐶 2 + 𝜖∥{𝐹𝑡 , 𝐺} ∘ Φ𝑡 ∥𝐶 2 ≤ 𝛿.
□
3.2. Normal form away from punctures. We will choose 𝛽 = 16 in Theorem 3.3.
Given a small 𝛿 > 0, 𝐾 = 𝐾(𝛿, 𝑛, ∥𝐻1 ∥𝐶 𝑟 ) be such that Theorem 3.3 applies. Let
(3)
Σ𝐾 = {𝑝 ∈ Γ ∩ 𝐵;
∃𝑘𝑛 , 𝑙 ∈ ℤ, ∣𝑘𝑛 ∣, ∣𝑙∣ ≤ 𝐾, 𝑘𝑛 ⋅ ∂𝑝𝑓 𝐻0 (𝑝) + 𝑙 = 0}.
for sufficiently small 𝜖, the set Γ ∩ 𝐵 ∖ 𝑈3𝜖 61 (Σ𝐾 ), (where 𝑈𝑟 (⋅) stands for the 𝑟 neighborhood of a set,) is a collection of disjoint segments. Let Γ0 = {(𝑝𝑠 = 𝑝∗ (𝑝𝑓 )) :
𝑝𝑓 ∈ [𝑎− , 𝑎+ ]} be one of those segments. We have the following normal form in a
neighborhood of Γ0 .
Corollary 3.2. Assume that 𝑟 ≥ 𝑛+5, we may choose the parameters in Theorem 3.3
in the following way:
−24
−12
6
(1) 𝜖 ≤ 𝜖0 (𝛿, 𝑛, ∥𝐻1 ∥𝐶 𝑟 ) := min{𝑐−6
𝑚 ∥𝐻0 ∥𝐶 4 ∥𝐻1 ∥𝐶 𝑟 , (𝛿/2) }.
1
(2) 𝐾(𝛿, 𝑛, ∥𝐻1 ∥𝐶 𝑟 ) = (𝑐𝜅𝑛+1 ∥𝐻1 ∥𝐶 𝑟 𝛿 −1 )− 𝑟−𝑛−4 .
We then have that there exists change of coordinates Φ defined on 𝕋𝑛 × 𝐵 × 𝕋, such
that on the set 𝕋𝑛 × 𝑈𝜖 61 (Γ0 ) × 𝕋
where 𝑍(𝜃𝑠 , 𝑝) =
1
𝐷′ 𝜖 6 .
∑
𝑁𝜖 := 𝐻 ∘ Φ = 𝐻0 + 𝜖𝑍(𝜃𝑠 , 𝑝) + 𝜖𝑅(𝜃, 𝑝, 𝑡)
𝑘𝑛 =𝑙=0
ℎ𝑘,0 (𝑝)𝑒2𝜋𝑖(𝑘⋅𝜃) and ∥𝑅∥𝐶 2 ≤ 32 𝛿. Furthermore ∥Φ − 𝑖𝑑∥ ≤
Proof. It is clear that by our choice of parameters Theorem 3.3 applies with the
1
additional
estimate that 𝜎 ≤ 𝜖 6 . To see the special normal form, we note that 𝑅1 =
∑
2𝜋(𝑘⋅𝜃+𝑙𝑡)
. Write (𝑘, 𝑙) = (𝑘1 , ⋅ ⋅ ⋅ , 𝑘𝑛1 , 𝑘𝑛 , 𝑙), we note that if 𝑘𝑛 = 𝑙 = 0,
∣𝑘∣≤𝐾 𝜌𝑘,𝑙 (𝑝)𝑒
𝜌𝑘,𝑙 (𝑝) = 1 for all 𝑝 ∈ 𝑈𝜖 16 (Γ). On the other hand, if (𝑘, 𝑙) = (0, 𝑘𝑛 , 𝑙), we have that
1
𝜌𝑘,𝑙 (𝑝) = 0 for all 𝑑(𝑝, Σ𝐾 ) ≥ 3𝜖 6 . It follows that
∑
𝑁𝜖 = 𝐻0 (𝑝) + 𝜖
ℎ𝑘,0 (𝑝)𝑒2𝜋𝑖(𝑘⋅𝜃) + 𝜖𝑅2 = 𝐻0 (𝑝) + 𝜖𝑍 + 𝜖𝑅2 + Π𝐾 𝑍.
𝑘𝑛 =𝑙=0,∣𝑘∣≤𝐾
Since ∥Π𝐾 𝑍∥𝐶 2 ≤ 12 𝛿 from the proof of Theorem 3.2, the statement is proved.
□
3.3. Smooth approximation. Finally we remove the restriction on 𝑟 by the following smooth approximation lemma:
Lemma 3.3. Let 𝑓 : ℝ𝑛 −→ ℝ𝑛 be a 𝐶 𝑟 function. Then there for 𝜏 > 0 there exists
an analytic function 𝑆𝜏 𝑓 and 𝑐 = 𝑐(𝑟) such that
∥𝑆𝜏 𝑓 − 𝑓 ∥𝐶 3 < 𝑐(𝑛, 𝑟)∥𝑓 ∥𝐶 3 𝜏 𝑟−3 ,
10
P. Bernard, V. Kaloshin, K. Zhang
′
∥𝑆𝜏 ∥𝐶 𝑟′ < 𝑐(𝑛, 𝑟)∥𝑓 ∥𝐶 𝑟 𝜏 −(𝑟 −𝑟) ,
where 𝑟′ > 𝑟.
If 𝑟 < 𝑛 + 5, we will using Lemma 3.3 to approximate it by a smoother function.
1
Given 𝛿 > 0, let 𝜏 = (𝛿/𝑐(𝑟)) 𝑟−3 , we can find 𝐻1∗ such that ∥𝐻1∗ − 𝐻1 ∥𝐶 3 < 𝛿 and
𝑟−𝑟 ′
𝑟 ′ −3
𝑟−𝑟 ′
∥𝐻1∗ ∥𝐶 𝑟′ ≤ 𝑐(𝑟)(𝛿/𝑐(𝑟)) 𝑟−3 = 𝑐(𝑟) 𝑟−3 𝛿 𝑟−3 , using ∥𝐻1 ∥𝐶 𝑟 = 1. Applying Corollary 3.2
to 𝐻0 + 𝜖𝐻1∗ and smoothness 𝑟′ we get for suitable parameters there exists Φ such
that (𝐻0 + 𝜖𝐻1∗∫∫
) ∘ Φ = 𝐻0 + 𝜖𝑍 ∗ + 𝜖𝑅∗ with ∥𝑅∗ ∥𝐶 2 ≤ 32 𝛿. On one hand, we have
∥𝑍 ∗ −𝑍∥𝐶 2 = ∥ (𝐻1 −𝐻1∗ )𝑑𝜃𝑓 𝑑𝑡∥𝐶 2 ≤ 𝛿, on the other hand we have (𝐻0 +𝜖𝐻1 )∘Φ =
𝐻0 + 𝜖𝑍 ∗ + 𝜖𝑅1∗ + 𝜖(𝐻1 − 𝐻1∗ ) ∘ Φ. Since
∥(𝐻1∗ − 𝐻1 ) ∘ Φ∥𝐶 2 ≤ 𝑐∥𝐻1∗ − 𝐻1 ∥∥Φ∥2𝐶 2 ≤ 𝑐𝛿.
We conclude that 𝐻𝜖 ∘ Φ = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 with 𝑅 = (𝑍 − 𝑍 ∗ ) + (𝐻1∗ − 𝐻1 ) ∘ Φ satisfying
∥𝑅∥𝐶 2 ≤ 𝑐𝛿.
Some discussion on the choice of parameters are in order. To apply Corollary 3.2,
we will take
𝑟 ′ −3
𝑟−𝑟 ′
1
1
𝑟 ′ −3
𝐾(𝛿, 𝑛) = (𝑐𝜅𝑛+1 𝑐(𝑛, 𝑟) 𝑟−3 𝛿 𝑟−3 𝛿 −1 )− 𝑟′ −𝑛−4 = (𝑐𝜅𝑛+1 )− 𝑟′ −𝑛−4 (𝑐(𝑟)−1 𝛿) (𝑟−3)(𝑟′ −𝑛−4) .
It is convenient to simply take 𝑟′ = 2𝑛 + 5. It is easy to see that 𝜅𝑛+1 ≤ 3𝑛+1 , we
1
conclude that (𝑐𝜅𝑛+1 )− 𝑟′ −𝑛−4 is bounded by an uniform constant. Let’s still call it
′ −3
𝑐. On the other hand 𝑟′𝑟−𝑛−4
= 2, we have that for this choice of 𝑟′ we can choose
2
𝐾(𝛿, 𝑛) = 𝑐(𝑐(𝑟)−1 𝛿) 𝑟−3 . The following proposition implies Theorem 3.1.
Proposition 3.4. Assume that ∥𝐻1 ∥𝐶 𝑟 = 1, then given 𝛿 > 0, choose parameters in
the following way:
2
(1) 𝐾(𝛿, 𝑛, 𝑟) = 𝑐(𝑐(𝑛, 𝑟)−1 𝛿)− 𝑟−3 .
12(2𝑛+5−𝑟)
𝑟−3
(2) 𝜖0 (𝛿, 𝑛, 𝑟) = min{𝑐(𝑛, 𝑟)∥𝐻0 ∥−24
, 𝑐(𝛿/2)6 }, where 𝑐(𝑛, 𝑟) is some con𝐶4 𝛿
stant depending on 𝑛 and 𝑟.
We have the normal form on 𝑈𝜖 16 (Γ0 ) is
𝑁𝜖 = 𝐻0 + 𝜖𝑍(𝜃𝑠 , 𝑝) + 𝜖𝑅(𝜃, 𝑝, 𝑡)
with ∥𝑅∥𝐶 2 ≤ 𝑐𝛿 for some constant 𝑐.
4. Normally hyperbolic cylinders
Recall that by the generic condition [𝐺0], for each 𝑝𝑓 ∈ [𝑎𝑗 − 𝜆, 𝑎𝑗+1 + 𝜆] (regular
values between 2 bifurcation points) we have that 𝑍(𝜃𝑠 , 𝑝𝑠 , 𝑝𝑓 ) has a unique maximum
at 𝜃𝑗𝑠 and that −∂𝜃2𝑠 𝜃𝑠 𝑍 ≥ 𝜆𝐼 as a quadratic form. In this section we show that for a
Arnold diffusion along normally hyperbolic invariant cylinders
11
sufficiently small 𝛿 the system in normal form 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 admits a normally
hyperbolic cylinder for each interval [𝑎𝑗 − 𝜆/2, 𝑎𝑗+1 + 𝜆/2].
Theorem 4.1. For the Hamiltonian 𝑁𝜖 = 𝐻0 (𝑝) + 𝜖𝑍 + 𝜖𝑅 such that 𝑍 satisfy
conditions [G0]-[G2], there exists 𝛿0 = 𝛿0 (𝜆) and 𝜖0 = 𝜖0 (𝛿, 𝜆) such that if 0 < 𝛿 < 𝛿0
and 0 < 𝜖 < 𝜖0 , there exists 𝜌1 = 𝜌1 (𝜆) and a 𝐶 1 function
(Θ𝑠𝑗 , 𝑃𝑗𝑠 )(𝜃𝑓 , 𝑝𝑓 , 𝑡) : 𝕋 × 𝐽 × 𝕋 −→ {∥(𝜃𝑠 , 𝑝𝑠 ) − (𝜃𝑗𝑠 , 𝑝𝑠∗ )∥ ≤ 𝜌1 }
satisfying the following estimates
( 𝑠(
( ∂Θ𝑗 (
√
− 52
(
(
( ∂𝑝𝑓 ( = 𝑂(𝜆 𝛿/ 𝜖),
The set
𝑋𝑗 = {(𝜃𝑠 , 𝑝𝑠 ) = (Θ𝑠𝑗 , 𝑃𝑗𝑠 )(𝜃𝑓 , 𝑝𝑓 , 𝑡));
(
(
( ∂Θ𝑠𝑗 (
− 52
(
(
( ∂(𝜃𝑓 , 𝑡) ( = 𝑂(𝜆 𝛿).
𝑝𝑓 ∈ [𝑎𝑗 − 𝜆/2, 𝑎𝑗+1 + 𝜆/2], (𝜃𝑓 , 𝑡) ∈ 𝕋 × 𝕋}
is weakly invariant with respect to the Hamiltonian flow of 𝑁𝜖 in the sense that the
flow is tangent to 𝑋. Moreover, 𝑋𝑗 is maximally invariant on 𝑉𝑗 := {(𝜃, 𝑝, 𝑡); 𝑝𝑓 ∈
[𝑎𝑗 − 𝜆/2, 𝑎𝑗+1 + 𝜆/2], ∥(𝜃𝑠 , 𝑝𝑠 ) − (𝜃𝑗𝑠 , 𝑝𝑠∗ )∥ ≤ 𝜌1 }in the sense that any 𝑁𝜖 −invariant
set contained in 𝑁 must also be contained in 𝑋𝑗 .
We will fix 0 ≤ 𝑗 ≤ 𝑠 − 1 and prove Theorem 4.1 for this particular 𝑗. Let us write
𝐽 = [𝑎𝑗 − 𝜆/2, 𝑎𝑗+1 + 𝜆/2] and 𝐽𝜆/2 = [𝑎𝑗 − 𝜆, 𝑎𝑗+1 + 𝜆] and write 𝜃∗𝑠 instead of 𝜃𝑗𝑠 with
for the rest of this section. We have that −∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) ≥ 𝜆𝐼 for all 𝑝𝑓 ∈ 𝐽𝜆/2 .
It is known classically that normally hyperbolic cylinders persists under small perturbation. However, the perturbation 𝜖𝑅 to 𝐻0 + 𝜖𝑍 is a singular perturbation, in
that with 𝜖 −→ 0, both the perturbation and the hyperbolicity approaches 0. Thus,
we need to do some shifts and scaling before classical theory can be applied.
4.1. Hyperbolic coordinates for the system. The estimates in this section uses
the following statements: There exists 𝐴 > 1 and 0 < 𝜆 < 1 such that
2
𝐴−1 𝐼 ≤ ∂𝑝𝑝
𝐻0 ≤ 𝐴𝐼,
∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) ≥ 𝜆𝐼 2
as quadratic forms; ∥𝑍∥𝐶 2 ≤ 1 and ∥𝑅∥𝐶 2 ≤ 𝑐𝛿. To simplify notations, we will be
using the 𝑂(⋅) notation, where 𝑓 = 𝑂(𝑔) means ∣𝑓 ∣ ≤ 𝐶𝑔 for a constant 𝐶 independent
of 𝜆, 𝛿, 𝑛 and 𝑟. In particular, we will not be keeping track of the parameter 𝐴, which
is considered fixed throughout the paper.
12
P. Bernard, V. Kaloshin, K. Zhang
The Hamiltonian flow admits the following equation of motion (with time component included)
⎧ 𝑠
𝜃˙ = ∂𝑝𝑠 𝐻0 + 𝜖∂𝑝𝑠 𝑍 + 𝜖∂𝑝𝑠 𝑅




𝑠

⎨𝑝˙ = −𝜖∂𝜃𝑠 𝑍 − 𝜖∂𝜃𝑠 𝑅
𝜃˙𝑓 = ∂𝑝𝑓 𝐻0 + 𝜖∂𝑝𝑓 𝑍 + 𝜖∂𝑝𝑓 𝑅



𝑝˙𝑓 = −𝜖∂𝜃𝑓 𝑅


⎩˙
𝑡=1
(4)
.
To help demonstrate the hyperbolic structure for this system, we make a series of
coordinate changes. The final coordinate system will take the form of (𝑥, 𝑦, 𝑧), where
𝑥 is the stable direction, 𝑦 is the unstable direction and 𝑧 is the central direction. The
hyperbolicity will be evident
√ from examining the vector field in (𝑥, 𝑦, 𝑧) coordinates,
with a rescaled time 𝜏 = 𝜖𝑡.
The first step of the coordinate changes involves scaling and shifting of the variables; the second change of coordinates is an auxiliary rescaling; the third change of
coordinates is a linear change of coordinates in a neighborhood. After the first change
of variables we write the equations using the new time variable 𝜏 while keeping 𝑡 as
part of the phase space. Here is a brief diagram:
⎡ 𝑠⎤
⎡ 𝑠⎤
⎡ 𝑠⎤
𝜃¯
𝜃
𝜃¯
⎡ ⎤
⎢ 𝐼¯𝑠 ⎥
⎢ 𝑝𝑠 ⎥
⎢ 𝐼¯𝑠 ⎥
𝑥
⎢ 𝑓⎥
⎢ 𝑓⎥
⎢ 𝑓⎥
⎢𝜃 ⎥ −→ ⎢𝜃 ⎥ −→ ⎢𝜃𝛾 ⎥ −→ ⎣𝑦 ⎦ .
⎢ 𝑓⎥
⎢ 𝑓⎥
⎢ 𝑓⎥
⎣𝐼 ⎦
⎣𝑝 ⎦
⎣𝐼 ⎦
𝑧
𝑡
𝑡
𝑡
The first change of coordinates takes the following form:
𝜃¯𝑠 = 𝜃𝑠 − 𝜃∗𝑠 ,
𝜃𝑓 = 𝜃𝑓 ,
√
𝐼¯𝑠 = (𝑝𝑠 − 𝑝𝑠∗ )/ 𝜖,
√
𝐼 𝑓 = 𝑝𝑓 / 𝜖.
√
The new time variable is given by 𝜏 = 𝜖𝑡 and we use ′ to denote the derivative in 𝜏 .
With this change of coordinates the critical points of ∂𝑝𝑠 𝐻0 and ∂𝜃𝑠 𝑍 is moved to the
origin 𝜃¯𝑠 = 𝐼¯𝑠 = 0. The second step is an additional rescaling by 𝜃𝛾𝑓 = 𝛾𝜃𝑓 , where 𝛾 is
a positive parameter to be determined later. In these coordinates the new equation
Arnold diffusion along normally hyperbolic invariant cylinders
of motion is the following:
⎤
⎡
⎡ 𝑠 ⎤′ ⎡
⎤
√
𝜃¯
∂ 𝑝 𝑠 𝐻0 / 𝜖
∂𝑝𝑠 (𝑍 + 𝑅) + ∂𝑝𝑓 𝜃∗𝑠 ∂𝜃𝑓 𝑅
⎢−∂𝜃𝑠 (𝑍 + 𝑅) − ∂𝑝𝑓 𝑝𝑠 ∂𝜃𝑓 𝑅⎥
⎥
⎢
⎢ 𝐼¯𝑠 ⎥
0
⎢
⎥
⎢ 𝑓⎥
⎥ √ ⎢
√ ∗
⎥
⎢ 𝜃𝛾 ⎥ = ⎢
⎥
⎢
𝛾∂𝑝𝑓 (𝑍 + 𝑅)
𝛾∂𝑝𝑓 𝐻0 / 𝜖
⎢
⎥
⎢ 𝑓⎥
⎥ + 𝜖⎢
⎣
⎦
⎣
⎣𝐼 ⎦
⎦
0
∂𝜃𝑓√𝑅
0
1/ 𝜖
𝑡
:= 𝐹 (𝑤) = 𝐹1 (𝑤) +
13
√
𝜖𝐹2 (𝑤),
where 𝑤 denotes a point in the domain and 𝐹 (𝑤) denotes the vector field. Note that
the extra terms are coming from the derivatives of 𝜃∗𝑠 and 𝑝𝑠∗ as a result of the shifting.
1
1
∥Ω∥, ∥Ω−1 ∥ = 𝑂(𝑚𝑍2 + 𝑚𝐻2 ), while sup𝑝𝑓 ∈𝐽 ∥∂𝑝𝑓 Ω∥ = 𝑂(𝑚3𝑍 𝑚3𝐻 ).
Here is an estimate on those derivatives:
Lemma 4.1.
(1) ∥𝑝𝑠∗ ∥𝐶 2 = 𝑂(1), ∥𝜃∗𝑠 ∥𝐶 1 = 𝑂(𝜆−1 ) and ∥𝜃∗𝑠 ∥𝐶 2 = 𝑂(𝜆−3 ).
(2) ∥𝐹2 ∥𝐶 0 = 𝑂(𝜆−1 ) and ∥𝐹2 ∥𝐶 1 = 𝑂(𝜆−3 ).
We now describe the final change of coordinates. Let us write
[
]
0
∂𝑝2𝑠 𝑝𝑠 𝐻0 (𝑝∗ (𝑝𝑓 ))
𝑓
𝐵0 (𝑝 ) =
.
−∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃∗𝑠 (𝑝𝑓 ), 𝑝∗ (𝑝𝑓 ))
0
We claim that there exists a family of invertible (2𝑛 − 2) × (2𝑛 − 2) matrices Ω(𝑝𝑓 )
such that Ω(𝑝𝑓 )𝐵0 (𝑝𝑓 )Ω(𝑝𝑓 )−1 = diag{−Λ(𝑝𝑓 ), Λ(𝑝𝑓 )}. Furthermore Λ is a symmetric
(𝑛−1)×(𝑛−1) matrix, and there exists 𝜆∗ > 0 such that for each 𝑝𝑓 ∈ 𝐽, Λ(𝑝𝑓 ) ≥ 𝜆∗ 𝐼
as a quadratic form.
√
Lemma 4.2.
(1) We have that 𝜆∗ ≤ 𝜆/𝐴.
1
(2) ∥Ω∥, ∥Ω−1 ∥ = 𝑂(𝜆− 2 ) and ∥∂𝑝𝑓 Ω∥, ∥∂𝑝𝑓 Ω−1 ∥ = 𝑂(𝜆−3 ).
Proof. Let us write 𝐶 = −∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃∗𝑠 (𝑝𝑓 ), 𝑝∗ (𝑝𝑓 )) and 𝐷 = ∂𝑝2𝑠 𝑝𝑠 𝐻0 (𝑝∗ (𝑝𝑓 )). Let 𝜆∗ be
the smallest eigenvalue of 𝐵0 and the eigenvector is [𝑣1 , 𝑣2 ]𝑇 . We have that 𝐶𝑣
√1 = 𝜆𝑣2
−1
and 𝐷𝑣2 = 𝜆𝑣1 . Using 𝐶 ≥ 𝜆𝐼 and 𝐷 ≥ 𝐴 𝐼 we obtain the estimate 𝜆∗ ≥ 𝜆/𝐴.
For the second estimate, we have that the matrix Ω has an explicit formula (see
[Be3])
[
]
1 𝐿 𝐿
Ω=
2 𝐿 −𝐿
1
( 1 1
)2
1 1
1
where 𝐿 = 𝐶 − 2 (𝐶 2 𝐷𝐶 2 ) 2 𝐶 − 2 . Using ∥𝐶∥, ∥𝐷∥ = 𝑂(1), ∥𝐶 −1 ∥ ≤ 𝜆−1 and
1
∥𝐷−1 ∥ = 𝑂(1), a calculation shows that ∥Ω∥, ∥Ω−1 ∥ = 𝑂(𝜆− 2 ).
14
P. Bernard, V. Kaloshin, K. Zhang
1
To estimate the derivative of Ω, we note that the function 𝑓 (𝑀 ) = 𝑀 2 is a local
diffeomorphism in a neighborhood of a strictly positive definite 𝑀 . Taking derivative
1
1
for the inverse of 𝑓 , a calculation shows that ∥∂(𝑀 2 )∥ = 𝑂(∥𝑀 − 2 ∥∥∂𝑀 ∥). Using
the explicit formula we have that ∥∂Ω∥, ∥∂Ω−1 ∥ = 𝑂(𝜆−3 ).
□
On the set ∥𝜃¯𝑠 ∥ < 12 , we define the final change of coordinates by first renaming
(𝜃𝑓 , 𝐼 𝑓 , 𝑡) to 𝑧, and then let
[ ]
[ 𝑠]
𝑥
𝜃¯
= Ω ¯𝑠 .
𝑦
𝐼
Since the new change of coordinates applies only to the first two coordinates, it
suffices to rewrite the equation for (𝑥, 𝑦) variables only. Let 𝐹˜ (𝑤) denote the vector field writen in the (𝑥, 𝑦, 𝑧) coordinates, let Π12 be the projection to the (𝜃¯𝑠 , 𝐼¯𝑠 )
components and Π𝑥𝑦 the projection to the (𝑥, 𝑦) components. We have the following:
[ ]′
[ 𝑠 ]′
[ 𝑠]
[ ]
¯
√ 𝜃¯𝑠
𝑥
𝜃¯
′ 𝜃
−3
˜
(5)
Π𝑥𝑦 𝐹 (𝑤) =
= Ω ¯𝑠 + Ω ¯𝑠 = ΩΠ12 𝐹 (𝑤) + 𝑂(𝜆
𝜖) ¯𝑠 .
𝑦
𝐼
𝐼
𝐼
The second equality follows from the estimates Ω′ = ∂𝑝𝑓 Ω ⋅ (𝑝𝑓 )′ , Lemma 4.2 and
√
(𝑝𝑓 )′ = 𝑂( 𝜖).
To study how the flow acts on the tangent space, we also need to examine the
variational equation. After the first and second coordinate changes, the variational
equation is given by 𝑣 ′ = 𝐷(𝜃¯𝑠 ,𝐼¯𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹 (𝑤)𝑣. We will use a context based definition
for the operator 𝐷 when no subscript is present: The derivative is assumed to be
with respect to the coordinate variables of the corresponding vector field, that is, 𝐷𝐹
means 𝐷(𝜃¯𝑠 ,𝐼¯𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹 and 𝐷𝐹˜ means 𝐷(𝑥,𝑦,𝑧) 𝐹˜ .
√
Let 𝐼 𝑠 = 𝑝𝑠 / 𝜖, we compute 𝐷𝐹 by writing
∂(𝜃𝑠 , 𝐼 𝑠 , 𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡)
𝐷(𝜃¯𝑠 ,𝐼¯𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹 = 𝐷(𝜃𝑠 ,𝐼 𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹
.
∂(𝜃¯𝑠 , 𝐼¯𝑠 , 𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡)
The variables (𝜃𝑠 , 𝐼 𝑠 , 𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡) includes only the rescaling but not the shifting. More√
over, since 𝐹 = 𝐹1 + 𝜖𝐹2 we focus on computing the derivative of 𝐹1 . The matrix
𝐷(𝜃𝑠 ,𝐼 𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹1 is given by
⎡
⎤
0
∂𝑝2𝑠 𝑝𝑠 𝐻0
0
∂𝑝2𝑓 𝑝𝑠 𝐻0
0
⎢−∂ 2𝑠 𝑠 𝑍 + 𝑂(𝛿)
0
𝑂(𝛾 −1 𝛿)
0
𝑂(𝛿)⎥
⎢ 𝜃𝜃
⎥
2
2
⎥,
⎢
0
𝛾∂
𝐻
0
𝛾∂
𝐻
0
𝑓
𝑠
𝑓
𝑓
0
0
𝑝 𝑝
𝑝 𝑝
⎢
⎥
−1
⎣
𝑂(𝛿)
0
𝑂(𝛾 𝛿)
0
𝑂(𝛿)⎦
0
0
0
0
0
Arnold diffusion along normally hyperbolic invariant cylinders
15
where we have taken advantage of the facts that ∥𝑅∥𝐶 2 = 𝑂(𝛿) and ∥𝑝𝑠∗ ∥𝐶 2 = 𝑂(1).
Since
⎡
⎤
𝐼
⎢
∂𝑝𝑓 𝑝𝑠∗ ⎥
⎥
√
∂(𝜃𝑠 , 𝐼 𝑠 , 𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡) ⎢ 𝐼
⎢
⎥ + 𝑂(𝜆−3 𝜖),
1
=⎢
⎥
𝑓
∂(𝜃¯𝑠 , 𝐼¯𝑠 , 𝜃𝛾 , 𝐼 𝑓 , 𝑡) ⎣
⎦
1
1
we have that 𝐷(𝜃¯𝑠 ,𝐼¯𝑠 ,𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) 𝐹1 is given by
⎡
⎤
0
∂𝑝2𝑠 𝑝𝑠 𝐻0
0
0
0
⎢−∂𝜃2𝑠 𝜃𝑠 𝑍 + 𝑂(𝛿)
0
𝑂(𝛿)
0
𝑂(𝛿)⎥
⎢
⎥
√
2
2
⎢
⎥ + 𝑂(𝜆−3 𝜖).
0
𝛾∂
𝐻
0
𝛾∂
𝐻
0
(6)
𝑓
𝑠
𝑓
𝑓
0
0
𝑝 𝑝
𝑝 𝑝
⎢
⎥
⎣
𝑂(𝛿)
0
𝑂(𝛿)
0
𝑂(𝛿)⎦
0
0
0
0
0
The calculation includes a cancellation at the first row, fourth column of the matrix.
This is due to ∂𝑝2𝑠 𝑝𝑠 𝐻0 ∂𝑝𝑓 𝑝𝑠∗ +∂𝑝2𝑠 𝑝𝑓 𝐻0 = 0 at 𝑝𝑠 = 𝑝𝑠∗ (𝑝𝑓 ), obtained from differentiating
both sides of the equation ∂𝑝𝑠 𝐻0 (𝑝𝑠∗ (𝑝𝑓 ), 𝑝𝑓 ) = 0.
We write the matrix in (6) in block form [𝐵1 , 𝐵2 ; 𝐵3 , 𝐵4 ] where
[
]
0
∂𝑝2𝑠 𝑝𝑠 𝐻0
𝐵1 (𝑤) =
−∂𝜃2𝑠 𝜃𝑠 𝑍 + 𝑂(𝛿)
0
and so on. We have that ∥𝐵2 ∥ = 𝑂(𝛿𝛾 −1 ) while ∥𝐵3 ∥, ∥𝐵4 ∥ = 𝑂(max{𝛿, 𝛾}) . We
will choose 𝛾 ≫ 𝛿, it then follows that max{∥𝐵2 ∥, ∥𝐵3 ∥, ∥𝐵4 ∥} = 𝑂(𝛾).
Finally, the variational equation under the variables (𝑥, 𝑦, 𝑧) is 𝑣 ′ = 𝐷𝐹˜ 𝑣, where
[
]
√
Ω𝐵1 Ω−1 Ω𝐵2
˜
𝐷𝐹 =
+ 𝑂(𝜆−4 𝜖).
−1
𝐵3 Ω
𝐵4
the remainder estimate uses the fact that ∥Ω∥ ⋅ ∥Ω−1 ∥ = 𝑂(𝜆−1 ).
4.2. Sufficient conditions for normally hyperbolic cylinder. We verify two sets
of properties for the vector field 𝐹˜ in Proposition 4.2 below. The first set of properties
implies the existence of an “isolating block” in the sense of Conley; the second set
of properties tells us that on the tangent space, the flow expands the 𝑥 direction,
contracts 𝑦 direction and is nearly neutral 𝑧 direction. Theorem 4.1 then follows from
an abstract result described in detail in Appendix B.
Proposition 4.2. Let 𝑈𝜌 = {∥𝑥∥ ≤ 𝜌, ∥𝑦∥ ≤ 𝜌}. There exists positive constants 𝑐1 ,
1
𝑐2 and 𝑐3 such that for 0 < 𝜖 ≤ 𝑐1 𝜆−9 , 0 < 𝜌 ≤ 𝑐2 𝜆 and 0 < 𝛿 < 𝑐3 𝜆 2 𝜌, the following
hold:
16
P. Bernard, V. Kaloshin, K. Zhang
(1) We have that for 𝑝𝑓 ∈ [𝑎1 − 𝜆/2, 𝑎2 + 𝜆/2]
∙ ⟨Π𝑥 𝐹˜ (𝑤), 𝑥⟩∣∣𝑥∣=𝜌,∣𝑦∣≤𝜌 ⩽ −𝜆∗ 𝜌2 /2;
∙ ⟨Π𝑦 𝐹˜ (𝑤), 𝑦⟩∣∣𝑦∣=𝜌,∣𝑥∣≤𝜌 ⩾ 𝜆∗ 𝜌2 /2,
∙ ∥Π𝑧 𝐹˜ ∥ = 𝑂(𝛿).
(2) We write 𝐷𝐹˜ in the following block form:
⎡
𝐿𝑥𝑥 𝐿𝑥𝑦
˜
⎣
𝐷𝐹 (𝑤) = 𝐿𝑦𝑥 𝐿𝑦𝑦
𝐿𝑧𝑥 𝐿𝑧𝑦
and 𝜆∗ defined before Lemma 4.2,
⎤
𝐿𝑥𝑧
𝐿𝑦𝑧 ⎦ .
𝐿𝑧𝑧
For all 𝑤 ∈ 𝑈2𝜌 the following holds:
∙ ⟨𝐿𝑥𝑥 (𝑤)𝑣𝑥 , 𝑣𝑥 ⟩ ⩽ −𝜆∗ ∥𝑣𝑥 ∥2 /2,
∙ ⟨𝐿𝑦𝑦 (𝑤)𝑣𝑦 , 𝑣𝑦 ⟩ ⩾ 𝜆∗ ∥𝑣𝑦 ∥2 /2,
1
∙ ∥𝐿𝑥𝑦 ∥, ∥𝐿𝑦𝑥 ∥ = 𝑂(𝜆−1 𝛿), ∥𝐿𝑥𝑧 ∥, ∥𝐿𝑦𝑧 ∥ = 𝑂(𝜆− 2 𝛾 −1 𝛿) and ∥𝐿𝑧𝑥 ∥, ∥𝐿𝑧𝑦 ∥,
1
∥𝐿𝑧𝑧 ∥ = 𝑂(𝜆− 2 𝛾).
Proof. Given 𝑤 = (𝑥, 𝑦, 𝑧) in the phase space, we denote 𝑤0 = (0, 0, 𝑧). Since [𝑥, 𝑦]𝑇 =
1
Ω[𝜃¯𝑠 , 𝐼¯𝑠 ]𝑇 , we have that for ∥𝑥∥, ∥𝑦∥ ≤ 𝜌, ∥(𝜃¯𝑠 , 𝐼¯𝑠 )∥ = 𝑂(∥Ω−1 ∥𝜌) = 𝑂(𝜆− 2 𝜌). By
(5), We have:
[ ]
√ 𝐼¯𝑠
−3
˜
Π𝑥𝑦 𝐹 = ΩΠ12 𝐹1 (𝑤) + ∥Ω∥𝑂(𝜆
𝜖) ¯𝑠
𝜃
[ 𝑠]
√
𝐼¯
= ΩΠ12 𝐹1 (𝑤0 ) + ΩΠ12 𝐷𝐹 (𝑤0 ) ¯𝑠 + 𝑂(𝜌2 ) + 𝑂(𝜆−4 𝜌 𝜖)
𝜃
[ ]
√
𝑥
= Ω𝐵1 Ω−1 (𝑤0 )
+ 𝑂(𝜆−1 𝛿𝜌) + 𝑂(𝜌2 ) + 𝑂(𝜆−4 𝜌 𝜖).
𝑦
In the last equality, we used the fact that Π12 𝐹1 (𝑤0 ) = 𝑂(𝛿). Since 𝐵1 (𝑤0 ) = 𝐵0 (𝑝𝑓 )+
𝑂(𝛿), it follows that for ∣𝑦∣ = 𝜌, ∣𝑥∣ ≤ 𝜌:
√
⟨Π𝑦 𝐹˜ , 𝑦⟩ = ⟨Λ𝑦, 𝑦⟩ + 𝑂(𝜆−1 𝛿𝜌2 ) + 𝑂(𝜌3 ) + 𝑂(𝜆−4 𝜌2 𝜖).
1
Since 𝜆∗ ≥ 𝐴−1 𝜆− 2 , for sufficiently small 𝑐1 , 𝑐2 and 𝑐3 , our assumptions imply that
all the 𝑂(⋅) terms in the last displayed formula are bounded by 𝜆∗ 𝜌2 /2. It follows
that ⟨Π𝑦 𝐹˜ , 𝑦⟩ ≥ 𝜆∗ 𝜌2 /2. The inequality for 𝑥 can be proved in the same way. The
third statement follows directly from the equation of motion in the new coordinates.
For the second set of properties, let us first prove the third statement. We have
that
[
]
√
Ω𝐵1 Ω−1 Ω𝐵2
˜
𝐷𝐹 =
+ 𝑂(𝜆−4 𝜖).
−1
𝐵3 Ω
𝐵4
Arnold diffusion along normally hyperbolic invariant cylinders
17
Using (6), we have that ∥𝐵2 ∥ = 𝑂(𝛿𝛾 −1 ) while ∥𝐵3 ∥, ∥𝐵4 ∥ = 𝑂(𝛾). It follows that
1
1
∥Ω𝐵2 ∥ = 𝑂(𝜆− 2 𝛿) while ∥𝐵3 Ω−1 ∥, ∥𝐵4 ∥ = 𝑂(𝜆− 2 𝛾). The estimates for ∥𝐿𝑧𝑦 ∥, ∥𝐿𝑦𝑧 ∥,
∥𝐿𝑧𝑥 ∥, ∥𝐿𝑥𝑧 ∥ and ∥𝐿𝑧𝑧 ∥ all follows. Furthermore, we have that
Ω𝐵1 Ω−1 (𝑤) = Ω𝐵1 Ω−1 (𝑤0 ) + 𝑂(𝜆−1 𝜌) = diag{−Λ, Λ} + 𝑂(𝜆−1 𝛿) + 𝑂(𝜆−1 𝜌).
√
Since 𝐿𝑥𝑦 and 𝐿𝑦𝑥 are the off-diagonal block of the matrix Ω𝐵1 Ω−1 (𝑤) + 𝑂(𝜆−4 𝜖),
we obtain ∥𝐿𝑥𝑦 ∥, ∥𝐿𝑦𝑥 ∥ = 𝑂(𝜆−1 𝛿). This implies the third statement. We now turn
to the first two statements of the list. We have that
(
√ )
𝜆∗
⟨𝐿𝑦𝑦 𝑣𝑦 , 𝑣𝑦 ⟩ = ⟨Λ𝑣𝑦 , 𝑣𝑦 ⟩ + 𝑂(𝜆−1 𝛿) + 𝑂(𝜆−1 𝜌) + 𝑂(𝜆−4 𝜖) ∥𝑣𝑦 ∥2 ≥ ∥𝑣𝑦 ∥2 .
2
The calculation for ⟨𝐿𝑥𝑥 𝑣𝑥 , 𝑣𝑥 ⟩ is the same.
□
Proposition 4.3. There exists positive constants 𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 (may be different
constants from those of Proposition 4.2) such that for 0 < 𝜖 ≤ 𝑐1 𝜆9 , 0 < 𝛿 ≤ 𝑐3 𝜆2 ,
𝜌 = 𝑐2 𝜆 and 𝛾 = 𝑐4 𝜆 the following hold.
There exists a normally hyperbolic cylinder
√
(𝑥, 𝑦) = (𝑋, 𝑌 )(𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡), 𝐼 𝑓 ∈ 𝐽/ 𝜖, 𝜃¯𝑓 ∈ 𝛾𝕋, 𝑡 ∈ 𝕋
contained in ∥𝑥∥, ∥𝑦∥ ≤ 𝜌 with ∥∂(𝜃𝛾𝑓 ,𝐼 𝑓 ,𝑡) (𝑋, 𝑌 )∥ = 𝑂(𝛿 −2 𝛿).
Proof. We will show that Proposition B.3 applies. We note that in our coordinate
system, 𝑥 is the stable direction, 𝑦 is the unstable direction, while 𝑧 = (𝜃𝛾𝑓 , 𝐼 𝑓 , 𝑡) is
the central direction. To apply Proposition B.3, we lift the (𝜃𝛾𝑓 , 𝑡) directions to the
√
universal cover, hence the domain for the central variables becomes Ω𝑐 = ℝ×𝐽/ 𝜖×ℝ.
We also need to extend the domain to an 𝑟 neighborhood of Ω𝑐 while keeping all
the estimates, this is√no restriction, as the isolation block conditions hold on 𝐼 𝑓 ∈
[𝑎1 − 𝜆/2, 𝑎2 + 𝜆/2]/ 𝜖, we may simply choose 𝑟 = 1 in Proposition B.3.
1
By Proposition 4.2, we have that in the notations of Proposition B.3, 𝑚 = 𝑂(𝜆− 2 𝛾 −1 𝛿),
1
𝑀 = 𝑂(𝜆− 2 𝛾) and 𝜆. Proposition B.3 applies if
𝜇=
𝑚
1
<√ .
𝜆∗ /2 − 2(𝑚 + 𝑀 )
2
Using our assumptions, for sufficiently small 𝑐1 , 𝑐2 , 𝑐3 and 𝑐4 we can make sure
3
that 2(𝑚 + 𝑀 ) < 𝜆∗ /4 and 𝑚 = 𝑂(𝜆− 2 𝛿). For sufficently small 𝑐3 we have that
3
−2
√1
𝜇 = 𝑂(𝜆− 2 𝛿𝜆−1
□
∗ ) = 𝑂(𝜆 𝛿) < 2 .
Let 𝜖, 𝛿, 𝜆 be such that the assumptions of Proposition 4.3 applies and let (Θ𝑠 , 𝑃 𝑠 )
be the function (𝑋, 𝑌 ) written in the original variables. Assume in addition that
18
P. Bernard, V. Kaloshin, K. Zhang
𝜖 ≤ 𝛿, we have that
(
( (
(
(
(
( ∂(𝑋, 𝑌 ) (
( ∂(Θ𝑠 , 𝐼 𝑠 ) ( ( 𝑑Ω (
√
− 52
− 25
( (
(
(
(
(
( ∂𝐼 𝑓 ( ≤ ( 𝑑𝐼 𝑓 ( ∥(𝑋, 𝑌 )∥ + ∥Ω∥ ( ∂𝐼 𝑠 ( = 𝑂( 𝜖𝜌) + 𝑂(𝜆 𝛿) = 𝑂(𝜆 𝛿)
and
(
(
(
(
(
(
( ∂(Θ𝑠 , 𝐼 𝑠 ) (
( = ∥Ω∥ ( ∂(𝑋, 𝑌 ) ( = 𝑂(𝜆− 52 𝛿).
(
( ∂(𝜃𝑓 , 𝑡) (
( ∂(𝜃𝑓 , 𝑡) (
√
Since 𝐼 𝑓 = 𝑝𝑓 / 𝜖, we conclude that:
( 𝑠(
( 𝑑Θ (
√
(
( = 𝑂(𝜆− 52 𝛿/ 𝜖)
( 𝑑𝑝𝑓 (
This concludes the proof of Theorem 4.1.
(
(
( 𝑑Θ𝑠 (
− 52
(
(
( 𝑑(𝜃𝑓 , 𝑡) ( = 𝑂(𝜆 𝛿).
5. Localization and Mather’s projected graph theorem
We study the normal form system 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 on the neighborhood of
the set {𝑝 = 𝑝∗ (𝑝𝑓 ), 𝑝𝑓 ∈ [𝑎− , 𝑎+ ] ⊂ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ]}. We assume that 𝑍 satisfies the
generic conditions ∪
[G0]-[G2] and that ∥𝑅∥𝐶 2 ≤ 𝛿. Recall that there exists a partition
𝑓
of [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] = 𝑠−1
𝑗=1 [𝑎𝑗 , 𝑎𝑗+1 ], such that for 𝑝 ∈ [𝑎𝑗 − 𝜆, 𝑎𝑗+1 + 𝜆] the function
𝑍(𝜃𝑠 , 𝑝𝑠 , 𝑝𝑓 ) as a nondegenerate local maximum at 𝜃𝑗𝑠 . It is clear that we can restrict
∪
this partition to [𝑎− , 𝑎+ ]. We abuse notation and still write [𝑎− , 𝑎+ ] = 𝑠−1
𝑗=1 [𝑎𝑗 , 𝑎𝑗+1 ].
We first point out the following consequences of the genericity conditions [G0]-[G2]:
there exists 0 < 𝑏 < 𝜆/4 depending on 𝐻1 such that
[G1’]
𝑍(𝜃𝑗𝑠 (𝑝𝑓 ), 𝑝∗ (𝑝𝑓 )) − 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) ≥ 𝑏∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝𝑓 )∥,
for each 𝑝𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏].
[G2’] For 𝑝𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏], 𝑗 = 0, ⋅ ⋅ ⋅ , 𝑠 − 2, we have
𝑠
max{𝑍(𝜃𝑗𝑠 , 𝑝∗ (𝑝𝑓 )), 𝑍(𝜃𝑗+1
, 𝑝∗ (𝑝𝑓 ))} − 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 ))
𝑠
≥ 𝑏 min{∥𝜃𝑠 − 𝜃𝑗𝑠 ∥, ∥𝜃𝑠 − 𝜃𝑗+1
∥}2 .
In the first case, the function 𝑍 has a single non-degenerate maximum, which we
will call the “single peak” case, while the second case will be called the “double peak”
case. The shape of the function 𝑍 allows us to localize the Aubry set and Mañe set
of the Hamiltonian 𝑁𝜖 .
According to Theorem 4.1, for each [𝑎𝑗 − 𝜆/2, 𝑎𝑗+1 + 𝜆/2] there exists
𝑋𝑗 = {(𝜃𝑠 , 𝑝𝑠 ) = (Θ𝑠𝑗 , 𝑃𝑗𝑠 )(𝜃𝑓 , 𝑝𝑓 , 𝑡));
𝜆
𝜆
𝑝𝑓 ∈ [𝑎𝑗 − , 𝑎𝑗+1 + ], (𝜃𝑓 , 𝑡) ∈ 𝕋 × 𝕋},
2
2
Arnold diffusion along normally hyperbolic invariant cylinders
19
which are maximally invariant set on 𝑁𝑗 := {(𝜃, 𝑝, 𝑡); 𝑝𝑓 ∈ [𝑎𝑗 − 𝜆2 , 𝑎𝑗+1 + 𝜆2 ], ∥(𝜃𝑠 , 𝑝𝑠 )−
(𝜃𝑗𝑠 , 𝑝𝑠∗ )∥ ≤ 𝜌1 }.
These information allows us to study the Mather set, Aubry set and Mañe set of
the Hamiltonian 𝑁𝜖 .
Theorem 5.1 (Localization). For 𝑁𝜖 = 𝐻0 +𝜖𝑍 +𝜖𝑅 such that 𝑍 satisfies [G0]-[G2],
then there exists 𝜖0 , 𝛿0 and 0 < 𝜌2 < 𝜌1 such that for 0 < 𝜖 < 𝜖0 and 0 < 𝛿 < 𝛿0 the
following hold.
(1) For any 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) such that 𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏], 𝒩˜ (𝑐) is contained in
√
{(𝜃, 𝑝, 𝑡), ∥𝑝 − 𝑐∥ ≤ 6𝐴 𝑛𝜖, ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝𝑓 )∥ ≤ 𝜌2 }.
(2) For 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) such that 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏], we have that 𝒜˜𝑁𝜖 (𝑐) is
contained in
√
𝑠
{(𝜃, 𝑝, 𝑡), ∥𝑝 − 𝑐∥ ≤ 6𝐴 𝑛𝜖, min{∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝𝑓 )∥, ∥𝜃𝑠 − 𝜃𝑗+1
(𝑝𝑓 )∥} ≤ 𝜌2 }.
Apply the statements of the previous theorem with Theorem 4.1, we may further
localize these sets on the normally hyperbolic cylinders. Moreover, locally these sets
are graphs over the 𝜃𝑓 component, which is a version of Mather’s projected graph
theorem.
Theorem 5.2 (Mather’s projected graph theorem). For any 𝑁𝜖 such that 𝑍 satisfies
[G0]-[G2], there exists 𝛿0 and 𝜖0 depending on 𝑏, 𝑛 and 𝑟 such that for 𝛿 ≤ 𝛿0 and
𝜖 < 𝜖0 we have:
(1) There exists 0 < 𝜌2 < 𝜌1 such that for for 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) with 𝑐𝑓 ∈ (𝑎𝑗 +
𝑏, 𝑎𝑗+1 − 𝑏) the Mañe set 𝒩˜𝑐 is contained in the normally hyperbolic cylinder
𝑋𝑗 .
Moreover, let 𝜋𝜃𝑓 be the projection to the 𝜃𝑓 component, we have that 𝜋𝜃𝑓 ∣𝒜˜𝑐
is one-to-one and the inverse is Lipshitz.
(2) For 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏], we have that 𝒜𝑐 ⊂ 𝑋𝑗 ∪ 𝑋𝑗+1 .
𝜋𝜃𝑓 ∣𝒜˜𝑐 ∩ 𝑋𝑗 and 𝜋𝜃𝑓 ∣𝒜˜𝑐 ∩ 𝑋𝑗+1 are both one-to-one and have Lipshitz inverses.
For the rest of this section, we will derive various estimates of quantities and set
arising from Mather theory. We deduce Theorem 5.1 and Theorem 5.2 from these
estimates.
5.1. Vertical estimates. We derive estimates on the Mather sets of a general Hamiltonian 𝐻(𝑡, 𝜃, 𝑝), depending on 𝜖, under the assumptions that
𝐼/𝐴 ⩽ ∂𝑝𝑝 𝐻 ⩽ 𝐴𝐼
20
P. Bernard, V. Kaloshin, K. Zhang
in the sense of quadratic forms, and
∥∂𝜃 𝐻∥𝐶 1 ⩽ 2𝜖.
Note that both Hamiltonians 𝐻𝜖 and 𝑁𝜖 satisfy these assumptions. The main result
in this section is:
Proposition 5.3. We assume that 𝜖 ⩽ 1. For each cohomology 𝑐 ∈ ℝ𝑛 and each
√
Weak KAM solution 𝑢 of 𝐻𝜖 at cohomology 𝑐, the√set ℐ̃(𝑢, 𝑐) is contained in a 36𝐴 𝜖Lipshitz graph, and in the domain ∥𝑝 − 𝑐∥ ⩽ 6𝐴 𝑛𝜖.
It is useful to use the Lagrangian 𝐿(𝑡, 𝜃, 𝑣) associated to 𝐻. Recalling the expressions
(
)−1
∂𝑣𝑣 𝐿(𝑡, 𝜃, 𝑣) = ∂𝑝𝑝 𝐻(𝑡, 𝜃, ∂𝑣 𝐿(𝑡, 𝜃, 𝑣) ,
(
)
∂𝜃𝑣 𝐿(𝑡, 𝜃, 𝑣) = −∂𝜃𝑝 𝐻 𝑡, 𝜃, ∂𝑣 𝐿(𝑡, 𝜃, 𝑣) ∂𝑣𝑣 𝐿(𝑡, 𝜃, 𝑣)
and
(
)
∂𝜃𝜃 𝐿(𝑡, 𝜃, 𝑣) = −∂𝜃𝜃 𝐻(𝑡, 𝜃, ∂𝑣 𝐿(𝑡, 𝜃, 𝑣)) − ∂𝜃𝑝 𝐻 𝑡, 𝜃, ∂𝑣 𝐿(𝑡, 𝜃, 𝑣) ∂𝜃𝑣 𝐿(𝑡, 𝜃, 𝑣),
we obtain the estimates
∥∂𝑣𝑣 𝐿∥𝐶 0 ⩽ 𝐴,
∥∂𝜃𝑣 𝐿∥𝐶 0 ⩽ 2𝐴𝜖,
∥∂𝜃𝜃 𝐿∥𝐶 0 ⩽ 3𝜖
when 𝜖 < 𝜖0 (𝐴).
We recall the concept of semi-concave function on 𝕋𝑛 . A function 𝑢 : 𝕋𝑛 −→ ℝ is
called 𝐾-semi-concave if the function
𝑥 ?−→ 𝑢(𝑥) − 𝐾∥𝑥∥2 /2
is concave on ℝ𝑛 , where 𝑢 is seen as a periodic function on ℝ𝑛 . It is equivalent
to require that, for each 𝜃 ∈ 𝕋𝑛 , there exists a linear form 𝑙 on ℝ𝑛 such that the
inequality
𝑢(𝜃 + 𝑦) ⩽ 𝑢(𝜃) + 𝑙 ⋅ 𝑦 + 𝐾∥𝑦∥2 /2
holds for each 𝑦 ∈ ℝ𝑛 . The following Lemma is a simple case of Lemma A.10 in [Be1]:
√
Lemma 5.1. If 𝑢 : 𝕋𝑛 −→ ℝ is 𝐾-semi-concave, then it is (𝐾 𝑛)-Lipshitz.
Proof. For each 𝑥 ∈ 𝕋𝑛 , there exists 𝑙𝑥 ∈ ℝ𝑛 such that
𝑢(𝑥 + 𝑦) ⩽ 𝑢(𝑥) + 𝑙𝑥 ⋅ 𝑦 + 𝐾∥𝑦∥2
for all 𝑦 ∈ ℝ𝑛 . By applying this inequality with 𝑦 = (±1, 0, 0, ⋅ ⋅ ⋅ , 0), we conclude
that the first component (𝑙𝑥 )1 of 𝑙𝑥 satisfies ∣(𝑙𝑥 )1 ∣ ⩽ 𝐾. Similar√estimates hold for
the other components
of 𝑙𝑥 , and we conclude that that ∥𝑙𝑥 ∥ ⩽ 𝐾 𝑛 for each 𝑥, and
√
thus that 𝑢 is 𝐾 𝑛-Lipshitz.
□
We will need the following regularity result of Fathi:
Arnold diffusion along normally hyperbolic invariant cylinders
21
Lemma 5.2. Let 𝑢 and 𝑣 be 𝐾-semiconcave functions, and let ℐ ⊂ 𝕋𝑛 be the set of
points where the sum 𝑢 + 𝑣 is minimal. Then the functions 𝑢 and 𝑣 are differentiable
at each point of ℐ, and the differential 𝑥 ?−→ 𝑑𝑢(𝑥) is 6𝐾-Lipshitz on ℐ.
Let us recall that the Weak KAM solutions of cohomology 𝑐 are defined as fixed
points of the operator 𝒯𝑐 : 𝐶(𝕋𝑛 ) −→ 𝐶(𝕋𝑛 ) defined by
∫ 1
𝒯𝑐 (𝑢)(𝜃) := min 𝑢(𝛾(0)) +
𝐿(𝑡, 𝛾(𝑡), 𝛾(𝑡))
˙
+ 𝑐 ⋅ 𝛾(𝑡)𝑑𝑡,
˙
𝛾
0
where the minimum is taken on the set of 𝐶 1 curves 𝛾 : [0, 1] −→ 𝕋𝑛 satisfying the
final condition 𝛾(𝑇 ) = 𝜃.
Proposition
5.4. For each√ 𝑐 ∈ ℝ𝑛 , each Weak KAM solution 𝑢 of 𝐿 + 𝑐 ⋅ 𝑣 is
√
6𝐴 𝜖-semi-concave and 6𝐴 𝑛𝜖-Lipshitz.
Proof. For each 𝑇 ∈ ℕ and 𝜃 ∈ 𝕋𝑛 , we have
∫ 𝑇
𝑢(𝜃) = min 𝑢(𝛾(0)) +
𝐿(𝑡, 𝛾(𝑡), 𝛾(𝑡))
˙
+ 𝑐 ⋅ 𝛾(𝑡)𝑑𝑡,
˙
𝛾
0
where the minimum is taken on the set of 𝐶 1 curves 𝛾 : [0, 𝑇 ] −→ 𝕋𝑛 satisfying the
final condition 𝛾(𝑇 ) = 𝜃. Let Θ(𝑡) be an optimal curve in that expression, meaning
that Θ(𝑡) = 𝜃 and that
∫ 𝑇
𝑢(𝜃) = 𝑢(Θ(0)) +
𝐿(𝑡, Θ(𝑡), Θ̇(𝑡))𝑑𝑡.
0
We lift Θ (and the point 𝜃 = Θ(𝑇 )) to a curve in ℝ𝑛 without changing its name, and
consider, for each 𝑥 ∈ ℝ𝑛 , the curve
Θ𝑥 (𝑡) := Θ(𝑡) + 𝑡𝑥/𝑇,
so that Θ𝑥 (𝑇 ) = 𝜃 + 𝑥. We have the inequality
∫ 𝑇
𝑢(𝜃 + 𝑥) − 𝑢(𝜃) ⩽
𝐿(𝑡, Θ𝑥 (𝑡), Θ̇𝑥 (𝑡)) − 𝐿(𝑡, Θ(𝑡), Θ̇(𝑡)) + 𝑐 ⋅ 𝑥/𝑇 𝑑𝑡.
0
The integrand can be estimated as follows:
𝐿(𝑡, Θ𝑥 (𝑡), Θ̇𝑥 (𝑡)) ⩽ 𝐿(𝑡, Θ(𝑡), Θ̇(𝑡))
(7)
+ ∂𝜃 𝐿(𝑡, Θ(𝑡), Θ̇(𝑡)) ⋅ 𝑡𝑥/𝑇 + ∂𝑣 𝐿(𝑡, Θ(𝑡), Θ̇(𝑡)) ⋅ 𝑥/𝑇
+ 3𝜖∣𝑡𝑥/𝑇 ∣2 + 2𝐴𝜖𝑡∣𝑥/𝑇 ∣2 + 𝐴∣𝑥/𝑇 ∣2 /2
Integrating, and using the Euler-Lagrange equation, we conclude that
𝑢(𝜃 + 𝑥) − 𝑢(𝜃) ⩽ (𝑐 + ∂𝑣 𝐿(𝑇, Θ(𝑇 ), Θ̇(𝑇 )) ⋅ 𝑥 + (𝜖𝑇 + 𝜖 + 1/2𝑇 )𝐴∣𝑥∣2
22
P. Bernard, V. Kaloshin, K. Zhang
√
√
for each 𝑇 ∈ ℕ. Taking 𝑇 ∈ [1/2 𝜖, 1/ 𝜖] (this interval contains an integer since
𝜖 ⩽ 1), we obtain
√
𝑢(𝜃 + 𝑥) − 𝑢(𝜃) ⩽ (𝑐 + ∂𝑣 𝐿(𝑇, Θ(𝑇 ), Θ̇(𝑇 )) ⋅ 𝑥 + 3𝐴 𝜖∣𝑥∣2 .
This ends the proof of the semi-concavity. The Lipshitz constant can then be obtained
from Lemma 5.1.
□
Let 𝑢 be a weak KAM solution, and let 𝑢ˇ be the conjugated dual weak KAM
solution. Then the set ℐ̃(𝑢, 𝑐) can be characterized as follows: Its projection ℐ(𝑢, 𝑐)
on 𝕋𝑛 is the set where 𝑢 = 𝑢ˇ, and
ℐ̃(𝑢, 𝑐) = {(𝑥, 𝑐 + 𝑑𝑢(𝑥)), 𝑥 ∈ ℐ(𝑢, 𝑐)}.
Since −ˇ
𝑢 is semi-concave, it is a consequence of Lemma 5.2 that the differential 𝑑𝑢(𝑥)
exists√for 𝑥 ∈ ℐ(𝑢, 𝑐). Moreover, we can prove exactly as in Proposition 5.4 that √
−˘
𝑢
is 6𝐴 𝜖-semi-concave. Lemma 5.2 then implies that the map 𝑥 ?−→ 𝑑𝑢(𝑥) is 36𝐴√ 𝜖Lipschitz on ℐ(𝑢, 𝑐). Moreover, 𝑑𝑢(𝑥) is bounded by the Lipschitz constant 6𝐴 𝑛𝜖
of 𝑢. This ends the proof of Proposition 5.3
□
5.2. Horizontal localization. For 𝜃𝑠 ∈ 𝕋𝑛−1 and 𝑟 > 0, let 𝐷(𝜃, 𝑟) denote the
closed Euclidean ball centered at 𝜃𝑠 with radius 𝑟.
Proposition 5.5. Let 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 with 𝑍 satisfying [𝐺0] − [𝐺2], then there
exists 𝜅 > 0 depending only on 𝐴, 𝑏 and 𝑛, 𝜖0 > 0 depending only on 𝐴 and 𝛿 such
that for each 𝜖 ∈]0, 𝜖0 [ and 𝑐 ∈ Γ(𝜖) we have the following results on the projected
Mañe set and the Aubry set.
(1) If 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏], then for each weak KAM solution
𝑢 of 𝑁𝜖 at cohomology 𝑐 we have that
(
)
1
𝑠 𝑓
4
ℐ(𝑢, 𝑐) ⊂ 𝐷 𝜃𝑗 (𝑐 ), 𝜅𝛿 × 𝕋,
as a consequence, we have
(
)
1
𝒩𝑁𝜖 (𝑐) ⊂ 𝐷 𝜃𝑗𝑠 (𝑐𝑓 ), 𝜅𝛿 4 × 𝕋.
(2) If 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏], then
( (
)
) ( (
)
)
1
1
𝑠
𝒜𝑁𝜖 (𝑐) ⊂ 𝐷 𝜃𝑗𝑠 (𝑐𝑓 ), 𝜅𝛿 4 × 𝕋 ∪ 𝐷 𝜃𝑗+1
(𝑐𝑓 ), 𝜅𝛿 4 × 𝕋 .
The Lagrangian 𝑁 ∗ (𝑡, 𝜃, 𝑣) associated to 𝑁𝜖 will play a central role in the proof.
We write it
𝑁 ∗ (𝑡, 𝜃, 𝑣) = 𝐿0 (𝑣) + 𝜖𝑍(𝜃𝑠 , ∂𝑣 𝐿0 (𝑣)) + 𝜖𝐿2 (𝑡, 𝜃, 𝑣, 𝜖),
where 𝐿0 is the Legendre dual of 𝐻0 . We have
2𝐼/𝐴 ⩽ ∂𝑣𝑣 𝐿0 ⩽ 𝐴𝐼/2
Arnold diffusion along normally hyperbolic invariant cylinders
23
in the sense of quadratic forms.
Lemma 5.3. We have the estimate
inf 𝐻2 ⩽ 𝐿2 (𝑡, 𝜃, 𝑣) ⩽ 𝐴𝜖 + sup 𝐻2 .
Proof. Let us first consider the truncated Hamiltonian
˜ 𝜃, 𝑝) = 𝐻0 (𝑝) + 𝜖𝑍(𝜃𝑠 , 𝑝)
𝐻(𝑡,
˜ 𝑠 , 𝑣). We claim that
and the associated Lagrangian 𝐿(𝜃
˜ 𝑠 , 𝑣) ⩽ 𝐿0 (𝑣) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣)) + 𝜖 .
𝐿0 (𝑣) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣)) ⩽ 𝐿(𝜃
8𝜅
In order to prove the left inequality, we write
˜ 𝑠 , 𝑣) = sup[𝑝 ⋅ 𝑣 − 𝐻0 (𝑝) − 𝜖𝑍(𝜃𝑠 , 𝑝)]
𝐿(𝜃
(8)
𝑝
⩾ ∂𝐿0 (𝑣) ⋅ 𝑣 − 𝐻0 (∂𝐿0 (𝑣)) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣))
⩾ 𝐿0 (𝑣) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣)).
The right inequality follows from the following computation:
˜ 𝑠 , 𝑣) = sup[𝑝 ⋅ 𝑣 − 𝐻0 (𝑝) − 𝜖𝑍(𝜃𝑠 , 𝑝)]
𝐿(𝜃
𝑝
[
⩽ sup 𝑝 ⋅ 𝑣 − 𝐻0 (∂𝐿0 (𝑣)) − 𝑣 ⋅ (𝑝 − ∂𝐿0 (𝑣)) − ∥𝑝 − ∂𝐿0 (𝑣)∥2 /𝐴
𝑝
]
− 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣)) + 𝜖∥𝑝 − ∂𝐿0 (𝑣)∥
[
⩽ sup 𝑣 ⋅ ∂𝐿0 (𝑣) − 𝐻0 (∂𝐿0 (𝑣)) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣))
𝑝
]
+ 𝜖∥𝑝 − ∂𝐿0 (𝑣)∥ − ∥𝑝 − ∂𝐿0 (𝑣)∥2 /𝐴
⩽ 𝐿0 (𝑣) − 𝜖𝑍(𝜃𝑠 , ∂𝐿0 (𝑣)) + sup[𝜖𝑦 − 𝑦 2 /𝐴)]
𝑦⩾0
⩽ 𝐿0 (𝑣) − 𝜖𝑍(𝜃 , ∂𝐿0 (𝑣)) + 𝐴𝜖2 .
𝑠
˜ we observe that
Now we have estimated 𝐿,
˜ 𝜃, 𝑝) − 𝜖 sup 𝐻2 ⩽ 𝑁𝜖 (𝑡, 𝜃, 𝑝) ⩽ 𝐻(𝑡,
˜ 𝜃, 𝑝) − 𝜖 inf 𝐻2
𝐻(𝑡,
from which follows that
˜ 𝜃, 𝑣) + 𝜖 inf 𝐻2 ⩽ 𝐿(𝑡, 𝜃, 𝑣) ⩽ 𝐿(𝑡,
˜ 𝜃, 𝑣) + 𝜖 sup 𝐻2 ,
𝐿(𝑡,
which implies the desired estimates in view of (8).
Let us now estimate the 𝛼 function of 𝑁𝜖 :
□
24
P. Bernard, V. Kaloshin, K. Zhang
Proposition 5.6. The 𝛼 function of Mather is estimated at the points 𝑐 ∈ Γ in the
following way:
𝐻0 (𝑐) + 𝜖𝑍(𝜃𝑠 (𝑐), 𝑐) − 𝜖 max𝑛+1 𝐿2 (𝑡, 𝜃, ∂𝐻0 (𝑐)) ⩽ 𝛼(𝑐) ⩽
(𝑡,𝜃)∈𝕋
⩽ 𝐻0 (𝑐) + 𝜖𝑍(𝜃𝑠 (𝑐), 𝑐) − 𝜖
min
(𝑡,𝜃)∈𝕋𝑛+1
𝐻2 (𝑡, 𝜃, 𝑐)
thus
𝐻0 (𝑐) + 𝜖𝑍(𝜃𝑠 (𝑐), 𝑐) − 𝜖∥𝐻2 ∥𝐶 0 − 𝐴𝜖2 ⩽ 𝛼(𝑐) ⩽ 𝐻0 (𝑐) + 𝜖𝑍(𝜃𝑠 (𝑐), 𝑐) + 𝜖∥𝐻2 ∥𝐶 0
Proof. We have
𝛼(𝑐) ⩽ max 𝐻(𝑡, 𝜃, 𝑐) ⩽ 𝐻0 (𝑐) + 𝜖 max
𝑍(𝜃𝑠 , 𝑐) − 𝜖
𝑠
(𝑡,𝜃)
𝜃
min
(𝑡,𝜃)∈𝕋𝑛+1
𝐻2 (𝑡, 𝜃, 𝑐)
which is the desired right hand side. On the other hand, let us set 𝜔 = ∂𝐻0 (𝑐) ∈ ℝ𝑛
and observe that 𝑐 = ∂𝐿0 (𝜔). We can consider the Haar measure 𝜇 of the torus
𝕋 × 𝕋 × {Θ𝑓 (𝑐)} × {𝜔}, this measure is not necessarily invariant but it is closed. We
thus have
∫
∫
∗
𝑓
𝛼(𝑐) ⩾ 𝑐 ⋅ 𝜔 − 𝑁 𝑑𝜇 = 𝑐 ⋅ 𝜔 − 𝐿0 (𝜔) + 𝜖𝑍(Θ (𝑐), 𝑐) − 𝜖 𝐿2 𝑑𝜇
⩾ 𝐻0 (𝑐) + 𝜖𝑍(Θ𝑓 (𝑐), 𝑐) − 𝜖 max𝑛+1 𝐿2 (𝑡, 𝜃, 𝜔)
(𝑡,𝜃)∈𝕋
□
Lemma 5.4. For each 𝑐 ∈ Γ, have the estimates
(9)
(10)
𝑁 ∗ (𝑡, 𝜃, 𝑣) − 𝑐 ⋅ 𝑣 + 𝛼(𝑐) ⩾ ∥𝑣 − ∂𝐻0 (𝑐)∥2 /(2𝐴) − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) − 𝜖𝜂
𝑁 ∗ (𝑡, 𝜃, 𝑣) − 𝑐 ⋅ 𝑣 + 𝛼(𝑐) ⩽ 𝐴∥𝑣 − ∂𝐻0 (𝑐)∥2 /2 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) + 𝜖𝜂
where 𝑍ˆ𝑐 (𝜃𝑠 ) = 𝑍(𝜃𝑠 , 𝑐) − max𝜃𝑠 𝑍(𝜃𝑠 , 𝑐) and
𝜂 = 2∥𝐻2 ∥𝐶 0 + (2𝐴 + 𝐴3 )𝜖.
Arnold diffusion along normally hyperbolic invariant cylinders
25
Proof. It is a direct computation:
𝑁 ∗ (𝑡, 𝜃, 𝑣)−𝑐 ⋅ 𝑣 + 𝛼(𝑐) ⩽
⩽ 𝐿0 (𝑣) − 𝑐 ⋅ 𝑣 + 𝐻0 (𝑐) − 𝜖𝑍(𝜃𝑠 , ∂𝑣 𝐿0 (𝑣))
+ 𝜖 max
𝑍(𝜃𝑠 , 𝑐) + 𝐴𝜖2 + 𝜖 sup 𝐻2 − 𝜖 min 𝐻2 (𝑡, 𝜃, 𝑐)
𝑠
𝜃
⩽ 𝐴∥𝑣 − ∂𝐻0 (𝑐)∥2 /4 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) + 𝜖∥∂𝐿0 (𝑣) − 𝑐∥ + 𝐴𝜖2 + 2𝜖∥𝐻2 ∥𝐶 0
⩽ 𝐴∥𝑣 − ∂𝐻0 (𝑐)∥2 /4 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) + 𝐴𝜖∥𝑣 − ∂𝐻0 (𝑐)∥ + 𝐴𝜖2 + 2𝜖∥𝐻2 ∥𝐶 0
⩽ 𝐴∥𝑣 − ∂𝐻0 (𝑐)∥2 /2 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) + 2𝐴𝜖2 + 2𝜖∥𝐻2 ∥𝐶 0
𝑁 ∗ (𝑡, 𝜃, 𝑣)−𝑐 ⋅ 𝑣 + 𝛼(𝑐) ⩾ 𝐿0 (𝑣) − 𝑐 ⋅ 𝑣 + 𝐻0 (𝑐) − 𝜖𝑍(𝜃𝑠 , ∂𝑣 𝐿0 (𝑣))
+ max
𝑍(𝜃𝑠 , 𝑐) − 2𝜖∥𝐻2 ∥𝐶 0 − 𝐴𝜖2
𝑠
𝜃
⩾ ∥𝑣 − ∂𝐻0 (𝑐)∥2 /𝐴 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) − 𝜖∥∂𝐿0 (𝑣) − 𝑐∥ − 2𝜖∥𝐻2 ∥𝐶 0 − 𝐴𝜖2
⩾ ∥𝑣 − ∂𝐻0 (𝑐)∥2 /𝐴 − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) − 𝐴𝜖∥𝑣 − ∂𝐻0 (𝑐)∥ − 2𝜖∥𝐻2 ∥𝐶 0 − 𝐴𝜖2
⩾ ∥𝑣 − ∂𝐻0 (𝑐)∥2 /(2𝐴) − 𝜖𝑍ˆ𝑐 (𝜃𝑠 ) − 𝐴𝜖2 − 𝐴3 𝜖2 − 2𝜖∥𝐻2 ∥𝐶 0
𝑠
We can now estimate the oscillation of a weak KAM solution near 𝜃𝑗𝑠 and 𝜃𝑗+1
.
□
Lemma 5.5. Let 𝑢(𝑡, 𝜃) be a weak KAM solution at cohomology 𝑐.
(1) If 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏], then for any (𝑡1 , 𝜃1 ), (𝑡2 , 𝜃2 ) ∈
𝕋 × 𝐷(𝜃𝑗𝑠 (𝑐𝑓 )) × 𝕋
√
𝑢(𝑡2 , 𝜃2 ) − 𝑢(𝑡1 , 𝜃1 ) ≤ 4𝑟1 𝑛𝐴𝜖
√
where 𝑟1 = 4𝜂/𝑏.
(2) If 𝑐 = (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗+1 −𝑏, 𝑎𝑗+1 +𝑏], then for either (𝑡1 , 𝜃1 ), (𝑡2 , 𝜃2 ) ∈
𝑠
𝕋 × 𝐷(𝜃𝑗𝑠 (𝑐𝑓 )) × 𝕋 or (𝑡1 , 𝜃1 ), (𝑡2 , 𝜃2 ) ∈ 𝕋 × 𝐷(𝜃𝑗+1
(𝑐𝑓 )) × 𝕋,
√
𝑢(𝑡2 , 𝜃2 ) − 𝑢(𝑡1 , 𝜃1 ) ≤ 4𝑟1 𝑛𝐴𝜖.
Proof. Using ∥𝑍∥𝐶 2 ≤ 1, we have that
1
𝑍ˆ𝑐 (𝜃𝑠 ) ⩾ − ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐𝑓 )∥2 .
2
We take two points (𝑡𝑖 , 𝜃𝑖 ), 𝑖 = 1 or 2 in this domain, and consider the curve
𝜃(𝑡) = 𝜃1 + (𝑡 − 𝑡˜1 )
𝜃˜2 − 𝜃˜1 + [(𝑇 + 𝑡˜2 − 𝑡˜1 )∂𝐻0 (𝑐)]
𝑡˜2 − 𝑡˜1 + 𝑇
26
P. Bernard, V. Kaloshin, K. Zhang
where 𝑇 ∈ ℕ is a parameter to be fixed later, and where 𝑡˜𝑖 ∈ [0, 1[ and 𝜃˜𝑖 ∈ [0, 1[𝑛
are representatives of the angular variables 𝑡𝑖 , 𝜃𝑖 , and [𝜔] ∈ ℤ𝑛 is the component-wise
integral part of 𝜔. Note that 𝜃(𝑡˜1 ) = 𝜃1 and 𝜃(𝑡˜2 + 𝑇 ) = 𝜃2 , hence
∫ 𝑡˜2 +𝑇
˙
˙ + 𝛼(𝑐)𝑑𝑡
𝑢(𝑡2 , 𝜃2 ) − 𝑢(𝑡1 , 𝜃1 ) ⩽
𝐿(𝑡, 𝜃(𝑡), 𝜃(𝑡))
− 𝑐 ⋅ 𝜃(𝑡)
⩽
⩽
∫
∫
∫
𝑡˜1
𝑡˜2 +𝑇
𝑡˜1
𝑡˜2 +𝑇
𝑡˜1
𝑡˜2 +𝑇
𝐴∥𝜃˙ − ∂𝐻0 (𝑐)∥2 /2 + 𝜖𝑍ˆ𝑐 (𝜃𝑠 (𝑡)) + 𝜖𝜂𝑑𝑡
2𝐴𝑛
+ 𝜖𝑟12 /2 + 𝜖𝜂𝑑𝑡
2
˜
˜
(𝑇 + 𝑡2 − 𝑡1 )
2𝐴𝑛
+ 𝜖𝑟12 𝑑𝑡
2
˜
˜
(𝑇 + 𝑡2 − 𝑡1 )
𝑡˜1
2𝐴𝑛
⩽
+ (𝑇 + 𝑡˜2 − 𝑡˜1 )𝜖𝑟12 .
˜
˜
(𝑇 + 𝑡2 − 𝑡1 )
⩽
This inequality holds for all 𝑇 ∈ ℕ, in particular, we can choose 𝑇 ∈ ℕ so that
√
√
𝑛𝐴
𝑛𝐴
⩽ 𝑇 + 𝑡˜2 − 𝑡˜1 ⩽ 2
2
𝜖𝑟1
𝜖𝑟12
and obtain
√
𝑢(𝑡2 , 𝜃2 ) − 𝑢(𝑡1 , 𝜃1 ) ⩽ 4𝑟1 𝑛𝐴𝜖.
□
Up to now, we used that ∥𝑍∥𝐶 2 ⩽ 1, but we used no information on the shape of
𝑍. Now we use properties [G1’] and [G2’] to prove Proposition 5.5.
5.2.1. The single peak case. This concerns the first statement of Proposition 5.5,
where
𝑐 = 𝑝∗ (𝑐𝑓 ), 𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏].
By [G1’] the function 𝑍(𝜃𝑠 , 𝑐) as a single peak at 𝜃𝑗𝑠 , as a consequence
𝑍ˆ𝑐 (𝜃𝑠 ) ≤ −𝑏∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐𝑓 )∥2 .
Let 𝜃(𝑡) : ℝ −→ 𝕋𝑚 be a curve calibrated by 𝑢. Then the function
˙
˙ + 𝛼(𝑐)
𝑡 ?−→ 𝐿(𝑡, 𝜃(𝑡), 𝜃(𝑡))
− 𝑐 ⋅ 𝜃(𝑡)
is integrable. Since
𝐿(𝑡, 𝜃, 𝑣) ⩾ −𝜖𝑍ˆ𝑐 (𝜃𝑠 ) − 𝜖𝜂 ⩾ 𝜖𝑏𝑟12 − 𝜖𝜂 ⩾ 𝜖𝑏𝑟12 /4
Arnold diffusion along normally hyperbolic invariant cylinders
27
if 𝜃𝑠 does not belong to 𝐷(𝜃𝑗𝑠 , 𝑟1 ), we conclude that the set of times 𝑡 for which 𝜃𝑠 (𝑡)
does not belong to 𝐷𝑐 (𝑟1 ) has finite measure, and is an open set. Let ]𝑡1 , 𝑡2 [ be a
connected component of this open set of times. Then 𝜃𝑠 (𝑡1 ) and 𝜃𝑠 (𝑡2 ) belong to
𝐷(𝜃𝑗𝑠 , 𝑟1 ) hence
∫ 𝑡2
√
˙
˙ + 𝛼(𝑐)𝑑𝑡 = 𝑢(𝑡2 , 𝜃(𝑡2 )) − 𝑢(𝑡1 , 𝜃(𝑡1 )) ⩽ 4𝑟1 𝑛𝐴𝜖.
𝐿(𝑡, 𝜃(𝑡), 𝜃(𝑡))
− 𝑐 ⋅ 𝜃(𝑡)
𝑡1
Now let 𝑟0 be the maximum of the distance ∥𝜃𝑠 (𝑡) − Θ𝑠 (𝑐)∥, assume that 𝑟0 ⩾ 2𝑟1 .
Let 𝑡4 be the smallest solution of the equation ∥𝜃𝑠 (𝑡) − Θ𝑠 (𝑐)∥ = 𝑟0 in ]𝑡1 , 𝑡2 [, and let
𝑡3 < 𝑡1 be the greatest solution of the equation ∥𝜃𝑠 (𝑡) − Θ𝑠 (𝑐)∥ = 𝑟0 /2 in [𝑡1 , 𝑡4 ]. Note
that
∫ 𝑡4
√
˙
˙ + 𝛼(𝑐)𝑑𝑡 ⩽ 4𝑟1 𝑛𝐴𝜖
𝐿(𝑡, 𝜃(𝑡), 𝜃(𝑡))
− 𝑐 ⋅ 𝜃(𝑡)
𝑡3
because the integrand is positive on ]𝑡1 , 𝑡2 [. We conclude that
∫ 𝑡4
√
∥𝜃˙𝑠 (𝑡)∥2 /(2𝐴) + 𝜖𝑏𝑟02 − 𝜖𝜂𝑑𝑡 ⩽ 4𝑟1 𝑛𝐴𝜖,
𝑡3
and, using the Cauchy-Schwartz inequality, that
(11)
(∫ 𝑡 4
)2
∫ 𝑡4
1
𝜖𝑏𝑟02
𝑠
2
2
𝑠
∥𝜃˙ (𝑡)∥ /(2𝐴) + 𝜖𝑏𝑟0 − 𝜖𝜂𝑑𝑡 ⩾
∥𝜃˙ (𝑡)∥𝑑𝑡 +
(𝑡4 − 𝑡3 )
2𝐴(𝑡4 − 𝑡3 )
2
𝑡3
𝑡3
√
𝑟02
𝜖𝑏𝑟02
𝑟02 𝑏𝜖
⩾
+
(𝑡4 − 𝑡3 ) ⩾ √ .
2𝐴(𝑡4 − 𝑡3 )
2
2 𝐴
Finally, we obtain
√
√
𝑟02 𝑏𝜖
√ ⩽ 4𝑟1 𝑛𝐴𝜖.
2 𝐴
or equivalently
√
√
8𝐴 2𝑛 √
2
𝑟0 ⩽ 𝑟1 8𝐴 𝑛/𝑏 =
𝜂.
𝑏
5.2.2. Double peak case. We now turn to the case of
𝑐 = 𝑝∗ (𝑐𝑓 ),
𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏],
where the function 𝑍(𝜃𝑠 , 𝑐) has two potential maxima. It follows from [G2’] that
(
)2
𝑠
𝑍ˆ𝑐 (𝜃𝑠 ) ≤ −𝑏 min{∥𝜃𝑠 − 𝜃𝑗𝑠 ∥, ∥𝜃𝑠 − 𝜃𝑗+1
∥} .
𝑠
Let 𝜃0 = (𝜃0𝑠 , 𝜃0𝑓 ) ∈ 𝒜𝑁𝜖 (𝑐) be where the function (of 𝜃) min{∥𝜃𝑠 − 𝜃𝑗𝑠 ∥, ∥𝜃𝑠 − 𝜃𝑗+1
∥}
achieves its maximum. This is possible since 𝒜𝑁𝜖 (𝑐) is a compact set. Since ℎ(𝜃0 , 𝜃0 ) =
28
P. Bernard, V. Kaloshin, K. Zhang
0, there exists an increasing sequence of integers 𝑛𝑘 and absolutely continuous curves
𝜃𝑘 : ℝ −→ 𝕋𝑛 satisfying 𝜃𝑘 (0) = 𝜃0 and 𝜃𝑘 (𝑡 + 𝑛𝑘 ) = 𝛾𝑘 (𝑡), and
∫ 𝑛𝑘
lim
𝐿(𝑡, 𝜃𝑘 (𝑡), 𝜃˙𝑘 (𝑡)) − 𝑐 ⋅ 𝜃𝑘 (𝑡) + 𝛼(𝑐)𝑑𝑡 = 0.
𝑘−→∞
0
𝑠
/ 𝐷(𝜃𝑗𝑠 , 𝑟1 )∪𝐷(𝜃𝑗+1
, 𝑟1 ), we conclude
Similar to the first case, 𝐿(𝑡, 𝜃, 𝑣) ⩾ 𝜖𝑏𝑟12 /4 for 𝜃𝑠 ∈
𝑠
𝑠
that for sufficiently large 𝑘, 𝜃𝑘 (ℝ) must intersect 𝐷(𝜃𝑗 , 𝑟1 ) ∪ 𝐷(𝜃𝑗+1 , 𝑟1 ). Without loss
of generality, we may assume that it intersect 𝐷(𝜃𝑗𝑠 , 𝑟1 ).
Let 𝑡1 = min{𝑡1 < 0, 𝜃𝑘 (𝑡1 ) ∈ 𝐷(𝜃𝑗𝑠 , 𝑟1 )} and 𝑡2 = min{0 < 𝑡2 , 𝜃𝑘 (𝑡2 ) ∈ 𝐷(𝜃𝑗𝑠 , 𝑟1 )}.
We first study the action of 𝜃𝑘 on the interval [0, 𝑡2 ]. If 𝜃𝑘 ([0, 𝑡2 ]) does not intersect
𝑠
𝐷(𝜃𝑗+1
, 𝑟1 ), write 𝑡3 = 𝑡4 = 𝑡2 , otherwise, Write 𝑡3 = min{0 < 𝑡3 ≤ 𝑡2 , 𝛾(𝑡3 ) ∈
𝑠
𝑠
𝐷(𝜃𝑗+1
, 𝑟1 )} and 𝑡4 = max{𝑡3 ≤ 𝑡4 ≤ 𝑡2 , 𝛾(𝑡4 ) ∈ 𝐷(𝜃𝑗+1
, 𝑟1 )}.
𝑠
We still use 𝑟0 to denote the maximal distance min{∥𝜃0𝑠 − 𝜃𝑗𝑠 ∥, ∥𝜃0𝑠 − 𝜃𝑗+1
∥}, assume
that 𝑟0 ≥ 2𝑏1 . Let 𝑡5 ∈ [0, 𝑡3 ] be the smallest solution in [0, 𝑡3 ] such that 𝑑(𝜃𝑠 (𝑡5 )) =
𝑟0 /2, then by the same calculation as in (11),
√
∫ 𝑡5
2
𝑏𝜖
𝑟
0
𝐿(𝑡, 𝜃𝑘 (𝑡), 𝜃˙𝑘 (𝑡)) − 𝑐 ⋅ 𝜃˙𝑘 (𝑡) + 𝛼(𝑐)𝑑𝑡 ⩾ √ .
2 𝐴
0
Furthermore for any weak KAM solution 𝑢(𝜃, 𝑡)
∫ 𝑡4
√
𝐿 − 𝑐 ⋅ 𝜃˙𝑘 (𝑡) + 𝛼(𝑐)𝑑𝑡 ⩾ 𝑢(𝑡4 , 𝜃𝑘 (𝑡4 )) − 𝑢(𝑡3 , 𝜃𝑘 (𝑡3 )) ≥ −4𝑟1 𝑛𝐴𝜖,
𝑡3
while the integrand is nonnegative on both [𝑡5 , 𝑡3 ] and [𝑡4 , 𝑡2 ]. We conclude that
√
∫ 𝑡2
2
√
𝑟
𝜖
0
𝐿 − 𝑐 ⋅ 𝜃˙𝑘 (𝑡) + 𝛼(𝑐)𝑑𝑡 ⩾ √ − 4𝑟1 𝑛𝐴𝜖.
2 𝐴
0
The same estimate can be made for the action on the interval [𝑡1 , 0]. In addition
∫ 𝑛𝑘 +𝑡1
√
𝐿 − 𝑐 ⋅ 𝜃˙𝑘 (𝑡) + 𝛼(𝑐)𝑑𝑡 ⩾ 𝑢(𝑛𝑘 + 𝑡1 , 𝜃𝑘 (𝑛𝑘 + 𝑡1 )) − 𝑢(𝑡2 , 𝜃𝑘 (𝑡2 )) ≥ −4𝑟1 𝑛𝐴𝜖,
𝑡2
note that 𝑡2 < 𝑛𝑘 + 𝑡1 and that 𝜃𝑘 is 𝑛𝑘 periodic.
Finally we conclude that
√
∫ 𝑛𝑘 +𝑡1
2
√
𝑟
𝑏𝜖
𝐿 − 𝑐 ⋅ 𝜃˙𝑘 + 𝛼(𝑐)𝑑𝑡 ⩾ 0√ − 12𝑏1 𝑛𝐴𝜖.
2 𝐴
𝑡1
Let 𝑘 −→ ∞, the integral on the left hand side approaches 0. We obtain
√
√
𝑟02 𝑏𝜖
√ ⩽ 12𝑏1 𝑛𝐴𝜖
2 𝐴
Arnold diffusion along normally hyperbolic invariant cylinders
and
𝑟02
√
24𝐴 𝑛 √
𝜂.
≤
𝑏
29
√
√
We choose 𝜖0 sufficiently small such that 𝜂 ≤ 2∥𝐻0 ∥𝐶0 , choose 𝜅 = 48𝐴𝜆 2𝑛 and
verify that we have proved the statements of Proposition 5.5 in both cases.
□
Before moving on we point out that the estimates in the double peak case indeed
implies that the curves 𝜃𝑘 , which are not calibrated, can be localized in the limit of
𝑘 −→ ∞. We state it in the following lemma for future use.
Lemma 5.6. For the double peak case, i.e. 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏],
consider any 𝜃0 ∈ 𝒜𝑁𝜖 (𝑐), let 𝑛𝑘 be an increasing sequence of integers, 𝜃𝑘 = (𝜃𝑘𝑠 , 𝜃𝑘𝑓 ) :
ℝ −→ 𝑀 be a sequence of 𝑛𝑘 −periodic absolute continuous curves such that 𝛾(0) = 𝜃0
and
∫ 𝑛𝑘
lim
𝐿(𝑡, 𝜃𝑘 , 𝜃˙𝑘 ) − 𝑐 ⋅ 𝜃˙𝑘 + 𝛼(𝑐)𝑑𝑡 = 0,
𝑘−→∞
0
then there exists 𝐾 ∈ ℕ such that for all 𝑘 > 𝐾
1
𝑠
max min{∥𝜃𝑘𝑠 (𝑡) − 𝜃𝑗𝑠 ∥, ∥𝜃𝑘𝑠 (𝑡) − 𝜃𝑗+1
∥} < 2𝜅𝛿 4 .
𝑡∈ℝ
𝑠
Proof. Fix a curve 𝜃𝑘 , write 𝑑(𝜃𝑘 (𝑡)) = min{∥𝜃𝑘𝑠 (𝑡) − 𝜃𝑗𝑠 ∥, ∥𝜃𝑘𝑠 (𝑡) − 𝜃𝑗+1
∥}. Let 𝜏 be
where the maximum of 𝑑(𝜃𝑘 (𝑡)) is reached. Consider the shifted curve 𝜃𝑘′ (𝑡) = 𝜃𝑘 (𝑡−𝜏 ),
the arguments in section 5.2.2 go through with 𝜃𝑘 replaced with 𝜃𝑘′ and 𝑟0 replaced
by max 𝑑(𝜃𝑘 (𝑡)). We have that
1
lim max 𝑑(𝜃𝑘 (𝑡)) ≤ 𝜅𝛿 4
𝑘−→∞ 𝑡∈ℝ
and the lemma follows.
□
5.3. Proof of Theorem 5.1. Proposition 5.3 provides the vertical part of the lo1
calization, while Proposition 5.5 provides the horizontal localization, with 𝜌2 = 𝜅𝛿 4 .
Clearly we can choose 𝛿0 small enough such that 𝜌2 < 𝜌1 .
□
5.4. Proof of Theorem 5.2. For the first case, where 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 +
𝑏, 𝑎𝑗+1 −𝑏]. For a sufficiently small choice of 𝜖0 , Theorem 5.1 implies that 𝒩𝑁𝜖 (𝑐) ⊂ 𝑉𝑗 ,
where 𝑉𝑗 = {(𝜃, 𝑝, 𝑡); 𝑝𝑓 ∈ [𝑎𝑗 − 𝑏, 𝑎𝑗+1 + 𝑏], ∥(𝜃𝑠 , 𝑝𝑠 ) − (𝜃𝑗𝑠 , 𝑝𝑠∗ )∥ ≤ 𝜌1 } was defined in
Theorem 4.1. Since 𝑉𝑗 is maximally invariant and that 𝒩𝑁𝜖 (𝑐) is an invariant set, we
conclude that 𝒩𝑁𝜖 (𝑐) ⊂ 𝑋𝑗 .
For the second case, where 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏], we can similarly
claim that 𝒜𝑁𝜖 (𝑐) ⊂ 𝑉𝑗 ∪ 𝑉𝑗+1 , moreover 𝒜𝑁𝜖 (𝑐) ∩ 𝑉𝑗 and 𝒜𝑁𝜖 (𝑐) ∩ 𝑉𝑗+1 must both be
invariant, and hence 𝒜𝑁𝜖 (𝑐) ∩ 𝑉𝑗 ⊂ 𝑋𝑗 and 𝒜𝑁𝜖 (𝑐) ∩ 𝑉𝑗+1 ⊂ 𝑋𝑗+1 .
30
P. Bernard, V. Kaloshin, K. Zhang
In order to prove the projection part of Theorem 5.2, let us consider a Weak KAM
solution 𝑢 of 𝑁𝜖 at cohomology 𝑐. Let (𝑡𝑖 , 𝜃𝑖 , 𝑝𝑖 ), 𝑖 = 1, 2 be two points in ℐ̃(𝑢, 𝑐). We
shall denote by the same symbol 𝜅 various different constants which are independent
of 𝛿 and 𝜖. By Proposition 5.3, we have
√
√
∥𝑝2 − 𝑝1 ∥ ⩽ 36𝐴 𝜖∥𝜃2 − 𝜃1 ∥ ⩽ 36𝐴 𝜖(∥𝜃2𝑓 − 𝜃1𝑓 ∥ + ∥𝜃2𝑠 − 𝜃1𝑠 ∥).
Assume that these two points belong to one of the NHICs 𝑋𝑗 , we also have
√
∥𝜃2𝑠 − 𝜃1𝑠 ∥ ⩽ (𝜅𝛿/ 𝜖)(∥𝜃2𝑓 − 𝜃1𝑓 ∥ + ∥𝑝2 − 𝑝1 ∥).
We get
√
∥𝑝2 − 𝑝1 ∥ ⩽ 𝜅 𝜖∥𝜃2𝑓 − 𝜃1𝑓 ∥ + 𝜅𝛿∥𝑝2 − 𝑝1 ∥
thus, if 𝛿 is small enough,
√
∥𝑝2 − 𝑝1 ∥ ⩽ 𝜅 𝜖∥𝜃2𝑓 − 𝜃1𝑓 ∥
and then
√
∥𝜃2𝑠 − 𝜃1𝑠 ∥ ⩽ (𝜅𝛿/ 𝜖)∥𝜃2𝑓 − 𝜃1𝑓 ∥.
We have proved that the restriction to ℐ̃(𝑢, 𝑐) of the coordinate map 𝜃𝑓 has a Lipschitz
inverse.
Note that the Mañe set 𝒩˜𝑁𝜖 (𝑐), as well as the components of the Aubry set 𝒜˜𝑁𝜖 (𝑐)∩
˜ 𝑐), since we have just proved that
𝑉𝑗 and 𝒜˜𝑁𝜖 (𝑐)∩𝑉𝑗+1 are both contained in some 𝐼(𝑢,
𝑓
they belong to NHIC, their projection to the 𝜃 component has a Lipshitz inverse. □
6. Variational Construction
More detailed information on these sets can be obtained, if we are allowed to make
an additional perturbation to avoid degenerate situations.
Theorem 6.1. Let 𝑁𝜖 = 𝐻0 +𝜖𝑍 +𝜖𝑅 be such that 𝑍 satisfy the genericity conditions
[G0]-[G2] and that the parameters 𝜖 and 𝛿 is such that Theorem ?? applies. Then
there exists arbitrarily small 𝐶 𝑟 perturbation 𝜖𝑅′ of 𝜖𝑅, such that the following hold
for 𝑁𝜖′ = 𝐻0 + 𝜖𝑍 + 𝜖𝑅′ :
∪
(1) There exists a partition of [𝑎− , 𝑎+ ] into 𝑙−1
𝑎𝑗 , 𝑎
¯𝑗+1 ], which is a refinement
𝑗=0 [¯
of the partition {[𝑎𝑖 , 𝑎𝑖+1 ]}. Each [¯
𝑎𝑗 , 𝑎
¯𝑗+1 ] still corresponds to an invariant
cylinder 𝑋𝑗 . We have that for 𝑐𝑓 ∈ (¯
𝑎𝑗 , 𝑎
¯𝑗+1 ), the Aubry set 𝒜˜𝑁𝜖′ (𝑐) is con𝑓
˜
tained in 𝑋𝑗 ; for 𝑐 = 𝑎
¯𝑗+1 , 𝒜𝑁𝜖′ (𝑐) has nonempty component in both 𝑋𝑗 and
𝑋𝑗+1 , if 𝑋𝑗 ∕= 𝑋𝑗+1 .
(2) The sets 𝒜˜𝑁𝜖′ (𝑐) ∩ 𝑋𝑗 , when nonempty, contains a unique minimal invariant
probability measure. In particular, this implies that 𝒜˜𝑁𝜖′ (𝑐) = 𝒩˜𝑁𝜖′ (𝑐) for 𝑐𝑓 ∕=
𝑎
¯𝑗 for any 𝑗.
Arnold diffusion along normally hyperbolic invariant cylinders
31
(3) An immediate consequence of part (2) is the following dichotomy, for 𝑐𝑓 ∕= 𝑎
¯𝑗 ,
𝑗 = 1, ⋅ ⋅ ⋅ , 𝑙, one of the two holds.
(a) 𝒜𝑐 = 𝒩𝑐 and 𝜋𝜃𝑓 𝒜𝑐 = 𝕋. In this case, 𝒜𝑐 is an invariant circle.
(b) 𝜋𝜃𝑓 𝒩𝑐 ⊊ 𝕋.
Using the information obtained from the normal form system 𝑁𝜖 , we now return to
the original Hamiltonian 𝐻𝜖 . Using the symplectic invariance of the Mather, Aubry
and Mañe set developed in [Be2], we have that the same conclusion as in Theorem 5.2
and Theorem 6.1 can be drawn about 𝐻𝜖 .
Theorem 6.2. Let 𝐻𝜖 = 𝐻0 + 𝜖𝐻1 such that the resonant component of 𝐻1 satisfy
conditions [G0]-[G2]. There exists 𝜖0 > 0 and an interval [𝑎− , 𝑎+ ] ⊂ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ]
depending only on 𝐻0 and 𝐻1 , and for each 0 < 𝜖 < 𝜖0 there exists arbitrarily small
𝐶 𝑟 perturbation 𝐻𝜖′ of 𝐻𝜖 , such that the conclusions of Theorem 5.2 and Theorem 6.1
holds for the Hamiltonian 𝐻𝜖′ at cohomologies 𝑐 = 𝑝∗ (𝑐𝑓 ), where 𝑐𝑓 ∈ [𝑎− , 𝑎+ ].
These information on the Mañé set allow to use the variational mechanism of [Be1].
Let us denote by Γ(𝜖) the set of cohomology classes 𝑐 ∈ Γ such that 𝑐𝑓 ∈ [𝑎− , 𝑎+ ]. We
would like to prove that each cohomology 𝑐 ∈ Γ(𝜖) is in the interior of its forcing class
in the terminology of [Be1], which implies that all the cohomology classes in Γ(𝜖) are
contained in a single forcing class. By proposition 5.3 in [Be1], we could conclude
the existence of an orbit (𝜃(𝑡), 𝑝(𝑡)) of 𝐻𝜖 such that 𝑝(0) = 𝑐 and 𝑝(𝑇 ) = 𝑐′ for some
𝑇 ∈ ℕ. Note that this implies the existence of various more complicated orbits, see
[Be1].
In order to carry out this program, we denote by Γ0 (𝜖) the set of cohomology classes
𝑐 ∈ Γ(𝜖) such that the set 𝜃𝑓 (ℐ̃(𝑐, 𝑢)) is properly contained in 𝕋 for each weak KAM
solution 𝑢 at level 𝑐. By Theorem 0.11 in [Be1], each cohomology 𝑐 ∈ Γ0 (𝜖) is in the
interior of its forcing class.
Let Γ2 (𝜖) denote that set of 𝑐 ∈ Γ(𝜖) such that the Aubry set 𝒜(𝑐) has exactly two
static classes. In this case the Mañe set 𝒩 (𝑐) ⊋ 𝒜(𝑐). To ensure that 𝑐 ∈ Γ2 (𝜖) is
in the interior of its forcing class, some further degeneracy conditions are needed. To
be more specific, let Γ∗2 (𝑐) denote the set of 𝑐 ∈ Γ2 (𝜖) such that the set
𝒩 (𝑐) − 𝒜(𝑐)
is totally disconnected. This can also be stated in terms of barrier function. Let 𝜃0
and 𝜃1 be contained in each of the two static classes of 𝒜(𝑐), we define
𝑏+
𝑐 (𝜃) = ℎ𝑐 (𝜃0 , 𝜃) + ℎ𝑐 (𝜃, 𝜃1 )
and
˜
𝑏−
𝑐 (𝜃) = ℎ𝑐 (𝜃1 , 𝜃) + ℎ𝑐 (𝜃, 𝜃0 ),
32
P. Bernard, V. Kaloshin, K. Zhang
where ℎ𝑐 is the Peierls barrier for cohomology class 𝑐. Then Γ∗2 (𝜖) is the set of 𝑐 ∈ Γ2 (𝜖)
such that the minima of each 𝑏+ and 𝑏− outside of 𝒜(𝑐) are totally isolated.
We call Γ1 (𝜖) the set of cohomology classes 𝑐 ∈ Γ such that there exists only one
weak KAM solution 𝑢 at level 𝑐, and 𝜃𝑓 (ℐ̃(𝑐, 𝑢)) = 𝕋. Note then that
˜ = ℐ̃(𝑐, 𝑢)
𝒩˜ (𝑐) = 𝒜(𝑐)
is an invariant circle. We have Γ0 (𝜖) ∩ Γ1 (𝜖) = ∅ for each 𝜖 ∈]0, 𝜖0 [, by definition. We
first consider the covering
𝜉 : 𝕋𝑛 −→ 𝕋𝑛
𝑠
𝑠
𝜃 = (𝜃𝑓 , 𝜃1𝑠 , 𝜃2𝑠 , ⋅ ⋅ ⋅ , 𝜃𝑛−1
) ?−→ 𝜉(𝜃) = (𝜃𝑓 , 2𝜃1𝑠 , 𝜃2𝑠 , ⋅ ⋅ ⋅ , 𝜃𝑛−1
).
This covering lifts to a a symplectic covering
Ξ : 𝑇 ∗ 𝕋𝑛 −→ 𝑇 ∗ 𝕋𝑛
(𝜃, 𝑝) = (𝜃, 𝑝𝑓 , 𝑝𝑠1 , 𝑝𝑠2 , . . . , 𝑝𝑠𝑛−1 ) ?−→ Ξ(𝜃, 𝑝) = (𝜉(𝜃), 𝑝𝑓 , 𝑝𝑠1 /2, 𝑝𝑠2 , . . . , 𝑝𝑠𝑛−1 ),
˜ = 𝐻 ∘ Ξ. It is known that
and we define the Lifted Hamiltonian 𝐻
(
)
𝒜˜𝐻˜ (˜
𝑐) = Ξ−1 𝒜˜𝐻˜ (𝑐)
where 𝑐˜ = 𝜉 ∗ 𝑐 = (𝑐𝑓 , 𝑐𝑠1 /2, 𝑐𝑠2 , . . . , 𝑐𝑠𝑛−1 ). On the other hand, the inclusion
)
(
)
(
𝒩˜𝐻˜ (˜
𝑐) ⊃ Ξ−1 𝒩˜𝐻˜ (𝑐) = Ξ−1 𝒜˜𝐻˜ (𝑐)
is not an equality for 𝑐 ∈ Γ1 (𝜖). More precisely, for 𝑐 ∈ Γ1 (𝜖), the set 𝒜˜𝐻˜ (˜
𝑐) is
the union of two circles, while 𝒩˜𝐻˜ (˜
𝑐) contains heteroclinic connections between these
circles. Similarly to the case of Γ2 (𝜖), we call Γ∗1 (𝜖) the set of cohomologies 𝑐 ∈ Γ1 (𝜖)
such that the set
𝒩𝐻˜ (˜
𝑐) − 𝒜𝐻˜ (˜
𝑐)
is totally disconnected. Alternatively, we can chose a point 𝜃0 in the projected Aubry
set 𝒜(𝑢, 𝑐) of 𝐻, and consider its two preimages 𝜃˜0 and 𝜃˜1 under 𝜉. We define
˜𝑏+ (𝜃) = ℎ̃(𝜃˜0 , 𝜃) + ℎ̃(𝜃, 𝜃˜1 )
𝑐
and
˜𝑏− (𝜃) = ℎ̃(𝜃˜1 , 𝜃) + ℎ̃(𝜃, 𝜃˜0 )
𝑐
˜ Γ∗ (𝜖) is then the set of cohomologies
where ℎ̃ is the Peierl’s barrier associated to 𝐻.
1
𝑐 ∈ Γ1 (𝜖) such that the minima of each of the functions 𝑏±
𝑐 located outside of the
Aubry set 𝒜𝐻˜ (˜
𝑐) are isolated.
The following theorem is proved in [Be1].
Arnold diffusion along normally hyperbolic invariant cylinders
33
Theorem 6.3. If 𝑐 and 𝑐′ belong to the same connected component of Γ0 (𝜖) ∪ Γ∗1 (𝜖) ∪
Γ∗2 (𝜖), then there exists an orbit (𝜃(𝑡), 𝑝(𝑡)) and of 𝐻𝜖 a time 𝑇 ∈ ℕ such that 𝑝(0) = 𝑐
and 𝑝(𝑇 ) = 𝑐′ .
We have proved the main result provided Γ(𝜖) = Γ0 (𝜖) ∪ Γ∗1 (𝜖) ∪ Γ∗ (𝜖).
Theorem 6.4. Let 𝐻𝜖 be a Hamiltonian such that Theorem 6.2 holds, then there
exists an arbitrarily small 𝐶 𝑟 perturbation 𝐻𝜖′′ to 𝐻𝜖′ , such that for the Hamiltonian
𝐻𝜖′′ we have that Γ(𝜖) = Γ0 (𝜖) ∪ Γ∗1 (𝜖) ∪ Γ∗2 (𝜖).
We note that the conclusions of Theorem 6.1 already implies that {𝑐 ∈ Γ(𝜖), 𝑐𝑓 ∕=
𝑎
¯𝑗 , 𝑗 = 2, ⋅ ⋅ ⋅ 𝑙 − 1} ⊂ Γ0 (𝜖) ∪ Γ1 (𝜖), while {𝑐 ∈ Γ(𝜖), 𝑐𝑓 = 𝑎
¯𝑗 , 𝑗 = 2, ⋅ ⋅ ⋅ 𝑙 − 1} ⊂ Γ2 (𝜖).
In other words, Γ(𝜖) = Γ0 (𝜖) ∪ Γ1 (𝜖) ∪ Γ2 (𝜖). It suffices to prove that Γ1 (𝜖) = Γ∗1 (𝜖)
and Γ2 (𝜖) = Γ∗2 (𝜖).
For the rest of this section, we prove Theorem 6.1 and Theorem 6.4.
6.1. Local extension of 𝛼(𝑐) and 𝒜𝑁𝜖 (𝑐). Consider the normal form system 𝑁𝜖
and pick 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎− , 𝑎+ ]. For such a 𝑐 the function 𝑍(𝜃𝑠 , 𝑐) has a single
peak. It follows from Theorem 5.2 that the Aubry set 𝒜˜𝑁𝜖 (𝑐) (which is a subset of
𝒩˜𝑁𝜖 (𝑐)) is contained in a single NHIC 𝑋𝑗 and the projected graph theorem holds.
For the rest of the cohomology classes, the double peak case, the picture is less clear
as 𝒜˜𝑁𝜖 (𝑐) are contained in the union of two NHICs. To get a more precise picture,
we will locally extend the set function (of 𝑐𝑓 )
𝒜˜𝑁𝜖 ((𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ))∣[𝑎 +𝑏,𝑎 −𝑏]
𝑗
𝑗+1
from [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏] to [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏]. The extended local Aubry set will still
be contained in 𝑋𝑗 . These definitions are inspired by Mather’s definitions of relative
𝛼−function and Aubry set.
Let
𝑠
𝜌0 =
min
∥𝜃𝑗𝑠 (𝑝𝑓 ) − 𝜃𝑗+1
(𝑝𝑓 )∥/3.
𝑝𝑓 ∈[𝑎𝑗+1 −2𝑏,𝑎𝑗+1 +2𝑏]
It follows from properties [G1’] and [G2’] for 𝑝𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏],
𝑍(𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) − 𝑍(𝜃𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) ≥ 𝑏∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2
for ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥ ≤ 𝜌0 . By choosing a smaller 𝛿 if necessary, we may make sure 𝜌0 > 𝜌1 ,
where 𝜌1 was defined in Theorem 4.1. We write 𝑈𝑗 (𝑝) = {∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝)∥ ≤ 𝜌0 }, our
choice of 𝜌0 guarantees that 𝑈𝑗 (𝑝) ∩ 𝑈𝑗+1 (𝑝) = ∅ for 𝑝𝑓 ∈ [𝑎𝑗+1 − 2𝑏, 𝑎𝑗+1 + 2𝑏]. To
define the extension, we introduce the following modification of the Hamiltonian 𝑁𝜖 .
Let 𝑍𝑗 (𝜃𝑠 , 𝑝) be a function 𝕋𝑛−1 × ℝ𝑛 −→ ℝ satisfying the following properties:
∙ There exists 𝐶 depending only on 𝜆, ∥𝑍∥𝐶 2 and 𝑛 such that ∥𝑍𝑗 ∥𝐶 2 ≤ 𝐶.
∙ 𝑍𝑗 (𝜃𝑠 , 𝑝) = 𝑍(𝜃𝑠 , 𝑝) whenever ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝)∥ ≤ 𝜌0 .
34
P. Bernard, V. Kaloshin, K. Zhang
∙ 𝑍𝑗 (𝜃𝑠 , 𝑝) ≤ 𝑍(𝜃𝑠 , 𝑝) for all 𝜃𝑠 and 𝑝.
∙ For 𝑝𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏], we have that 𝑍𝑗 (𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) − 𝑍𝑗 (𝜃𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) ≥
𝑏
∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2 hold for all 𝜃𝑠 ∈ 𝕋𝑛−1 .
2
To see that such a modification exists, let 𝜌¯ > 𝜌0 be such that 𝑍𝑗 (𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) −
𝑍𝑗 (𝜃𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) ≥ 2𝑏 ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2 on ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥ ≤ 𝜌. How large 𝜌¯ − 𝜌0 is depends only
on 𝜆 and ∥𝑍∥𝐶 2 . Let 𝑄 : 𝕋𝑛−1 × ℝ𝑛 −→ ℝ be a smooth function such that
𝑍(𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) − 𝑄 = 2𝑏 ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2 for ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥ ≤ 𝜌¯ and 𝑍(𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) − 𝑄 ≥ 2𝑏 ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2
for all 𝑝𝑓 and 𝜃𝑠 . The norm of 𝑄 only depends on 𝜆 and 𝑛. Let 𝜑𝜌0 ,¯𝜌 : 𝕋𝑛 × ℝ𝑛 −→ ℝ
be a smooth function such that 𝜑𝜌0 ,¯𝜌 = 1 on {𝜃𝑠 , ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥ ≤ 𝜌0 } and 𝜑𝜌0 ,¯𝜌 = 0 on
{𝜃𝑠 , ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥ > 𝜌¯}. The norm of 𝜑𝜌0 ,¯𝜌 depends only on 𝑛, 𝜌0 and 𝜌¯. Then we can
choose
𝑍𝑗 = (1 − 𝜑𝜌0 ,¯𝜌 )𝑍 + 𝜑𝜌0 ,¯𝜌 𝑄.
We write 𝑁𝜖,𝑗 = 𝐻0 + 𝜖𝑍𝑗 + 𝜖𝑅.
For each 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏] we define
𝛼𝑗 (𝑐) = 𝛼𝑁𝜖,𝑗 (𝑐),
𝒜𝑁𝜖 ,𝑗 (𝑐) = 𝐴˜𝑁𝜖,𝑗 (𝑐).
It is not clear that these definitions are independent of the choice of the modification
𝑍𝑗 or the decomposition 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅. We resolve these questions, and provide
some more properties of these definition in the following proposition.
Proposition 6.5. Let 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 be a Hamiltonian satisfying the genericity
conditions [G0]-[G2] and that ∥𝑅∥𝐶 2 ≤ 𝛿. There exists 𝜖0 , 𝛿0 > 0 such that for
0 < 𝜖 < 𝜖0 and 0 < 𝛿 < 𝛿0 the following hold.
(1) The definitions of 𝛼𝑗 and 𝒜˜𝑁𝜖 ,𝑗 (𝑐) are independent of the decomposition 𝑁𝜖 =
𝐻0 + 𝜖𝑍 + 𝜖𝑅 as long as 𝑍 satisfies [G0]-[G2] and ∥𝑅∥𝐶 2 ≤ 𝛿; the definitions
are also independent of the modification 𝑍𝑗 , as long as it satisfies the same 4
bullet point properties.
(2) For each 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏], we have the local Aubry set
𝒜˜𝑁𝜖 ,𝑗 (𝑐) is contained in the set {∥𝜃𝑠 −𝜃𝑗𝑠 ∥ ≤ 𝜌2 } where 𝜌2 is as in Theorem 5.1.
It follows that 𝒜˜𝑁𝜖 ,𝑗 (𝑐) ⊂ 𝑋𝑗 and 𝜋𝜃𝑓 ∣𝒜˜𝑁𝜖 ,𝑗 (𝑐) is one-to-one with Lipshitz
inverse.
(3) For 𝑐 = 𝑝∗ (𝑐𝑓 ), 𝛼(𝑐) = 𝛼𝑗 (𝑐) if 𝑐𝑓 ∈ [𝑎𝑗 +𝑏, 𝑎𝑗+1 −𝑏]; 𝛼(𝑐) = max{𝛼𝑗 (𝑐), 𝛼𝑗+1 (𝑐)}
if 𝑐𝑓 ∈ (𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏). In particular, 𝛼𝑗 (𝑐) > 𝛼𝑗+1 (𝑐) for 𝑐𝑓 = 𝑎𝑗+1 − 𝑏
and 𝛼𝑗+1 (𝑐) > 𝛼𝑗 (𝑐) for 𝑐𝑓 = 𝑎𝑗+1 + 𝑏.
(4) For any 𝑐𝑓 ∈ [𝛼𝑗+1 − 𝑏, 𝛼𝑗+1 + 𝑏], if 𝛼(𝑐) = 𝛼𝑗 and 𝛼(𝑐) ∕= 𝛼𝑗+1 (𝑐), then
𝒜˜𝑁𝜖 (𝑐) = 𝒜˜𝑁𝜖 ,𝑗 (𝑐). Similar statement hold with 𝑗 and 𝑗 + 1 exchanged.
Arnold diffusion along normally hyperbolic invariant cylinders
35
Proof. We will prove the second statement first. The modified Hamiltonian 𝑁𝜖,𝑗 is
such that the single peak case of Theorem 5.1 applies, with 𝑏 replaced by 𝑏/2. By
choosing a smaller 𝛿 if necessary, we can guarantee that 𝜌2 can be chosen the same as
in Theorem 5.1. Theorem 5.2 also applies, where we obtain the projection property.
We will now show that the set 𝒜˜𝑁𝜖 ,𝑗 (𝑐) depends only on the value of 𝑁𝜖 on the
set {(𝜃, 𝑝), ∥𝜃𝑠 − 𝜃𝑠 (𝑝)∥ ≤ 𝜌0 }, which will imply that the definition of 𝒜˜𝑁𝜖 ,𝑗 (𝑐) is
independent of decomposition or choice of the modification, since for all different
decompositions and modifications, the Hamiltonian agree on this neighborhood. As
∗
before, we denote by 𝑁𝜖∗ (𝜃, 𝑣, 𝑡) the Lagrangian corresponding to 𝑁𝜖 and 𝑁𝜖,𝑗
the
∗
Lagrangian corresponding to 𝑁𝜖,𝑗 . The projected Aubry set 𝒜𝑁𝜖 ,𝑗 (𝑐) is defined by
∗ ,𝑐 (𝜃, 𝜃) = 0, where the subscript is added to stress
the set of 𝜃 ∈ 𝕋𝑛 such that ℎ𝑁𝜖,𝑗
the Lagrangian and cohomology class in the definition. It follows from the second
∗ ,𝑐 (𝜃, 𝜃) = 0 must be contained
statement of the proposition that any 𝜃 such that ℎ𝑁𝜖,𝑗
𝑠
𝑠
in {∥𝜃 − 𝜃 (𝑐)∥ ≤ 𝜌2 }. The following lemma implies independence of the local Aubry
set on the docomposition or the choice of the modification.
¯𝜖,𝑗 = 𝐻0 + 𝜖𝑍¯𝑗 + 𝜖𝑅
¯ be such that
Lemma 6.1. Let 𝑁𝜖,𝑗 = 𝐻0 + 𝜖𝑍𝑗 + 𝜖𝑅 and 𝑁
¯𝜖,𝑗 for ∥𝜃𝑠 − 𝜃𝑠 (𝑝)∥ ≤ 𝜌0 .
∙ 𝑁𝜖,𝑗 = 𝑁
∙ For 𝑝𝑓 ∈ [𝑎𝑗 −2𝑏, 𝑎𝑗+1 +2𝑏], we have that 𝑍𝑗 (𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 )−𝑍𝑗 (𝜃𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) ≥ 2𝑏 ∥𝜃𝑠 −
𝜃𝑗𝑠 ∥2 and that 𝑍¯𝑗 (𝜃𝑗𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) − 𝑍¯𝑗 (𝜃𝑠 , 𝑝𝑠∗ , 𝑝𝑓 ) ≥ 2𝑏 ∥𝜃𝑠 − 𝜃𝑗𝑠 ∥2 for all 𝜃𝑠 ∈ 𝕋𝑛−1 .
¯ 𝐶 2 ≤ 𝛿.
∙ ∥𝑅∥𝐶 2 , ∥𝑅∥
Then for sufficiently small 𝜖, 𝛿, and for 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏]
∗ ,𝑐 (𝜃, 𝜃) = 0 ⇐⇒ ℎ ¯ ∗
ℎ𝑁𝜖,𝑗
𝑁𝜖,𝑗 ,𝑐 (𝜃, 𝜃) = 0.
Proof of Lemma 6.1. Let 𝜃0 ∈ 𝒜𝑁𝜖 ,𝑗 (𝑐), we refer to Lemma 5.6 before and note that
there exists an increasing sequence of integers 𝑛𝑘 , 𝜃𝑘 = (𝜃𝑘𝑠 , 𝜃𝑘𝑓 ) : ℝ −→ 𝑀 a sequence
of 𝑛𝑘 −periodic absolutely continuous curves such that 𝜃𝑘 (0) = 𝜃0 and
∫ 𝑛𝑘
∗
lim
𝑁𝜖,𝑗
(𝑡, 𝜃𝑘 , 𝜃˙𝑘 ) − 𝑐 ⋅ 𝜃˙𝑘 + 𝛼𝑗 (𝑐)𝑑𝑡 = 0.
𝑘−→∞
0
Moreover, the curves 𝜃𝑘 can be chosen to be minimizing, i.e. they minimizes the
integral in the above displayed formula among the 𝑛𝑘 −periodic absolutely continuous
curves such that 𝜃𝑘 (0) = 𝜃0 . In particular, 𝜃𝑘 ∣[0, 𝑛𝑘 ] must be trajectories of the EulerLagrange flow. Lemma 5.6 states that for sufficiently large 𝑘 we may assume that the
whole curve 𝜃𝑘 ’s are contained in ∥𝜃𝑠 − 𝜃𝑠 (𝑐)∥ ≤ 𝜌0 (choose a smaller 𝛿 if necessary).
Let (𝜃𝑘 , 𝑝𝑘 ) be √
the corresponding Hamiltonian
trajectory to (𝜃𝑘 , 𝜃˙𝑘 ), we will show
√
that for 𝑛𝑘 > 1/ 𝜖, ∥𝑝𝑘 (𝑡) − 𝑐∥ ≤ 𝐶 𝜖, where 𝐶 is a constant depending only on
𝐴 and 𝑛. Let 𝜏 ∈ [0, 𝑛𝑘 ] be where ∥𝑝𝑘 (𝑡)∥ takes its maximum. Consider a shift
36
P. Bernard, V. Kaloshin, K. Zhang
√
𝜃𝑘′ (𝑡) = 𝜃𝑘 (𝑡 + 𝜏 − 1/ 𝜖) of 𝜃𝑘 , and
let 𝑝′𝑘 be the corresponding
action variable, then
√
√
′
∥𝑝𝑘 ∥ reaches maximum at 𝑡 = 1/ 𝜖. We will write 𝑇 = 1/ 𝜖 in the rest of the proof.
Similar to the proof of Proposition 5.4, we lift 𝜃𝑘′ to a curve in ℝ𝑛 without changing
its name, and define
′
𝜃𝑘,𝑥
(𝑡) = 𝜃𝑘 (𝑥) + 𝑡𝑥/𝜏.
We have the following
∫ 𝑇
∫ 𝑇
∗
′
′
∗
𝑁𝜖,𝑗 (𝑡, 𝜃𝑘,𝑥 , 𝜃˙𝑘,𝑥 ) − 𝑐 ⋅ 𝜃˙𝑘 + 𝛼𝑗 (𝑐)𝑑𝑡 −
𝑁𝜖,𝑗
(𝑡, 𝜃𝑘′ , 𝜃˙𝑘′ ) − 𝑐 ⋅ 𝜃˙𝑘′ + 𝛼𝑗 (𝑐)𝑑𝑡
0
0
√
√
∗
⩽ (−𝑐 + ∂𝑣 𝑁𝜖,𝑗 (𝜏, 𝜃𝑘 (𝜏 ), 𝛿𝜃𝑘 (𝜏 ))) ⋅ 𝑥 + 3𝐴 𝜖∣𝑥∣2 = (−𝑐 + 𝑝′𝑘 (𝑇 )) ⋅ 𝑥 + 3𝐴 𝜖∣𝑥∣2 ,
the computation is identical to (7) and the two formulas that follows it. Assume
∥𝑝′𝑘 (𝑇 ) − 𝑐∥ > 0 (otherwise there is nothing to prove), and we choose 𝑥 to be a unit
integer vector that minimizes 𝑥 ⋅ (−𝑐 + 𝑝′𝑘 (𝑇 )) among unit integer vectors. We have
that there exists 𝐶 ′ > 0 depending on 𝑛 that (−𝑐 + 𝑝′𝑘 (𝑇 )) ⋅ 𝑥 ⩽ −𝐶 ′ ∥𝑝′𝑘 (𝑇 ) − 𝑐∥.
Since 𝜃𝑘′ (𝑇 ) and 𝜃𝑘 (𝑇 ) projects to the same point on the torus, by minimality of 𝜃𝑘
we have that
√
√
0 ⩽ (−𝑐 + 𝑝𝑘 (𝑇 )) ⋅ 𝑥 + 3𝐴 𝜖∣𝑥∣2 ⩽ −𝐶 ′ ∥𝑝′𝑘 (𝑇 ) − 𝑐∥ + 3𝐴 𝜖,
√
it follows that ∥𝑝′𝑘 (𝑇 ) − 𝑐∥ ≤ 3𝐴/𝐶 ′ 𝜖. Choose 𝐶 = 3𝐴/𝐶 ′ and we have proved our
claim.
To summarize, we have proved that for√𝑛𝑘 sufficiently large, the curves (𝜃𝑘 , 𝑝𝑘 )
satisfy ∥𝜃𝑘𝑠 − 𝜃𝑗𝑠 (𝑐)∥ ≤ 𝜌2 and ∥𝑝𝑘 − 𝑐∥ ≤ 𝐶 𝜖. By choosing a sufficiently small 𝜖, we
can guarantee that ∥𝜃𝑘𝑠 − 𝜃𝑗𝑠 (𝑝𝑘 )∥ < 𝜌0 . This implies that the Hamiltonians 𝑁𝜖,𝑘 and
¯𝜖,𝑘 take the same values on the curves (𝜃𝑘 , 𝑝𝑘 ), by taking the Legendre transform,
𝑁
∗
¯ ∗ must take the same values as well.
we can conclude that the Lagrangian 𝑁𝜖,𝑘
and 𝑁
𝜖,𝑘
It follows that
∫ 𝑛𝑘
∗
0 = ℎ𝑁𝜖,𝑗 ,𝑐 (𝜃0 , 𝜃0 ) = lim inf
𝑁𝜖,𝑗
(𝑡, 𝜃𝑘 , 𝜃˙𝑘′ ) − 𝑐 ⋅ 𝜃˙𝑘 + 𝛼𝑗 (𝑐)𝑑𝑡 ⩾ ℎ𝑁¯𝜖,𝑗 ,𝑐 (𝜃0 , 𝜃0 ) ⩾ 0.
𝑘−→∞
0
Hence ℎ𝑁𝜖,𝑗 ,𝑐 (𝜃0 , 𝜃0 ) = 0 =⇒ ℎ𝑁¯𝜖,𝑗 ,𝑐 (𝜃0 , 𝜃0 ) = 0. The other direction also holds since
the argument is completely symmetric. This concludes the proof of the lemma. □
˙
The alpha function of a Lagrangian 𝐿 can be defined by 𝛼(𝑐) = − inf 𝜇 (𝐿 − 𝑐 ⋅ 𝜃)𝑑𝜇,
where 𝜇 is taken over all invariant probability measures supported on the Aubry set
˜
¯𝜖,𝑗 as before, since the Aubry sets are
𝒜(𝑐).
Consider two Hamiltonians 𝑁𝜖,𝑗 and 𝑁
identical for these Hamiltonians, and the Hamiltonians coincide on a neighborhood
of the Aubry sets, the alpha function 𝛼𝑗 (𝑐) defined for these Hamiltonians must also
be the same. This conclude the proof of the first statement of our proposition.
Arnold diffusion along normally hyperbolic invariant cylinders
37
We now prove statements 3 and 4. Consider the cohomology classes 𝑐 = 𝑝∗ (𝑐𝑓 ) with
𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏], we note that the function 𝑍 already satisfies the conditions
that we require of the modification, and since the local Aubry set is independent of
specific modifications, we conclude that 𝒜˜𝑁𝜖 (𝑐) = 𝒜˜𝑁𝜖 ,𝑗 (𝑐) and 𝛼𝑗 (𝑐) = 𝛼(𝑐).
We now focus on the cohomology classes 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏].
Using Theorem 5.1, for these cohomology classes
the Aubry set 𝒜˜𝑁𝜖 (𝑐) is contained in
√
the vertical neighborhood {∥𝑝 − 𝑐∥ ≤ 36𝐴 𝜖}, and horizontally in the neighborhood
𝑠
{∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐)∥ ≤ 𝜌2 } ∪ {∥𝜃𝑠 − 𝜃𝑗+1
(𝑐)∥ ≤ 𝜌2 }. Take a point 𝜃0 ∈ 𝒜𝑁𝜖 (𝑐) ∩ {∥𝜃𝑠 −
𝜃𝑗𝑠 (𝑐)∥ ≤ 𝜌2 }, by going through the same argument as in the proof of Lemma 6.1,
we can conclude that ℎ𝑁𝜖 ,𝑐 (𝜃0 , 𝜃0 ) = 0 implies that ℎ𝑁𝜖,𝑗 ,𝑐 (𝜃0 , 𝜃0 ) = 0. It follows that
𝒜˜𝑁𝜖 (𝑐) ∩ 𝑈𝑗 (𝑐) ⊂ 𝒜˜𝑁𝜖 ,𝑗 (𝑐); the same holds for 𝑗 + 1. We have that
∗
˙
˙
𝛼(𝑐) = − min{inf (𝑁𝜖∗ − 𝑐 ⋅ 𝜃)𝑑𝜇
1 , inf (𝑁𝜖 − 𝑐 ⋅ 𝜃)𝑑𝜇2 } ⩽ max{𝛼𝑗 (𝑐), 𝛼𝑗+1 (𝑐)}
𝜇1
𝜇2
where 𝜇1 is supported on 𝒜˜𝑁𝜖 (𝑐)∩𝑈𝑗 (𝑐) while 𝜇2 is supported on 𝒜˜𝑁𝜖 (𝑐)∩𝑈𝑗+1 (𝑐). On
˙
the other hand, since 𝛼(𝑐) = − inf 𝜇 (𝑁𝜖∗ − 𝑐 ⋅ 𝜃)𝑑𝜇
with 𝜇 taken over all probability invariant measures, 𝛼(𝑐) ⩾ 𝛼𝑗 (𝑐), 𝛼𝑗+1 (𝑐). We conclude that 𝛼(𝑐) = max{𝛼𝑗 (𝑐), 𝛼𝑗+1 (𝑐)}.
We have proved statement 3.
Moreover, assume that 𝒜𝑁𝜖 (𝑐) ∩ 𝑈𝑗 (𝑐) ∕= ∅, then there exists 𝜃0 in this set such
that ℎ𝑁𝜖 ,𝑐 (𝜃0 , 𝜃0 ) = 0, as well as a sequence of localized periodic curves 𝜃𝑘 converging
to it. By taking any weak-*-limit of probability measures supported on these curves,
we∫ obtain at least one measure 𝜈 supported on 𝒜𝑁𝜖 (𝑐) ∩ 𝑈𝑗 (𝑐) such that 𝛼(𝑐) =
˙
− (𝑁𝜖∗ −𝑐⋅ 𝜃)𝑑𝜈.
This implies that 𝛼(𝑐) ≤ 𝛼𝑗 (𝑐), hence 𝛼(𝑐) = 𝛼𝑗 (𝑐). As a conclusion,
if 𝛼(𝑐) ∕= 𝛼𝑗 (𝑐) then 𝒜𝑁𝜖 (𝑐) ∩ 𝑈𝑗 (𝑐) = ∅. This proves statement 4 and concludes the
proof of Proposition 6.5.
□
6.2. Generic property of 𝒜˜𝑁𝜖 (𝑐). In this section we discuss the property of the
sets 𝒜˜𝑁𝜖 (𝑐) for 𝑐 = (𝑝𝑠∗ (𝑐𝑓 )) with 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏] and their properties when
we allowed to subject the Hamiltonian to an additional perturbation. It is convenient
for us to fix a modified Hamiltonian 𝑁𝜖,𝑗 and base all discussions on this system.
From Proposition 6.5, we have that the sets 𝒜˜𝑁𝜖 ,𝑗 (𝑐) (we will write 𝒜˜𝑗 (𝑐) for short
in this section) are contained in the NHIC 𝑋𝑗 , and 𝜋𝜃𝑓 ∣𝒜˜𝑗 (𝑐) is one-to-one. We will
study finer structures of the Aubry sets, by relating to the Aubry-Mather theory of
two dimensional area preserving twist maps. We will prove the following statement.
Proposition 6.6. There exists 𝜖0 , 𝛿0 > 0 such that for 0 < 𝜖 < 𝜖0 and 0 < 𝛿 < 𝛿0 ,
there exists arbitrarily small 𝐶 𝑟 perturbation 𝑁𝜖′ of 𝑁𝜖 , such that for each 𝑐𝑓 ∈ [𝑎𝑗 −
2𝑏, 𝑎𝑗+1 + 2𝑏], 𝒜˜𝑁𝜖′ (𝑝∗ (𝑐𝑓 )) supports a unique 𝑐−minimal measure.
38
P. Bernard, V. Kaloshin, K. Zhang
We note that the time-one-map of the Hamiltonian flow is a twist map defined on
𝕋 × ℝ𝑛 . The generating function of this twist map is 𝐺𝑗 (𝑥, 𝑥′ ) : ℝ𝑛 × ℝ𝑛 −→ ℝ ,
∫ 1
′
∗
𝐺𝑗 (𝑥, 𝑥 ) =
inf
𝑁𝜖,𝑗
(𝑡, 𝛾, 𝛾)𝑑𝑡.
˙
′
𝑛
𝛾(0)=𝑥,𝛾(1)=𝑥
0
Consider an orbit {(𝜃(𝑡), 𝑝(𝑡))} of the Hamiltonian flow, its trajectory in the configuration space can be lifted to a curve 𝑥(𝑡) ∈ ℝ𝑛 , which is unique modulo integer
translation. The sequence 𝑥𝑘 = 𝑥(𝑘), 𝑘 ∈ ℤ will be called a configuration. A configuration’s rotation number is defined by lim𝑘−→∞ (𝑥𝑎+𝑘 − 𝑥𝑎 )/𝑘, if such a limit exists.
Let {𝑥𝑘 } = {(𝑥𝑠𝑘 , 𝑥𝑓𝑘 )} be a configuration corresponding to an orbit in 𝒜˜𝑗 (𝑐), we will
say that this configuration belong to the Aubry set for short. Since 𝒜˜𝑗 (𝑐) ⊂ 𝑋𝑗 , we
have that the slow component 𝑥𝑠 stays bounded all the time. Take two configurations
{𝑥𝑘 } and {𝑦𝑘 }, we say that they intersect in the fast direction (in short, intersect,
as this is the only type of intersection we will consider) if there exists an integer 𝑚
and indices 𝑘1 , 𝑘2 such that 𝑥𝑓𝑘1 > 𝑦𝑘𝑓1 + 𝑚 and 𝑥𝑓𝑘2 < 𝑦𝑘𝑓2 + 𝑚. We have the following
statements, analogous to the twist map case.
Lemma 6.2.
(1) Any two distinct configurations {𝑥𝑘 } and {𝑦𝑘 } in 𝒜˜𝑗 (𝑐) does
not intersect.
(2) Any configuration {𝑥𝑘 } in 𝒜˜𝑗 (𝑐) has a uniquely defined rotation number 𝜌 =
(0, 𝜌𝑓 ).
Proof. For the first statement, we prove by contradiction. Assume that 𝑥(𝑡) and 𝑦(𝑡)
are the lifts of two distinct trajectories such that {𝑥(𝑘)} and {𝑦(𝑘)} intersect. It
follows that there exists 𝑚 and 𝑘1 . 𝑘2 such that 𝑥𝑓 (𝑘1 ) > 𝑦 𝑓 (𝑘1 ) + 𝑚 and 𝑥𝑓 (𝑘2 ) <
𝑦 𝑓 (𝑘2 ) + 𝑚. It follows that there exists 𝜏 ∈ ℝ such that 𝑥𝑓 (𝜏 ) = 𝑦 𝑓 (𝜏 ) + 𝑚. Let
𝜃1 (𝑡) and 𝜃2 (𝑡) be the projections of 𝑥(𝑡) and 𝑦(𝑡) to 𝕋𝑛 , we have that 𝜃1𝑓 (𝜏 ) = 𝜃2𝑓 (𝜏 ).
Assume that 𝑝𝑖 (𝑡), 𝑖 = 1, 2 are the corresponding action variables for trajectories 𝜃𝑖 .
Let 𝑘 ≤ 𝜏 < 𝑘 + 1, we have that (𝜃𝑖 (𝑘), 𝑝𝑖 (𝑘)) ∈ 𝒜˜𝑗 (𝑐). From the graph theorem,
we have that (𝜃𝑖 (𝑘), 𝑝𝑖 (𝑘)) is a function of 𝜃𝑖𝑓 (𝑘). Applying the flow, we have that
(𝜃𝑖 (𝑡), 𝑝𝑖 (𝑡)) is a function of (𝜃𝑖𝑓 (𝑡), 𝑡). It follows that (𝜃1 (𝜏 ), 𝑝1 (𝜏 )) = (𝜃2 (𝜏 ), 𝑝2 (𝜏 )),
hence (𝜃1 (𝑡), 𝑝1 (𝑡)) = (𝜃2 (𝑡), 𝑝2 (𝑡)) for all 𝑡, a contradiction.
For the second statement, since any trajectory from 𝒜˜𝑗 (𝑐) must contained in 𝑋𝑗 ,
we have that any lift 𝑥(𝑡) of such a trajectory must have its slow component uniformly
bounded. Hence lim𝑘−→∞ 𝑥𝑠 (𝑘)/𝑘 = 0. It suffices to consider only {𝑥𝑓𝑘 }. Since 𝑥𝑓
is one-dimensional, most argument from the standard Aubry-Mather theory applies,
once we establish the non-intersecting property. We refer to [MF], section 11, where
existence of rotation number was proved under a weaker assumption (the Aubry
crossing lemma).
□
Arnold diffusion along normally hyperbolic invariant cylinders
39
Let 𝜇 be a 𝑐−minimal measure for 𝑁𝜖,𝑗 , we know that it is necessarily supported
on 𝐴˜𝑗 (𝑐). The rotation number of 𝜇 is 𝜌(𝜇) ∈ 𝐻1 (𝕋𝑛 , ℝ) ≃ ℝ𝑛 , defined by
∫
⟨𝑐, 𝑣⟩𝑑𝜇(𝜃, 𝑣) = ⟨𝑐, 𝜌(𝜇)⟩.
𝕋𝑛 ×ℝ𝑛
Using the no-intersection property (Lemma 6.2, 1), most of the statements we will be
need follows from standard Aubry-Mather theory. Most of the arguments presented
here are variations of those found in see [MF].
Proposition 6.7. For any 𝑐 = 𝑝∗ (𝑐𝑓 ), 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏], the following hold.
(1) All 𝑐−minimal measures supported on 𝒜˜𝑗 (𝑐) have a common rotation number
𝜌(𝑐) = (0, 𝜌𝑓 (𝑐)). Moreover, the function 𝛼𝑗 𝑝∗ (𝑐𝑓 ) as a function of 𝑐𝑓 is 𝐶 1 .
(2) If 𝜌𝑓 (𝑐) = 𝑝/𝑞 ∈ ℚ, written in lowest terms, then all minimal measures are
supported on 𝑞−periodic orbits. These orbits corresponds to (𝑝, 𝑞)−periodic
configurations {𝑥𝑘 } in the sense that (𝑥𝑠𝑘+𝑞 , 𝑥𝑓𝑘+𝑞 ) = (𝑥𝑠𝑘 , 𝑥𝑓𝑘 ) + (0, 𝑝). Furthermore, they are the minima of the functional
𝑞−1
∑
𝐺𝑗 (𝑥𝑘 , 𝑥𝑘+1 )
𝑘=0
over the set of configurations that are (𝑝, 𝑞)−periodic.
(3) If 𝜌𝑓 (𝑐) ∈
/ ℚ, then there is one unique 𝑐−minimal measure.
Proof. First we show that all the configurations on 𝒜˜𝑗 (𝑐) has the same rotation number. To see this, consider any two configurations with different rotation numbers,
since they must intersect, Lemma 6.2 implies that they cannot both be contained in
𝒜˜𝑗 (𝑐).
We now look at the function 𝛼𝑗 𝑝∗ (𝑐𝑓 ). It is known (add reference) that 𝛼𝑗 (𝑐) is a
convex function and any rotation number 𝜌 of a 𝑐−minimal measure is a subderivative
of 𝛼𝑗 at 𝑐. If for some 𝑐 the subderivative is unique, then 𝛼 is differentiable at 𝑐. It
follows 𝛼𝑗 (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) is differentiable for each 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏]. The fact that
it is 𝐶 1 follows from the following statement: let 𝑓 (𝑥) be convex, 𝑥𝑛 is a sequence
that converges to 𝑥∗ , 𝑝𝑛 is a subderivative of 𝑓 (𝑥) at 𝑥𝑛 and 𝑝𝑛 converges to 𝑝∗ , then
𝑝∗ is a subderivative of 𝑓 (𝑥) at 𝑥∗ . This concludes the proof of the first statement.
We now prove the second statement. Consider any configuration {𝑥𝑘 } with rotation
number 𝑝/𝑞, we have that 𝑥𝑓𝑘+𝑞 − 𝑥𝑓𝑘 − 𝑝 does not change sign for this configuration.
Assume that it does, say 𝑥𝑓𝑘1 +𝑞 − 𝑥𝑓𝑘1 − 𝑝 > 0 and 𝑥𝑘2 +𝑞 − 𝑥𝑘2 − 𝑝 < 0, then the
configurations {𝑥𝑘 } and 𝑥𝑘+𝑘2 −𝑘1 intersects, contradiction. On the other hand, since
the rotation number is 𝑝/𝑞, we have that lim𝑘−→∞ 𝑥𝑓𝑘+𝑞 − 𝑥𝑓𝑘 − 𝑝 = 0. It follows that
40
P. Bernard, V. Kaloshin, K. Zhang
any 𝑥𝑘 such that 𝑥𝑓𝑘+𝑞 − 𝑥𝑓𝑘 − 𝑝 ∕= 0 does not project to a point on the support of an
invariant measure, since this point is not recurrent. By the same argument, we can
show that 𝑥𝑠𝑘+𝑞 − 𝑥𝑠𝑘 = 0 for any point that projects to the support of an invariant
measure.
We have proved that any point on the support of an invariant measure lifts to a
configuration with 𝑥𝑘+𝑞 − 𝑥𝑘 = (0, 𝑝). Let 𝜇 be a 𝑐−minimal measure supported on
(𝜃(𝑘), 𝑝(𝑘)), 𝑘 = 0, ⋅ ⋅ ⋅ 𝑞 − 1, and let 𝑥𝑘 be the corresponding configuration. Since
∫
∗
(𝑁𝜖,𝑗
∫
˙
− 𝑐 ⋅ 𝜃)𝑑𝜇
=
∫
∗
𝑁𝜖,𝑗
𝑑𝜇
+𝑐⋅𝜌=
𝑞−1
∑
𝑘=0
𝐺𝑗 (𝑥𝑘 , 𝑥𝑘+1 ) + 𝑐 ⋅ 𝜌,
∑𝑞−1
∗
˙
𝜇 minimizes (𝑁𝜖,𝑗
− 𝑐 ⋅ 𝜃)𝑑𝜇
implies that {𝑥𝑘 } minimizes 𝑘=0
𝐺𝑗 (𝑥𝑘 , 𝑥𝑘+1 ).
For the irrational rotation number case, we refer to [MF], section 12. Consider
˜
𝒜𝑗 (𝑐) as a subset of 𝕋 and the dynamics on this subset. It is proved that the system
is semi-conjugate to a rigid rotation of irrational rotation number, and the semiconjugacy is not one-to-one on at most countably many points. It follows that the
dynamics on 𝒜˜𝑗 (𝑐) has one unique invariant measure, since irrational rotation is
uniquely ergodic.
□
For irrational rotation numbers, we have that the corresponding minimal measure
is unique. For rational rotation numbers, it is well known that for the twist map,
generically there exists only one minimal periodic orbit of rotation number 𝑝/𝑞. We
have the same conclusions here. The following statement and Lemma 6.2 imply
Proposition 6.6.
Proposition 6.8.
(1) By subjecting the generating function 𝐺𝑗 (𝑥, 𝑥′ ) to an arbitrarily small 𝐶 𝑟 perturbation, we have that for any rational rotation number
𝑝/𝑞, there are exactly 𝑞 periodic configurations of type (𝑝, 𝑞). (In this case
there exists a unique minimal periodic orbit with rotation number 𝑝/𝑞.)
(2) The perturbation to 𝐺𝑗 in part 1 can be realized by an arbitrarily small 𝐶 𝑟
perturbation to the Hamiltonian 𝑁𝜖,𝑗 , localized in the set {(𝜃, 𝑝) : ∥𝜃𝑠 −𝜃𝑗𝑠 (𝑝)∥ <
𝜌0 }. As a result, this perturbation can be realized by a small perturbation to
the original Hamiltonian 𝑁𝜖 .
Proof. Let {𝑥𝑘 } be a minimizing configuration of type (𝑝, 𝑞), let 𝑈 be an open set that
contains 𝑥0 but none of the 𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑞−1 . Let 𝑔𝑈 (𝑥) be nonnegative periodic function
that is supported on 𝑈 , 𝑔𝑈 (𝑥0 ) = 0 is the unique minimum and ∂ 2 𝑔 is positive definite.
If we consider the new generating function
𝐺𝑗 (𝑥, 𝑥′ ) + 𝑔𝑈 (𝑥),
Arnold diffusion along normally hyperbolic invariant cylinders
41
∑
the action 𝑞−1
𝑘=0 𝐺𝑗 (𝑥𝑘 , 𝑥𝑘+1 ) is unaffected, while the action increases for other configurations. It follows that {𝑥𝑘 } and its translations are the unique minimal configurations. However, this perturbation cannot be realized by a localized perturbation to
the Hamiltonian (to be more precise, it is localized horizontally, but not vertically).
We consider the following modification of the above construction.
Let Φ denote a lift of the time-one-map of the Hamiltonian flow. The generating
function uniquely determines the map Φ in the sense that given 𝑥, 𝑥′ ∈ ℝ𝑛 , write
𝑝 = −∂1 𝐺𝑗 and 𝑝2 = ∂2 𝐺𝑗 then Φ(𝑥, 𝑝) = (𝑥′ , 𝑝′ ). On the other hand, Theorem 5.1
√
implies that any orbit in the Aubry set 𝒜˜𝑗 (𝑐) is localized
{∥𝑝−𝑐∥ ≤ 6𝑛𝐴 𝜖},
√ in the set
′
𝑛
′
which leads
𝑗 (𝑥, 𝑥 ) −
√ us to the following definition. Let 𝑉𝑥 (6𝐴 𝑛𝜖) = {𝑥 ∈ ℝ , ∥∂1 𝐺√
𝑐∥ ≤ 6𝐴 𝑛𝜖}, and let 𝜌𝑥√be a smooth function that takes value 1 on 𝑉𝑥 (6𝐴 𝑛𝜖) and
takes value 0 on 𝑉𝑥 (12𝐴 𝑛𝜖). We have that the generating function
𝐺𝑗 (𝑥, 𝑥′ ) + 𝑔𝑈 (𝑥)𝜌𝑥 (𝑥′ )
will make {𝑥𝑘 } and its translation the unique minimizing configurations of type (𝑝, 𝑞).
The norm of the perturbation can be arbitrarily small since the norm of 𝑔𝑈 can be
arbitrarily small.
To treat all rational rotation numbers, we consider a sequence of such perturbations 𝑔𝑈𝑖 (𝑥)𝜌𝑥 (𝑥′ ), each subsequent perturbation can be chosen to be small enough,
such that the result of earlier perturbations are not destroyed. The final perturbed
generating function is
∑
𝐺′𝑗 (𝑥, 𝑥′ ) = 𝐺𝑗 (𝑥, 𝑥′ ) +
𝑔𝑈𝑖 (𝑥)𝜌𝑥 (𝑥′ ).
𝑖≥1
We now show that the perturbation
∑ can be realized by a localized perturbation of
the Hamiltonian. Write 𝑔(𝑥, 𝑥′ ) = 𝑖≥1 𝑔𝑈𝑖 (𝑥)𝜌𝑥 (𝑥′ ) and let Φ′ denote ∪
the perturbed
time-one-map
of the Hamiltonian flow. Since ∂1 𝑔 = 0 for all 𝑥 ∈
/ 𝑖 𝑈𝑖 or 𝑥′ ∈
/
√
′
𝑉𝑥 (12𝐴 ∪
𝑛𝜖), the perturbed time-one-map
Φ
is
identical
to
the
original
Φ
for
any
∪
√
(𝑥, 𝑝) ∈
/ 𝑖 𝑈𝑖 × {∥𝑝 − 𝑐∥ ≤ 12𝐴 𝑛𝜖}. Since we can choose 𝑈𝑖 such that 𝑖 𝑈𝑖 ⊂
{∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐)∥ < 𝑑 < 𝜌0 }, for sufficiently small 𝜖 we can guarantee
∪
√
𝑈𝑖 × {∥𝑝 − 𝑐∥ ≤ 12𝐴 𝑛𝜖} ⊂ {∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝)∥ < 𝜌0 }.
𝑖
It follows that Φ = Φ′ for any (𝜃, 𝑝) ∈
/ {∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝)∥ < 𝜌0 }. This perturbation of the
time-one-map can be realized by a perturbation to the Hamiltonian localized in the
same neighborhood.
□
6.3. Generic property of 𝛼(𝑐) and proof of Theorem 6.1. After obtaining the
desired properties for the local Aubry set, we now return to the Hamiltonian 𝑁𝜖 . If
𝑐𝑓 ∈ [𝑎𝑗 + 𝑏, 𝑎𝑗+1 − 𝑏], we have that 𝒜˜𝑁𝜖 (𝑐) = 𝒜˜𝑁𝜖 ,𝑗 (𝑐). For 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏],
42
P. Bernard, V. Kaloshin, K. Zhang
Proposition 6.5, statement 3 and 4 shows that it suffices to identify whether 𝛼(𝑐) is
equal to 𝛼𝑗 (𝑐) or 𝛼𝑗+1 (𝑐).
Proposition 6.9. Assume that 𝑁𝜖 = 𝐻0 + 𝜖𝑍 + 𝜖𝑅 is such that 𝑍 satisfy [G0][G2] and that ∥𝑅∥𝐶 2 ≤ 𝛿. Then there exists 𝜖0 , 𝛿0 > 0 such that for 0 < 𝜖 < 𝜖0
and 0 < 𝛿 < 𝛿0 , there exists an arbitrarily small perturbation 𝑁𝜖′ of 𝑁𝜖 , with the
following properties. For the Hamiltonian 𝑁𝜖′ Proposition 6.5 and Proposition 6.6
still hold, in addition, there exists only finitely many 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏] such that
𝛼𝑗 (𝑝𝑠∗ (𝑐𝑓 ), 𝑐𝑓 ) = 𝛼𝑗+1 𝑝∗ (𝑐𝑓 ).
Proof. By taking a small perturbation if necessary, let us assume that we start with a
Hamiltonian 𝑁𝜖 such that Proposition 6.5 and Proposition 6.6 already hold. Consider
the interval 𝑐𝑓 ∈ [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏] first. Let 𝑃𝑗𝜂 (𝜃, 𝑝, 𝑡) : 𝕋𝑛 × ℝ𝑛−1 × [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 +
2𝑏] −→ ℝ be a family of smooth functions such that
⎧
𝑠
𝑠 𝑓
𝑓

⎨𝜂, ∥𝜃 − 𝜃𝑗 (𝑝 )∥ ⩽ 𝜌0 and 𝑝 ∈ [𝑎𝑗 − 3𝑟/2, 𝑎𝑗+1 + 3𝑟/2]
𝑃𝑗𝜂 (𝜃, 𝑝, 𝑡) = 0, ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝𝑓 )∥ ⩾ 4𝜌0 /3
.

⎩0, 𝑝𝑓 ∈ {𝑎 − 2𝑏, 𝑎 + 2𝑏}
𝑗
𝑗+1
Clearly ∥𝑃𝑗𝜂 ∥𝐶 𝑟 can be arbitrarily close to 0 by choosing 𝜂 close to 0.
Let 𝑁 𝜂 = 𝑁𝜖 + 𝑃𝑗𝜂 . The new perturbation can be considered part of 𝑅 and if 𝜂 is
sufficiently close to 0, Proposition 6.5 still hold. This implies that the local Aubry sets
still depends only on the value of the Hamiltonian on the set ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑝𝑓 )∥ ⩽ 𝜌0 , on
which the perturbation is simply a constant (for 𝑐𝑓 ∈ [𝑎𝑗 −3𝑟/2, 𝑎𝑗+1 +3𝑟/2]). We have
that 𝛼𝑁 𝜂 ,𝑗 (𝑐) = 𝛼𝑁𝜖 ,𝑗 +𝜂 and that 𝒜˜𝑁 𝜂 ,𝑗 (𝑐) = 𝒜˜𝑁𝜖 ,𝑗 (𝑐) for 𝑐𝑓 ∈ [𝑎𝑗 −3𝑟/2, 𝑎𝑗+1 +3𝑟/2].
It follows that all properties of the local Aubry set 𝒜˜𝑁𝜖 ,𝑗 (𝑐) is intact, while 𝛼𝑗 (𝑐)
undergoes a shift.
On the other hand, Consider the functions 𝛼𝑗 𝑝∗ (𝑐𝑓 ) and 𝛼𝑗+1 𝑝∗ (𝑐𝑓 ) as functions
on [𝑎𝑗+1 − 3𝑟/2, 𝑎𝑗+1 + 3𝑟/2]. Since they are both 𝐶 1 , by Sard’s lemma, the critical
values of 𝛼𝑗 − 𝛼𝑗+1 has zero measure. It follows that there exists a full measure set
′
of 𝜂 ∈ ℝ such that 𝛼𝑗′ − 𝛼𝑗+1
= 0 implies 𝛼𝑗 − 𝛼𝑗+1 + 𝜂 ∕= 0. In other words, the two
functions 𝛼𝑗 + 𝜂 and 𝛼𝑗+1 intersect transversally, which implies that there are only
finitely many values where 𝛼𝑗 − 𝛼𝑗+1 + 𝜂 = 0.
We can perform this modification for each [𝑎𝑗 − 2𝑏, 𝑎𝑗+1 + 2𝑏], and 𝜂 can be chosen
to be arbitrarily close to 0.
□
Proof of Theorem 6.1. Since there are only finitely many 𝑐𝑓 ∈ [𝑎𝑗+1 − 𝑏, 𝑎𝑗+1 + 𝑏] on
which 𝛼𝑗 = 𝛼𝑗+1 , we add these points to the set {𝑎0 , ⋅ ⋅ ⋅ , 𝑎𝑠 } to form a new partition
{[¯
𝑎𝑗 , 𝑎
¯𝑗+1 ]}. On each open interval (¯
𝑎𝑗 , 𝑎
¯𝑗+1 ) 𝛼(𝑐) is only equal to one of the 𝛼𝑗 and
𝛼𝑗+1 . Use Proposition 6.5 and the first statement follows.
The second statement follows from Proposition 6.6.
□
Arnold diffusion along normally hyperbolic invariant cylinders
43
6.4. nondegeneracy of the barrier functions. In this section we prove Theorem 6.4. We have concluded that in order to prove Theorem 6.4, it suffices to show
that Γ1 (𝜖) = Γ∗1 (𝜖) and Γ2 (𝜖) = Γ∗2 (𝜖). We show that this is the case by proving the
following equivalent statement.
Proposition 6.10. Let 𝐻𝜖′ be a perturbation of 𝐻𝜖 such that the conclusions of Theorem 6.2 holds, then there exists an arbitrarily small 𝐶 𝑟 perturbation 𝐻𝜖′′ to 𝐻𝜖′ , such
that for the Hamiltonian 𝐻𝜖′′ Theorem 6.2 still hold, in addition:
(1) Consider 𝑐𝑓 ∈ (¯
𝑎𝑗 , 𝑎
¯𝑗+1 ) such that 𝒜𝑐 = 𝒩𝑐 and 𝜋𝜃𝑓 𝒜𝑐 = 𝕋. Take 𝜁 ∈ ℳ𝑐 ,
and let 𝜁1 and 𝜁2 be its lifts to the double cover. We have that the functions
ℎ̃𝑐 (𝜁1 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜁2 ) and ℎ̃𝑐 (𝜁2 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜁1 ) have isolated minima outside of the
lifts of 𝒜𝑐 .
(2) For 𝑐 = 𝑎
¯𝑗+1 , take 𝜁 ∈ 𝒜𝑐 ∩ 𝑋𝑗 and 𝜂 ∈ 𝒜𝑐 ∩ 𝑋𝑗+1 . We have that both
ℎ𝑐 (𝜁, 𝜃) + ℎ𝑐 (𝜃, 𝜂) and ℎ𝑐 (𝜂, 𝜃) + ℎ𝑐 (𝜃, 𝜁) has isolated minima outside of 𝒜𝑐 .
This proposition is essentially proved by Cheng and Yan in [CY2], here we briefly
describe their approach.
Consider the Hamiltonian 𝐻𝜖′ such that conclusions of Theorem 6.2 holds. In the
rest of the section, let’s refer to 𝐻𝜖′ simply as 𝐻. For now, let us also fix an interval
(¯
𝑎𝑗 , 𝑎
¯𝑗+1 ) and consider only cohomology classes with 𝑐𝑓 in that interval. Let Γ𝑗1 =
Γ1 (𝜖) ∩ {𝑐, 𝑐𝑓 ∈ (¯
𝑎𝑗 , 𝑎
¯𝑗+1 )}, we would like to show by perturbing the Hamiltonian, we
can make the functions ℎ̃𝑐 (𝜉1 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜉2 ) and ℎ̃𝑐 (𝜉2 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜉1 ) nondegenerate.
Recall that ℎ̃𝑐 is the barrier function defined on the covering space (2𝕋)𝑛 × ℝ𝑛 , and
˜ is the Hamiltonian lifted to the
𝜉 : (2𝕋)𝑛 × ℝ𝑛 −→ 𝕋𝑛 × ℝ𝑛 is the covering map. 𝐻
covering space.
Define the generating function 𝐺(𝑥, 𝑥′ ) : ℝ𝑛 × ℝ𝑛 −→ ℝ by
∫ 1
′
𝐺(𝑥, 𝑥 ) =
inf
𝐿(𝑡, 𝛾, 𝛾),
˙
′
𝛾(0)=𝑥,𝛾(1)=𝑥
0
where 𝐿 is the Lagrangian corresponding to 𝐻. A convenient way of introducing
perturbations to the functions ℎ̃𝑐 is by perturbing the generating functions. Denote
by 𝜋 : ℝ𝑛 −→ 𝕋𝑛 the standard projection.
We consider the following perturbation
𝐺′ (𝑥, 𝑥′ ) = 𝐺(𝑥, 𝑥′ ) + 𝐺1 (𝑥′ )
and denote by ℎ̃′𝑐 the corresponding perturbed barrier function. We have the following
statement.
Lemma 6.3. ([CY2], Lemma 7.1) For 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 ∈ (¯
𝑎𝑗 , 𝑎
¯𝑗+1 ), the following
hold.
44
P. Bernard, V. Kaloshin, K. Zhang
(1) There exists a family of open sets 𝑂𝑐 ⊂ (2𝕋)𝑛 such that the full orbit of any
˜ 𝑝˜) ∈ 𝒩˜ ˜ (𝑐) ∖ 𝒜˜ ˜ (𝑐) must intersect 𝑂𝑐 in the 𝜃˜ component.
(𝜃,
𝐻
𝐻
(2) There exists 𝜌 > 0 such that if we perturb 𝐺(𝑥, 𝑥′ ) by function 𝐺1 (𝑥′ ) with
supp 𝐺1 ⊂ 𝐵𝜌 (𝑢), where 𝐵𝜌 (𝑢) is the ball of radius 𝜌 centered at 𝑢, then for
each 𝑐 such that 𝐵𝜌 (𝑢) ⊂ 𝜋 −1 𝑂𝑐 the corresponding barrier function
ℎ̃′𝑐 (𝜉1 , 𝜃) + ℎ̃′𝑐 (𝜃, 𝜉2 ) = ℎ̃𝑐 (𝜉1 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜉2 ) + 𝐺1 (𝜃)
for each 𝜃 ∈ 𝑂𝑐 .
(3) 𝜉𝑂𝑐 ∩ {𝜃 : ∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐)∥ ≤ 𝜌2 } = ∅, in particular, 𝜉𝑂𝑐 ∩ 𝒩𝐻 (𝑐) = ∅. Moreover
∪
𝑈˜ = 𝑐𝑓 ∈(¯𝑎𝑗 ,¯𝑎𝑗+1 ) 𝑂𝑐 is an open set.
˜−
As before, let us write ˜𝑏+
𝑐 (𝜃) = ℎ̃𝑐 (𝜉1 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜉2 ) and 𝑏𝑐 (𝜃) = ℎ̃𝑐 (𝜉2 , 𝜃) + ℎ̃𝑐 (𝜃, 𝜉1 ).
Elements of 𝒩𝐻˜ (𝑐) ∖ 𝒜𝐻˜ (𝑐) coincide with the minimal set of the functions ˜𝑏±
𝑐 . To
prove that this set is isolated, it suffices to prove its intersection with 𝑂𝑐 is isolated,
as any accumulation point of 𝒩𝐻˜ (𝑐) ∖ 𝒜𝐻˜ (𝑐) has a diffeomorphic image in 𝑂𝑐 . We say
that the function ˜𝑏±
𝑐 (𝜃) is degenerate if its minimal set has at least one accumulation
point. Cheng and Yan proved that it is possible to introduce a perturbation to make
˜𝑏± nondegenerate for all 𝑐 ∈ Γ𝑗 simultaneously.
1
𝑐
This is not possible in general, if the functions ˜𝑏±
𝑐 behave badly as 𝑐 varies. Since
regularity of ˜𝑏±
in
𝑐
is
hard
to
prove,
Cheng
and
Yan
resolves this problem nicely by
𝑐
introducing an additional parameter. Recall that for each 𝑐 ∈ Γ𝑗1 , the Aubry set 𝒜˜
is an invariant curve on the time-zero section of the invariant cylinder 𝑋𝑗 , call it 𝛾𝑐 .
Fix an arbitrary curve 𝛾0 = {𝑝𝑓 = 𝑝𝑓0 } ∩ {𝑡 = 0} ∩ 𝑋𝑗 , we introduce a parameter 𝜎
which is the area between 𝛾𝑐 and 𝛾0 on the cylinder 𝑋𝑗 ∩ {𝑡 = 0}. 𝜎 is monotone in
𝑐𝑓 and is only defined for 𝑐 ∈ Γ𝑗1 . Cheng and Yan proved that
Lemma 6.4. ([CY2], Lemma 6.4) There exists constant 𝐶 > 0 such that, for 𝜎 and
𝜎 ′ such that 𝑐(𝜎), 𝑐(𝜎 ′ ) ∈ Γ𝑗1 , 𝜁 ∈ 𝒜𝐻 (𝑐) and 𝑚 ∈
/ {∥𝜃𝑠 − 𝜃𝑗𝑠 (𝑐(𝜎))∥ ≤ 𝜌2 } ∪ {∥𝜃𝑠 −
𝜃𝑗𝑠 (𝑐(𝜎 ′ ))∥ ≤ 𝜌2 },
√
∣ℎ𝑐(𝜎) (𝜁, 𝑚) − ℎ𝑐(𝜎′ ) (𝜁, 𝑚)∣ ≤ 𝐶( ∣𝜎 − 𝜎 ′ ∣ + ∣𝑐(𝜎) − 𝑐(𝜎 ′ )∣),
∣ℎ𝑐(𝜎) (𝑚, 𝜁) − ℎ𝑐(𝜎′ ) (𝑚, 𝜁)∣ ≤ 𝐶(
√
∣𝜎 − 𝜎 ′ ∣ + ∣𝑐(𝜎) − 𝑐(𝜎 ′ )∣).
It follows that the function ℎ𝑐(𝜎) can be extended to ℎ𝑐,𝜎 that is 21 −Hölder in 𝑐
and 𝜎, this regularity turns out to be enough. To see how this is carried out, let us
consider a subset 𝐵𝑑′ (𝑐𝑓∗ )×𝑅𝑑 (𝑢), where 𝐵𝑑′ (𝑐𝑓∗ ) = {𝑐 : ∣𝑐𝑓 −𝑐𝑓∗ ∣ < 𝑑′ } and 𝑅𝑑 (𝑢) ⊂ 𝕋𝑛
is an open cube centered at 𝑢 with edge 𝑑.
Arnold diffusion along normally hyperbolic invariant cylinders
45
Lemma 6.5. ([CY2], Lemma 7.2) There is a residue set of functions 𝐺1 ∈ 𝐶0𝑟 (𝑅𝑑 (𝑢), ℝ)
such that
˜𝑏± (𝜃) + 𝐺1 (𝜃)
𝑐
has isolated minima in 𝑅𝑑 (𝑢) for each 𝑐 ∈ Γ𝑗1 ∩ 𝐵𝑑′ (𝑐). (𝐶0𝑟 stands for 𝐶 𝑟 functions
with compact support).
Remark 6.1. The nontrivial part of this statement is that the nondegeneracy of ˜𝑏±
𝑐
can be achieved for all 𝑐 ∈ Γ𝑗1 ∩ 𝐵𝑑′ (𝑐) simultaneously. The regularity acquired in
Lemma 6.4 is crucial to the proof. We refer to [CY2] for details.
To construct the desired perturbation to the barrier function, let us state another
lemma, which is a consequence of the upper semi-continuity of the Mañe set on the
Lagrangian.
𝑓
Lemma 6.6. The property that the functions ˜𝑏±
𝑐 are non-degenerate on the set 𝐵𝑑′ (𝑐∗ )×
𝑅𝑑 (𝑢) survives under sufficiently small perturbation.
We proceed to prove Proposition 6.10. Let 𝐵𝑑′𝑖 (𝑐𝑓𝑖 ) × 𝑅𝑑𝑖 (𝑢𝑖 ) ⊂ 𝑈˜ , be a se∪
quence of sets such that 𝑈˜ = 𝑖 𝐵𝑑′𝑖 (𝑐𝑓𝑖 ) × 𝑅𝑑𝑖 (𝑢𝑖 ). We may choose a sequence of
∑
perturbations 𝐺𝑖 : 𝑅𝑑𝑖 (𝑢𝑖 ) −→ ℝ, and let 𝐺′𝑘 (𝑥, 𝑥′ ) = 𝐺(𝑥, 𝑥′ ) + 𝑘𝑖=1 𝐺𝑖 (𝑥′ ) and
˜𝑏± be the corresponding barrier functions corresponding to the generating function
𝑐,𝑘
𝐺′𝑘 . We can choose the sequence 𝐺𝑖 inductively such that ˜𝑏±
𝑐 is non-degenerate on
∪𝑘
𝑓
𝑗
(𝑐, 𝜃) ∈ 𝑖=1 (𝐵𝑑′𝑖 (𝑐𝑖 ) ∩ Γ1 ) × 𝑅𝑑𝑖 (𝑢𝑖 ), because new perturbations can be added that
does not disrupt the nondegeneracy already established in the previous steps. By
repeat this process for each interval (¯
𝑎𝑗 , 𝑎
¯𝑗+1 ), we have constructed a perturbation to
the generating function 𝐺, such that the first statement of Proposition 6.10 holds.
For the second statement, using the same arguments for Lemma 6.3, one can show
that the same type of conclusions apply to 𝑏±
𝑐 as well.
Lemma 6.7. For each 𝑐 = 𝑝∗ (𝑐𝑓 ) with 𝑐𝑓 = 𝑎
¯𝑗 , 𝑗 = 2, ⋅ ⋅ ⋅ , 𝑙 − 1 the following hold.
(1) There exists a family of open sets 𝑂𝑐 ⊂ (2𝕋)𝑛 such that the full orbit of any
˜ 𝑝˜) ∈ 𝒩˜𝐻 (𝑐) ∖ 𝒜˜𝐻 (𝑐) must intersect 𝑂𝑐 in the 𝜃˜ component.
(𝜃,
(2) There exists 𝑏 > 0 such that if we perturb 𝐺(𝑥, 𝑥′ ) by function 𝐺1 (𝑥′ ) with
supp 𝐺1 ⊂ 𝐵𝑏 (𝑢), where 𝐵𝑏 (𝑢) is the ball of radius 𝑏 centered at 𝑢, then for
each 𝑐 such that 𝐵𝑏 (𝑢) ⊂ 𝜋 −1 𝑂𝑐 the corresponding barrier function
ℎ′𝑐 (𝜁, 𝜃) + ℎ′𝑐 (𝜃, 𝜂) = ℎ𝑐 (𝜁, 𝜃) + ℎ𝑐 (𝜃, 𝜂) + 𝐺1 (𝜃)
for each 𝜃 ∈ 𝑂𝑐 . The same conclusion holds for ℎ𝑐 (𝜂, 𝜃) + ℎ𝑐 (𝜃, 𝜁).
46
P. Bernard, V. Kaloshin, K. Zhang
For a fixed 𝑐, it is easy to see 𝑏±
𝑐 (𝜃) + 𝐺1 (𝜃) has isolated minimal set in 𝑅𝑑 (𝑢) for
an open and dense set of 𝐺1 . Repeat the arguments for ˜𝑏±
𝑐 , we obtain a perturbation
for which the both statements of Proposition 6.10 hold.
□
Appendix A. Generic conditions
We prove Theorem 2.1 in this section. Consider the following (degeneracy) conditions on the function 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) : 𝕋𝑛−1 × [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] −→ ℝ.
[T0] For 𝑝𝑓 ∈ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ], all local maxima of 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) is nondegenerate.
[T1] For each 𝑝𝑓 ∈ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] and there are at most two distinct 𝜃1𝑠 , 𝜃2𝑠 ∈ 𝕋𝑛−1 such
that ∂𝜃𝑠 𝑍(𝜃𝑗𝑠 , 𝑝∗ (𝑝𝑓 )) = 0 for 𝑗 = 1, 2 and that 𝑍(𝜃1𝑠 , 𝑝∗ (𝑝𝑓 )) = 𝑍(𝜃2𝑠 , 𝑝∗ (𝑝𝑓 )).
[T2] For any 𝑝𝑓 ∈ [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] and distinct 𝜃1𝑠 , 𝜃2𝑠 ∈ 𝕋𝑛−1 such that ∂𝜃𝑠 𝑍(𝜃𝑗𝑠 , 𝑝∗ (𝑝𝑓 )) =
0 for 𝑗 = 1, 2, we have that
∂𝑝𝑓 𝑍(𝜃1𝑠 , 𝑝∗ (𝑝𝑓 )) ∕= ∂𝑝𝑓 𝑍(𝜃2𝑠 , 𝑝∗ (𝑝𝑓 )).
Let 𝒰 ′ denote the set of functions in 𝑆𝑟 that satisfies one or more of the conditions
[T0]-[T2].
Proposition A.1. 𝒰 ′ is open and dense.
Proof of Theorem 2.1. The set 𝒰 is open, since if 𝐻1 satisfies conditions [G0]-[G2]
with some 𝜆 > 0, any 𝐻1′ sufficiently close to 𝐻1 in 𝐶 𝑟 norm satisfies these conditions
with a slightly smaller 𝜆′ > 0.
We now prove that 𝒰 is dense by showing that 𝒰 ⊃ 𝒰 ′ . The conditions [T0]-[T2]
implies the statement that any 𝑝𝑓 is either a nondegenerate regular point or a nondegenerate bifurcation point, and that there are at most finitely many bifurcation
𝑠−1
points. To see that [T0]-[T2] also imply conditions [G0]-[G2], let {[𝑎𝑗 , 𝑎𝑗+1 ]}𝑗−0
be
𝑓
the partition of [𝑎𝑚𝑖𝑛 , 𝑎𝑚𝑎𝑥 ] by bifurcation points. Each 𝑝 ∈ (𝑎𝑗 , 𝑎𝑗+1 ) defines a
unique global maximum 𝜃𝑗𝑠 (𝑝𝑓 ). The function 𝜃𝑠 (𝑝𝑓 ) is continuous since any converging sequence 𝜃𝑠 (𝑝𝑓𝑘 ) also converges to a global maximum, and it must be smooth by
implicit function theorem. The function extends to [𝑎𝑗 , 𝑎𝑗+1 ] by continuity, and using
the nondegeneracy of the maximum and implicit function theorem, we can extend 𝜃𝑗𝑠
smoothly to the interval [𝑎𝑗 −𝑑, 𝑎𝑗+1 +𝑑], such that each 𝜃𝑗𝑠 (𝑝𝑓 ) is a nondegenerate local
maxima. Assume that for each 𝑝𝑓 ∈ [𝑎𝑗 −𝑑, 𝑎𝑗+1 +𝑑] we have −∂𝜃2𝑠 𝜃𝑠 𝑍(𝜃𝑠 , 𝑝∗ (𝑝𝑓 )) ≥ 𝑑′ 𝐼
as a quadratic form, hence 𝑍 satisfies [G0] with 𝜆 = min{𝑑, 𝑑′ }. [G1] and [G2] are
direct consequences of [T0]-[T2].
□
Appendix B. Normally hyperbolic manifold
Let 𝐹 : ℝ𝑛 −→ ℝ𝑛 be a 𝐶 1 vector field. We give sufficient conditions for the
existence of a Normally hyperbolic invariant graph of 𝐹 . We split the space ℝ𝑛 as
Arnold diffusion along normally hyperbolic invariant cylinders
47
ℝ𝑛𝑢 ×ℝ𝑛𝑠 ×ℝ𝑛𝑐 , and denote by 𝑥 = (𝑢, 𝑠, 𝑐) the points of ℝ𝑛 . We denote by (𝐹𝑢 , 𝐹𝑠 , 𝐹𝑐 )
the components of 𝐹 :
𝐹 (𝑥) = (𝐹𝑢 (𝑥), 𝐹𝑠 (𝑥), 𝐹𝑐 (𝑥)).
We study the flow of 𝐹 in the domain
Ω = 𝐵 𝑢 × 𝐵 𝑠 × Ω𝑐
where 𝐵 𝑢 and 𝐵 𝑠 are the open Euclidean balls of radius 𝑟𝑢 and 𝑟𝑠 in ℝ𝑛𝑢 and ℝ𝑛𝑠 ,
and Ω𝑐 is a convex open subset of ℝ𝑛𝑐 . We denote by
⎡
⎤
𝐿𝑢𝑢 (𝑥) 𝐿𝑢𝑠 (𝑥) 𝐿𝑢𝑐 (𝑥)
𝐿(𝑥) = 𝑑𝐹 (𝑥) = ⎣ 𝐿𝑠𝑢 (𝑥) 𝐿𝑠𝑠 (𝑥) 𝐿𝑠𝑐 (𝑥) ⎦
𝐿𝑐𝑢 (𝑥) 𝐿𝑐𝑠 (𝑥) 𝐿𝑐𝑐 (𝑥)
the linearized vector field at point 𝑥. We assume that ∥𝐿(𝑥)∥ is bounded on Ω, which
implies that each trajectory of 𝐹 is defined until it leaves Ω. We denote by 𝑊 𝑐 the
union of full orbits contained in Ω. In other words, this is the set of initial conditions
𝑥 ∈ Ω such that there exists a solution 𝑥(𝑡) : ℝ −→ Ω of the equation 𝑥˙ = 𝐹 (𝑥)
satisfying 𝑥(0) = 0. We denote by 𝑊 𝑠𝑐 the set of points whose positive orbit remains
inside Ω. In other words, this is the set of initial conditions 𝑥 ∈ Ω such that there
exists a solution 𝑥(𝑡) : [0, ∞) −→ Ω of the equation 𝑥˙ = 𝐹 (𝑥) satisfying 𝑥(0) = 0.
Finally, we denote by 𝑊 𝑢𝑐 the set of points whose negative orbit remains inside Ω. In
other words, this is the set of initial conditions 𝑥 ∈ Ω such that there exists a solution
𝑥(𝑡) : (∞, 0] −→ Ω of the equation 𝑥˙ = 𝐹 (𝑥) satisfying 𝑥(0) = 0. These sets have
specific features under the following assumptions:
Hypothesis B.1 (Isolating block). We have:
∙ 𝐹𝑐 = 0 on 𝐵 𝑢 × 𝐵 𝑠 × ∂Ω𝑐 .
¯ 𝑠 × Ω̄𝑐 .
∙ 𝐹𝑢 (𝑢, 𝑠, 𝑐) ⋅ 𝑢 > 0 on ∂𝐵 𝑢 × 𝐵
𝑢
¯ × ∂𝐵 𝑠 × Ω̄𝑐 .
∙ 𝐹𝑠 (𝑢, 𝑠, 𝑐) ⋅ 𝑠 < 0 on 𝐵
Hypothesis B.2. There exist positive constants 𝜆, 𝑚 and 𝑀 such that:
∙ 𝐿𝑢𝑢 (𝑥) ⩾ 𝜆𝐼, 𝐿𝑠𝑠 (𝑥) ⩽ −𝜆𝐼 for each 𝑥 ∈ Ω in the sense of quadratic forms.
∙ ∥𝐿𝑢𝑠 (𝑥)∥ + ∥𝐿𝑢𝑐 (𝑥)∥ + ∥𝐿𝑠𝑠 (𝑥)∥ + ∥𝐿𝑠𝑐 (𝑥)∥ ⩽ 𝑚 for each 𝑥 ∈ Ω.
∙ ∥𝐿𝑐𝑢 (𝑥)∥ + ∥𝐿𝑐𝑠 (𝑥)∥ + ∥𝐿𝑐𝑐 (𝑥)∥ ⩽ 𝑀 for each 𝑥 ∈ Ω.
Theorem B.1. Assume that Hypotheses B.1 and B.2 hold, and that
𝑚
1
𝐾 :=
⩽√ .
𝜆 − 2(𝑀 + 𝑚)
2
Then the set 𝑊 𝑠𝑐 is the graph of a 𝐶 1 function
𝑤𝑠𝑐 : 𝐵 𝑠 × Ω𝑐 −→ 𝐵 𝑢 ,
48
P. Bernard, V. Kaloshin, K. Zhang
the set 𝑊 𝑢𝑐 is the graph of a 𝐶 1 function
𝑤𝑢𝑐 : 𝐵 𝑢 × Ω𝑐 −→ 𝐵 𝑠 ,
and the set 𝑊 𝑐 is the graph of a 𝐶 1 function
𝑤𝑐 : Ω𝑐 −→ 𝐵 𝑢 × 𝐵 𝑠 .
Moreover, we have the estimates
∥𝑑𝑤𝑠𝑐 ∥ ⩽ 𝐾,
∥𝑑𝑤𝑢𝑐 ∥ ⩽ 𝐾,
∥𝑑𝑤𝑐 ∥ ⩽ 2𝐾.
Proof. This results can be reduced to several already existing ones or proved directly
by well-known methods. We shall use Theorem 1.1 in [Ya] which is the closest to
our needs because it is expressed in terms of vector fields. We first derive some
conclusions from the isolating block conditions. We denote by 𝜋 𝑠𝑐 the projection
(𝑢, 𝑠, 𝑐) ?−→ (𝑠, 𝑐), and so on.
Lemma B.1. If Hypothesis B.1 holds, then
𝜋 𝑢𝑐 (𝑊 𝑢𝑐 ) = 𝐵 𝑢 × Ω𝑐
and
𝜋 𝑠𝑐 (𝑊 𝑠𝑐 ) = 𝐵 𝑠 × Ω𝑐 .
Moreover, the closures of 𝑊 𝑠𝑐 and 𝑊 𝑢𝑐 satisfy
¯ 𝑠𝑐 ⊂ 𝐵 𝑠 × 𝐵
¯ 𝑐 × Ω̄𝑐 , 𝑊
¯ 𝑢𝑐 ⊂ 𝐵
¯ 𝑠 × 𝐵 𝑐 × Ω̄𝑐 .
𝑊
Proof. Let us define 𝑇 + (𝑥) ∈ [0, ∞] as the first positive time where the orbit of 𝑥
hits the boundary ∂Ω. Let us denote by 𝜑(𝑡, 𝑥) the flow of 𝐹 . If 𝑇 (𝑥) < ∞, we have
𝜑(𝑇 (𝑥), 𝑥) ∈ ∂𝐵 𝑢 × 𝐵 𝑠 × Ω, as follows from Hypothesis B.1. Then, it is easy to check
that the function 𝑇 is continuous, and even 𝐶 1 , at 𝑥.
We prove the first equality of the Lemma by contradiction, and assume that there
exists a point (𝑢, 𝑐) ∈ 𝐵 𝑢 ×Ω𝑐 such that 𝑊 𝑢𝑐 does not intersect the disc {𝑢}×𝐵 𝑠 ×{𝑐}.
Then, the first exit map
𝐵 𝑠 ∋ 𝑠 ?−→ 𝜑(𝑇 (𝑥), 𝑥) ∈ ∂𝐵 𝑠
¯ 𝑠 to its boundary ∂𝐵 𝑠 . Such
extends by continuity to a continuous retraction from 𝐵
a retraction does not exist. The proof of the other equality is similar.
Finally, we have
)∪( 𝑢
)
(
¯ 𝑢𝑐 ⊂ 𝐵
¯𝑢 × 𝐵
¯ 𝑠 × Ω̄𝑐 = 𝐵 𝑢 × 𝐵
¯ 𝑠 × Ω̄𝑐
¯ 𝑠 × Ω̄𝑐 .
𝑊
∂𝐵 × 𝐵
¯ 𝑠 × Ω̄𝑐 has a neighborhood formed
Hypothesis B.1 implies that each point of ∂𝐵 𝑢 × 𝐵
¯ 𝑠 × Ω̄𝑐
of points which leave Ω after a small time. As a consequence, the set ∂𝐵 𝑢 × 𝐵
𝑢𝑐
𝑢𝑐
𝑢
𝑠
𝑐
¯ , and we have proved that 𝑊
¯ ⊂ 𝐵 ×𝐵
¯ × Ω̄ . The other inclusion
can’t intersect 𝑊
can be proved in a similar way.
□
Arnold diffusion along normally hyperbolic invariant cylinders
49
In order to prove the statement of the Theorem concerning 𝑊 𝑠𝑐 , we apply Theorem
1.1 of [Ya]. More precisely, using the notation of that paper, we set
𝑎 = 𝑠/𝐾,
𝑧 = (𝑢, 𝑐),
𝑓 (𝑎, 𝑧) = 𝐹𝑢 (𝐾𝑎, 𝑧)/𝐾,
We have the estimates
∂𝑎 𝑓 = 𝐿𝑢𝑢 ⩾ 𝜆,
𝑔(𝑎, 𝑧) = (𝐹𝑠 (𝐾𝑎, 𝑧), 𝐹𝑐 (𝐾𝑎, 𝑧)).
[
]
𝐿𝑠𝑠 𝐿𝑠𝑐
∂𝑧 𝑔 =
⩽𝑚+𝑀
𝐿𝑐𝑠 𝐿𝑐𝑐
in the sense of quadratic forms. Moreover, we have the estimates
𝑚
∥∂𝑧 𝑓 ∥ ⩽ , ∥∂𝑎 𝑔∥ ⩽ 𝐾(𝑚 + 𝑀 ).
𝐾
Since
𝑚 + 𝑀 + 𝑚/𝐾 + 𝐾(𝑚 + 𝑀 ) < 2(𝑚 + 𝑀 ) + 𝑚/𝐾 = 𝜆
we conclude that Hypothesis 2 of [Ya] is satisfied. Hypothesis 1 of [Ya] is verified
by the domain Ω, and Hypothesis 3 is precisely the conclusion of Lemma B.1. As a
consequence, we can apply Theorem 1.1 of [Ya], and conclude that the set 𝑊 𝑠𝑐 is the
graph of a 𝐶 1 and 1-Lipschitz map above 𝐵 𝑠 × Ω𝑐 in (𝑎, 𝑧) coordinates, and therefore
the graph of a 𝐾-Lipschitz 𝐶 1 map 𝑤𝑠𝑐 : 𝐵 𝑠 × Ω𝑐 −→ 𝐵 𝑢 in (𝑢, 𝑠, 𝑐) coordinates.
In order to prove the statement concerning 𝑊 𝑢𝑐 , we apply Theorem 1.1 of [Ya] with
𝑎 = 𝑢/𝐾,
𝑧 = (𝑠, 𝑐),
𝑓 (𝑎, 𝑧) = −𝐹𝑠 (𝐾𝑎, 𝑧)/𝐾, 𝑔(𝑎, 𝑧) = −(𝐹𝑢 (𝐾𝑎, 𝑧), 𝐹𝑐 (𝐾𝑎, 𝑧)).
It is easy to check as above that all hypotheses are satisfied.
Let us now study the set 𝑊 𝑐 = 𝑊 𝑠𝑐 ∩ 𝑊 𝑢𝑐 . First, let us prove that 𝑊 𝑐 is a 𝐶 1
graph above Ω𝑐 . We know that 𝑊 𝑠𝑐 is the graph of a 𝐾-Lipshitz 𝐶 1 function 𝑤𝑠𝑐 (𝑠, 𝑐)
and that 𝑊 𝑢𝑐 is the graph of a 𝐾-Lipshitz 𝐶 1 function 𝑤𝑢𝑐 (𝑢, 𝑐). The point (𝑢, 𝑠, 𝑐)
belongs to 𝑊 𝑐 if and only if
𝑢 = 𝑤𝑠𝑐 (𝑠, 𝑐) and 𝑠 = 𝑤𝑢𝑐 (𝑢, 𝑐),
or in other words if and only if (𝑢, 𝑠) is a fixed point of the 𝐾-Lipschitz 𝐶 1 map
(𝑢, 𝑠) ?−→ (𝑤𝑠𝑐 (𝑠, 𝑐), 𝑤𝑢𝑐 (𝑢, 𝑐)).
¯𝑢 × 𝐵
¯ 𝑠 , which corFor each 𝑐, this contracting map has a unique fixed point in 𝐵
𝑠𝑐
𝑢𝑐
¯ ∩𝑊
¯ . It follows from Lemma B.1 that this point is
responds to a point of 𝑊
𝑢
𝑠
contained in 𝐵 × 𝐵 . Then, it depends in a 𝐶 1 way of the parameter 𝑐. We have
proved that 𝑊 𝑐 is the graph of a 𝐶 1 function 𝑤𝑐 . In order to estimate the Lipschitz
constant of this graph, we consider two points (𝑢𝑖 , 𝑠𝑖 , 𝑐𝑖 ), 𝑖 = 0, 1 in Γ. We have
∥𝑢1 − 𝑢0 ∥2 ⩽ 𝐾 2 (∥𝑠1 − 𝑠0 ∥2 + ∥𝑐1 − 𝑐0 ∥2 )
50
P. Bernard, V. Kaloshin, K. Zhang
and
∥𝑠1 − 𝑠0 ∥2 ⩽ 𝐾 2 (∥𝑢1 − 𝑢0 ∥2 + ∥𝑐1 − 𝑐0 ∥2 ).
Taking the sum gives
and
(1 − 𝐾 2 )(∥𝑢1 − 𝑢0 ∥2 + ∥𝑠1 − 𝑠0 ∥2 ) ⩽ 2𝐾 2 ∥𝑐1 − 𝑐0 ∥2
√
2𝐾 2
∥(𝑢1 , 𝑠1 ) − (𝑢0 , 𝑠0 )∥ ⩽
∥𝑐1 − 𝑐0 ∥ ⩽ 2𝐾∥𝑐1 − 𝑐0 ∥,
1 − 𝐾2
√
□We need an addendum
since 𝐾 ⩽ 1/ 2. We conclude that 𝑤𝑐 is 2𝐾-Lipschitz.
for applications:
Proposition B.2. Assume in addition that there exists a translation 𝑔 of ℝ𝑛𝑐 such
that
𝑔(Ω𝑐 ) = Ω𝑐 and 𝐹 ∘ (𝑖𝑑 ⊗ 𝑖𝑑 ⊗ 𝑔) = 𝐹.
Then we have
𝑤𝑠𝑐 ∘ (𝑖𝑑 ⊗ 𝑔) = 𝑤𝑠𝑐 ,
𝑤𝑢𝑐 ∘ (𝑖𝑑 ⊗ 𝑔) = 𝑤𝑢𝑐 ,
𝑤𝑐 ∘ 𝑔 = 𝑤𝑐 .
Proof. It follows immediately from the definition of the sets 𝑊 𝑠𝑐 , 𝑊 𝑢𝑐 and 𝑊 𝑐 that
𝑔(𝑊 𝑠𝑐 ) = 𝑊 𝑠𝑐 , 𝑔(𝑊 𝑢𝑐 ) = 𝑊 𝑢𝑐 and 𝑔(𝑊 𝑐 ) = 𝑊 𝑐 .
□
In applications the first condition of Hypothesis B.1 is usually not satisfied, except
in the case where Ω𝑐 = ℝ𝑛𝑐 . It is thus useful to state a more ”applicable” variant of
the result. given a positive parameter 𝑟, let Ω𝑐𝑟 be the set of points 𝑐 ∈ ℝ𝑛𝑐 such that
𝑑(𝑐, Ω𝑐 ) < 𝑟. This is a convex open subset of ℝ𝑛𝑐 containing Ω𝑐 . We denote by Ω𝑟
the product 𝐵 𝑢 × 𝐵 𝑠 × Ω𝑐𝑟 .
Proposition B.3. Assume that there exists 𝜆, 𝑚, 𝑀, 𝑟 > 0 such that
¯ 𝑠 × Ω̄𝑐𝑟 .
∙ 𝐹𝑢 (𝑢, 𝑠, 𝑐) ⋅ 𝑢 > 0 on ∂𝐵 𝑢 × 𝐵
𝑢
¯ × ∂𝐵 𝑠 × Ω̄𝑐𝑟 .
∙ 𝐹𝑠 (𝑢, 𝑠, 𝑐) ⋅ 𝑠 < 0 on 𝐵
∙ 𝐿𝑢𝑢 (𝑥) ⩾ 𝜆𝐼, 𝐿𝑠𝑠 (𝑥) ⩽ −𝜆𝐼 for each 𝑥 ∈ Ω𝑟 in the sense of quadratic forms.
∙ ∥𝐿𝑢𝑠 (𝑥)∥ + ∥𝐿𝑢𝑐 (𝑥)∥ + ∥𝐿𝑠𝑠 (𝑥)∥ + ∥𝐿𝑠𝑐 (𝑥)∥ ⩽ 𝑚 for each 𝑥 ∈ Ω𝑟 .
∙ ∥𝐿𝑐𝑢 (𝑥)∥ + ∥𝐿𝑐𝑠 (𝑥)∥ + ∥𝐿𝑐𝑐 (𝑥)∥ + 2∥𝐹𝑐 (𝑥)∥/𝑟 ⩽ 𝑀 for each 𝑥 ∈ Ω𝑟 .
Assume Furthermore that
1
𝑚
⩽√ .
𝐾 :=
𝜆 − 2(𝑀 + 𝑚)
2
1
𝑐
Then, there exists a 𝐶 function 𝜌 : Ω𝑟 −→ [0, 1] which is equal to 1 on Ω𝑐 and such
that the vector field
𝐹˜ (𝑢, 𝑠, 𝑐) := (𝐹𝑢 (𝑢, 𝑠, 𝑐), 𝐹𝑐 (𝑢, 𝑠, 𝑐), 𝜌(𝑐)𝐹𝑐 (𝑢, 𝑠, 𝑐))
satisfies all the hypotheses of Theorem B.1 on Ω𝑟 . Note that 𝐹˜ = 𝐹 on Ω.
Arnold diffusion along normally hyperbolic invariant cylinders
51
Proof. We take a function 𝜌(𝑐) such that :
∙ 𝜌 = 0 near the boundary of Ω𝑐𝑟 ,
∙ 𝜌 = 1 on Ω𝑐 ,
∙ ∥𝑑𝜌∥ ⩽ 2/𝑟.
˜ ∗∗ the variational matrix associated to 𝐹˜ , we see that
Denoting by 𝐿
˜ 𝑐𝑢 (𝑢, 𝑠, 𝑐) = 𝜌(𝑐)𝐿𝑐𝑢 (𝑢, 𝑠, 𝑐), 𝐿
˜ 𝑐𝑠 (𝑢, 𝑠, 𝑐) = 𝜌(𝑐)𝐿𝑐𝑠 (𝑢, 𝑠, 𝑐)
𝐿
and
˜ 𝑐𝑐 (𝑢, 𝑠, 𝑐) = 𝜌(𝑐)𝐿𝑐𝑐 (𝑢, 𝑠, 𝑐) + 𝑑𝜌(𝑢, 𝑠, 𝑐) ⊗ 𝐹𝑐 (𝑢, 𝑠, 𝑐).
𝐿
As a consequence, we have
˜ 𝑐𝑢 (𝑥)∥+∥𝐿
˜ 𝑐𝑠 (𝑥)∥ + ∥𝐿
˜ 𝑐𝑐 (𝑥)∥ =
∥𝐿
(
)
= 𝜌(𝑐) ∥𝐿𝑐𝑢 (𝑥) + ∥𝐿𝑐𝑠 (𝑥)∥ + ∥𝐿𝑐𝑐 (𝑥)∥ + ∥𝐹𝑐 (𝑥)∥∥𝑑𝜌(𝑐)∥
⩽ ∥𝐿𝑐𝑢 (𝑥) + ∥𝐿𝑐𝑠 (𝑥)∥ + ∥𝐿𝑐𝑐 (𝑥)∥ + 2∥𝐹𝑐 (𝑥)∥/𝑟 ⩽ 𝑀.
□
˜ 𝑠𝑐 , 𝑊
˜ 𝑢𝑐 , 𝑊
˜ 𝑐 associated to 𝐹˜
Under the hypotheses of Proposition B.3, the sets 𝑊
are graphs of 𝐶 1 functions
𝑤˜ 𝑠𝑐 : 𝐵 𝑠 × Ω𝑐𝑟 −→ 𝐵 𝑢 ,
which satisfying the estimates
𝑤˜ 𝑢𝑐 : 𝐵 𝑢 × Ω𝑐𝑟 −→ 𝐵 𝑠 ,
∥𝑑𝑤˜ 𝑠𝑐 ∥ ⩽ 𝐾,
The restrictions to Ω
˜ 𝑠𝑐 ∩ Ω,
𝒲 𝑠𝑐 = 𝑊
∥𝑑𝑤˜ 𝑢𝑐 ∥ ⩽ 𝐾,
˜ 𝑢𝑐 ∩ Ω,
𝒲 𝑢𝑐 = 𝑊
𝑤˜ 𝑐 : Ω𝑐𝑟 −→ 𝐵 𝑢 × 𝐵 𝑠
∥𝑑𝑤˜ 𝑐 ∥ ⩽ 2𝐾.
˜ 𝑐 ∩ Ω,
𝒲 𝑐 = 𝒲 𝑠𝑐 ∩ 𝒲 𝑢𝑐 = 𝑊
are weakly invariant by 𝐹 in the sense that this vector field is tangent to them. They
satisfy various interesting properties. For example, each 𝐹 -invariant set contained in
Ω is contained in 𝒲 𝑐 .
References
[Ar1]
[Ar2]
[Be1]
[Be2]
[Be3]
Arnold, V. Instabilities in dynamical systems with several degrees of freedom, Sov Math Dokl
5 (1964), 581–585.
Arnold, V. Mathematical methods of classical mechanics, 2nd edition, Graduate Texts in
Mathematics, Springer, 1989.
Bernard, P. The dynamics of pseudographs in convex Hamiltonian systems. J. Amer. Math.
Soc. 21 (2008), no. 3, 615–669.
Bernard, P. Symplectic aspects of Mather theory. Duke Math. J. 136 (2007), 401–420
Bernard, P. Large normally hyperbolic cylinders in a priori stable Hamiltonian systems.
Annales Henri Poincaré 11 (2010), No. 5, 929–942.
52
[BK]
[CY1]
[CY2]
[DH]
[Fa]
[KZZ]
[Ma1]
[Ma2]
[Ma3]
[Ma4]
[Ma5]
[MF]
[Ya]
P. Bernard, V. Kaloshin, K. Zhang
Bourgain J. and Kaloshin V. On diffusion in high-dimensional Hamiltonian systems. J.
Funct. Anal. 229(1) (2005), 1–61.
Cheng, Ch.-Q. and Yan, J. Existence of diffusion orbits in a priori unstable Hamiltonian
systems. J. Diff Geom. 67 (2004), 457–517
Cheng, Ch.-Q. and Yan, J. Arnold diffusion in Hamiltonian systems a priori unstable case.
J. Diff Geom. 82 (2009), 229–277
Delshams, A. and Huguet, G. Geography of resonances and Arnold diffusion in a priori
unstable Hamiltonian systems. Nonlinearity 22 (2009), no. 8, 1997–2077.
Fathi, A. A. Weak KAM theorem in Lagrangian dynamics, fifth prelimiary edition, book
preprint.
Kaloshin, V. Zhang, K. Zheng, Y. Almost dense orbit on energy surface, Proceedings of
XVITH International Congress on Mathmatical Physics. Prague, Czech Republic, 2009.
Edited by Pavel Exner (Doppler Institute, Prague, Czech Republic). Published by World
Scientific Publishing Co. Pte. Ltd., 314-322.
Mather, J. Action minimizing invariant measures for positive definite Lagrangian systems,
Math. Z. 207 (1991), 169–207.
Mather J. Variational construction of connecting orbits, Ann. Inst. Fourier, 43 (1993), 13491386.
Mather, J. Arnold diffusion. I. Announcement of results. (Russian) Sovrem. Mat. Fundam.
Napravl. 2 (2003), 116–130 (electronic); translation in J. Math. Sci. (N. Y.) 124 (2004), no.
5, 5275–5289.
Mather, J. Arnold diffusion II, prerpint, 2008, 183 pp.
Mather, J. Lecture course on Arnold diffusion, Maryland, spring 2010, 15 lectures.
Mather, J and Forni, G. Action minimizing orbits in Hamiltonian systems. Transition to
chaos in classical and quantum mechanics (Montecatini Terme, 1991), 92186, Lecture Notes
in Math., 1589, Springer, Berlin, 1994.
Yang, D. An invariant manifold for ODEs and its applications, preprint,
http://arxiv.org/abs/0909.1103v1.
∗ - Université Paris - Dauphine,
# - Penn State University, University of Maryland at College Park,
∗∗ - University of Toronto.