A strong form of Arnold diffusion for two and a half degrees of freedom

2.1 MB·English

by  Vadim Kaloshin

A strong form of Arnold diffusion for two and a half
degrees of freedom
V. Kaloshin∗, K. Zhang†
3
1
0
January 29, 2013
2

n
a
J

8
2
Abstract

]
S In the present paper we prove a strong form of Arnold diffusion. Let T2 be
D the two torus and B2 be the unit ball around the origin in R2. Fix ρ > 0. Our
. mainresultsaysthatfora“generic”time-periodicperturbationofanintegrable
h
t system of two degrees of freedom
a
m
H (p)+εH (θ,p,t), θ ∈ T2, p ∈ B2, t ∈ T = R/Z,
[ 0 1

2 withastrictlyconvexH ,thereexistsaρ-denseorbit(θ ,p ,t)(t)inT2×B2×T,
0 (cid:15) (cid:15)
v namely, a ρ-neighborhood of the orbit contains T2×B2×T.
0
5 Our proof is a combination of geometric and variational methods. The
1 fundamental elements of the construction are usage of crumpled normally hy-
1
perbolic invariant cylinders from [13], flower and simple normally hyperbolic
.
2
invariantmanifoldsfrom[47]aswellastheirkissingpropertyatastrongdouble
1
2 resonance. This allows us to builda “connected” net of 3-dimensional normally
1 hyperbolic invariant manifolds. To construct diffusing orbits along this net we
:
v employ a version of Mather variational method [54] equipped with weak KAM
i theory [34], proposed by Bernard in [9].
X
r
a
∗University of Maryland at College Park (vadim.kaloshin gmail.com)
†University of Toronto (kzhang math.utoronto.edu)
1
Contents
1 Introduction 5
1.1 Statement of the result . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Preliminary splitting of the proof of Theorem 1 into ten Key Theo-
rems 11
2.1 Decomposition of the diffusion path into single and double resonances. 11
2.2 Genericity conditions at single resonances . . . . . . . . . . . . . . . . 13
3 Theorems on existence of normally hyperbolic invariant manifolds 19
3.1 Description of single resonances . . . . . . . . . . . . . . . . . . . . . 19
3.2 Description of double resonances and generic properties at high energy 22
3.3 Description of double resonances (low energy) . . . . . . . . . . . . . 25
3.4 Generic properties of homoclinic orbits and genericity at low energy . 27
3.5 Heuristics of the diffusion across double resonances . . . . . . . . . . 34
3.5.1 Crossing through along a simple loop γ0 (cid:51) 0 . . . . . . . . . . 34
h
3.5.2 Crossing through along a simple loop γ0 (cid:54)(cid:51) 0 . . . . . . . . . . 35
h
3.5.3 Crossing through along a non-simple loop γ0 . . . . . . . . . . 35
h
3.5.4 Turning a corner from Γ to Γ(cid:48) . . . . . . . . . . . . . . . . . . 35
4 Localization of the Aubry sets and the Man˜e sets and Mather’s
projected graph theorems 37
4.1 Localization and Mather’s projected graph theorem for single resonances 37
4.2 LocalizationandMather’sprojectedgraphtheoremfordoubleresonances 40
4.2.1 NHIMs near a double resonance . . . . . . . . . . . . . . . . . 41
4.2.2 Choice of the cohomology classes . . . . . . . . . . . . . . . . 42
4.2.3 Properties of the Aubry sets and Man˜e sets . . . . . . . . . . 44
4.3 Choice of auxiliary cohomology classes for a non-simple homology . . 46
5 Description of c-equivalence and a variational λ-lemma 49
5.1 Heuristic descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Forcing relation and shadowing . . . . . . . . . . . . . . . . . . . . . 50
5.3 Choice of cohomology classes for global diffusion . . . . . . . . . . . . 52
6 Equivalent forcing classes 56
6.1 Equivalent forcing class along single resonances . . . . . . . . . . . . 56
6.2 Equivalent forcing class along cylinders of the same homology class . 58
6.3 Equivalent forcing class between kissing cylinders . . . . . . . . . . . 63
2
7 Normally hyperbolic invariant cylinders through the transition zone
into double resonances 65
7.1 Normally hyperbolic invariant manifolds going into double resonances 66
7.2 A Normal form in the transition zone . . . . . . . . . . . . . . . . . . 68
7.2.1 Autonomous case and slow-fast coordinates . . . . . . . . . . 68
7.2.2 Time-periodic setting and reduction to d = 3 . . . . . . . . . . 72
7.3 Construction of an isolating block . . . . . . . . . . . . . . . . . . . . 73
7.3.1 Auxiliary estimates on the vector field. . . . . . . . . . . . . . 73
7.3.2 Constructing the isolation block . . . . . . . . . . . . . . . . . 78
8 Proof of Key Theorem 3 about existence of invariant cylinders at
double resonances 81
8.1 Normal form near the hyperbolic fixed point . . . . . . . . . . . . . . 82
8.2 Behavior of a family of orbits passing near 0 and Shil’nikov boundary
value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.3 Properties of the local maps . . . . . . . . . . . . . . . . . . . . . . . 88
8.4 Conley-McGehee isolating blocks . . . . . . . . . . . . . . . . . . . . 93
8.5 Single leaf cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.6 Double leaf cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.7 Normally hyperbolic invariant cylinders the slow mechanical system . 100
8.8 Smooth approximations . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.9 Proof of Lemma 3.3 on cyclic concatenations of simple geodesics . . . 104
9 Diffusion mechanisms, weak KAM theory and forcing relation 106
9.1 Duality of Hamiltonian and Lagrangian, homology and cohomology . 106
9.2 Overlapping pseudographs . . . . . . . . . . . . . . . . . . . . . . . . 108
9.3 Evolution of pseudographs and the Lax-Oleinik mapping . . . . . . . 109
9.4 Aubry, Mather, Man˜e sets in Hamiltonian setting and properties of
forcing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.5 Symplectic invariance of the Mather, Aubry, and Man˜e sets . . . . . . 112
9.6 Mather’s α and β-functions, Legendre-Fenichel transform and barrier
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.7 Forcing relation of cohomology classes and its dynamical properties . 114
9.8 Other diffusion mechanisms and apriori unstable systems . . . . . . . 115
10 Properties of the barrier functions 117
10.1 The Lagrangian setting . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.2 Properties of the action and barrier functions . . . . . . . . . . . . . 122
10.2.1 Uniform family and Tonelli convergence . . . . . . . . . . . . 122
10.2.2 Properties of the action function . . . . . . . . . . . . . . . . . 123
3
10.2.3 Properties of the barrier function . . . . . . . . . . . . . . . . 125
10.2.4 Semi-continuity of the Aubry and Man˜e set and continuity of
the barrier function . . . . . . . . . . . . . . . . . . . . . . . . 126
11 Diffusion along the same homology at double resonance 129
11.1 Localization and graph theorem . . . . . . . . . . . . . . . . . . . . . 129
11.2 Forcing relation along the same homology class . . . . . . . . . . . . 131
11.3 Local extensions of the Aubry sets and a proof of Theorem 23 . . . . 134
˜
11.4 Generic property of the Aubry sets A (c¯ (E)) . . . . . . . . . . . . 137
Hs h
ε
11.5 Generic property of the α-function and proof of Theorem 13 . . . . . 137
11.6 Nondegeneracy of the barrier functions . . . . . . . . . . . . . . . . . 138
12 Equivalent forcing class between kissing cylinders 143
12.1 Variational problem for the slow mechanical system . . . . . . . . . . 143
12.2 Variational problem in the original coordinates . . . . . . . . . . . . . 146
12.3 Scaling limit of the barrier function . . . . . . . . . . . . . . . . . . . 148
12.4 Proof of forcing relation . . . . . . . . . . . . . . . . . . . . . . . . . 150
A Generic properties of mechanical systems on the two-torus 155
A.1 Generic properties of periodic orbits . . . . . . . . . . . . . . . . . . . 155
A.2 Generic properties of minimal orbits . . . . . . . . . . . . . . . . . . . 161
A.3 Proof of Theorem 4 about genericity of [DR1]-[DR3] . . . . . . . . . . 168
A.4 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Derivation of the slow mechanical system 172
B.1 Normal forms near double resonances . . . . . . . . . . . . . . . . . . 172
B.2 Affine coordinate change and rescaling . . . . . . . . . . . . . . . . . 177
B.3 Variational properties of the coordinate changes . . . . . . . . . . . . 180
C Variational aspects of the slow mechanical system 184
C.1 Relation between the minimal geodesics and the Aubry sets . . . . . 184
C.2 Characterization of the channel and the Aubry sets . . . . . . . . . . 187
C.3 The width of the channel . . . . . . . . . . . . . . . . . . . . . . . . . 189
C.4 The case E = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
D Transition between single and double resonance 195
E Notations 197
4
1 Introduction
The famous question called the ergodic hypothesis, formulated by Maxwell and Boltz-
mann, suggests that for a typical Hamiltonian on a typical energy surface all, but a
set of zero measure of initial conditions, have trajectories dense in this energy surface.
However, KAM theory showed that for an open set of nearly integrable systems there
isasetofinitialconditionsofpositivemeasurewithalmostperiodictrajectories. This
disproved the ergodic hypothesis and forced to reconsider the problem.
A quasi-ergodic hypothesis, proposed by Ehrenfest [33] and Birkhoff [17], asks if a
typical Hamiltonian on a typical energy surface has a dense orbit. A definite answer
whether this statement is true or not is still far out of reach of modern dynamics.
There was an attempt to prove this statement by E. Fermi [36], which failed (see [37]
for a more detailed account). To simplify the problem, Arnold [5] asks:
Does there exist a real instability in many-dimensional problems of perturbation
theory when the invariant tori do not divide the phase space?
For nearly integrable systems of one and a half and two degrees of freedom the
invariant tori do divide the phase space and an energy surface respectively. This
implies that instability do not occur. We solve a weaker version of this question
for systems with two and a half and 3 degrees of freedom. This corresponds to
time-periodic perturbations of integrable systems with two degrees of freedom and
autonomous perturbations of integrable systems with three degrees of freedom.
1.1 Statement of the result
Let (θ,p) ∈ T2 ×B2 be the phase space of an integrable Hamiltonian system H (p)
0
with T2 being 2-dimensional torus T2 = R2/Z2 (cid:51) θ = (θ ,θ ) and B2 being the unit
1 2
ball around 0 in R2, p = (p ,p ) ∈ B2. Assume that H is strictly convex, i.e. Hessian
1 2 0
∂2 H is strictly positive definite.
pipj 0
Consider a smooth time periodic perturbation
H (θ,p,t) = H (p)+εH (θ,p,t), t ∈ T = R/T.
ε 0 1
We study a strong form of Arnold diffusion for this system, namely,
existence of orbits {(θ ,p )(t)} going from one open set p (0) ∈ U to another
(cid:15) (cid:15) t (cid:15)
p (t) ∈ U(cid:48) for some t = t > 0.
(cid:15) (cid:15)
Arnold [3] proved existence of such orbits for an example and conjectured that they
exist for a typical perturbation (see e.g. [4, 5, 7]).
Integer relations (cid:126)k ·(∂ H ,1) = 0 with (cid:126)k = ((cid:126)k ,k ) ∈ (Z2 \0)×Z and · being
p 0 1 0
the inner product define a resonant segment. The condition that the Hessian of H is
0
5
non-degenerate implies that ∂ H : B2 −→ R2 is a diffeomorphism and each resonant
p 0
line defines a smooth curve embedded into action space
Γ = {p ∈ B2 : (cid:126)k ·(∂ H ,1) = 0}.
(cid:126)k p 0
If curves Γ and Γ are given by two linearly independent resonances vectors {(cid:126)k,(cid:126)k(cid:48)},
(cid:126)k (cid:126)k(cid:48)
they either have no intersection or intersect at a single point in B2.
We call a vector (cid:126)k = ((cid:126)k ,k ) = (k1,k2,k ) ∈ (Z2 \0)×Z and the corresponding
1 0 1 1 0
resonance Γ = Γ space irreducible if either (k1,k2) = (1,0) or (0,1) or gcd((cid:126)k1) = 1,
(cid:126)k 1 1
i.e. |k1| and |k2| are relatively prime.
1 1
Consider now two open sets U,U(cid:48) ⊂ B2. Select a finite collection of space irre-
ducible resonant segments {Γ = Γ }N for some collection of {(cid:126)k }N
j (cid:126)kj j=1 j j=1
(cid:126) (cid:126)
• with neighbors k and k being linearly independent,
j j+1
• Γ ∩Γ (cid:54)= ∅ for j = 1,...,N −1 and so that
j j+1
• Γ ∩U (cid:54)= ∅ and Γ ∩U(cid:48) (cid:54)= ∅.
1 N
We would like to construct diffusing orbits along a connected path formed by
segments inside Γ ’s, i.e. we select a connected piecewise smooth curve Γ∗ ⊂ ∪N Γ
j j=1 j
so that Γ∗ ∩U (cid:54)= ∅ and Γ∗ ∩U(cid:48) (cid:54)= ∅ (see Figure 2).
Consider the space of Cr perturbations Cr(T2×B2×T) with a natural Cr norm
given by maximum of all partial derivatives of order up to r, here r < +∞. Denote
by Sr = {H ∈ Cr(T2 ×B2 ×T) : (cid:107)H (cid:107) = 1} the unit sphere in this space.
1 1 Cr
Theorem 1. In the above notations fix the piecewise smooth segment Γ∗ and 4 ≤ r <
+∞. Then there is an open and dense set U = U ⊂ Sr and a nonnegative function
Γ∗
ε = ε (H ) with ε | > 0. Let V = {(cid:15)H : H ∈ U, 0 < ε < ε }, then for an open
0 0 1 0 U 1 1 0
and dense set of (cid:15)H ∈ W (cid:40) V the Hamiltonian system H = H +(cid:15)H has an orbit
1 (cid:15) 0 1
{(θ ,p )(t)} whose action component satisfies
(cid:15) (cid:15) t
p (0) ∈ U, p (t) ∈ U(cid:48) for some t = t > 0
(cid:15) (cid:15) (cid:15)
√
Moreover, for all 0 < t < t the action component p (t) stays O( (cid:15))-close to the
(cid:15) (cid:15)
union of resonances Γ∗.
Remark 1.1. The open and dense set of perturbation U ⊂ Sr will be defined by three
sets of non-degeneracy conditions.
• In section 2.2, conditions [G0]-[G2] defines the quantitative non-degeneracy con-
dition for λ > 0. The set of λ−non-degenerate perturbations Uλ is open, and
SR
the union (cid:83) Uλ is open and dense.
λ>0 SR
6
• Each λ > 0 determines a finite set of double resonances on Γ∗. For each double
resonance p , we define two sets of non-degeneracy conditions.
0
In section 3.2, conditions [DR1]-[DR3] defines the non-degeneracy at a dou-
ble resonance p away from singularities, the non-degenerate set UE (p ) is open
0 DR 0
and dense.
In section 3.4, conditions [A0]-[A4] defines the non-degeneracy at double
resonances p near singularities. The non-degenerate set UCrit(p ) is open and
0 DR 0
dense.
We have
(cid:32) (cid:33)
(cid:91) (cid:92)
U = Uλ ∩ UE (p )∩UCrit(p ) ,
SR DR 0 DR 0
λ>0 p0
where p ranges over all double resonances. This set is clearly open and dense.
0
For any H ∈ U we prove that the Hamiltonian H + (cid:15)H has a connected
1 0 1
collection of 3-dimensional normally hyperbolic (weakly) invariant manifolds in the
phase space T2×B2×T which “shadow” a collection of segments Γ∗ connecting U and
√
U(cid:48) in the sense that the natural projection of these manifolds onto B2 is O( ε)-close
to each point in Γ∗. Note that Marco [49, 50] announced a similar result.
Once this structure established we impose further non-degeneracy condition W (cid:40)
V. For each εH ∈ W we prove existence of orbits diffusing along this collection of
1
invariant manifolds. Our proof relies on Mather variational method [54] equipped
with weak KAM theory of Fathi [34]. The crucial element is Bernard’s notion [9] of
forcing relation (see section 5.2 for more details).
Notice that the notion of genericity we use is not standard. We show that in a
neighborhood of perturbations of H the set of good directions U is open dense in
0
Sr. Around each exceptional (nowhere dense) direction we remove a cusp and call
the complement V. For this set of perturbations we establish connected collection
of invariant manifolds. Then in the complement to some exceptional perturbations
W we show that there are diffusing orbits “shadowing” these cylinders. Mather calls
such a set of perturbations cusp residual. See Figure 1.
1. Autonomous version
Let n = 3, p˜ = (p˜ ,p˜ ,p˜ ) ∈ B3, and H˜ (p) be a strictly convex Hamiltonian.
1 2 3 0
Consider the region ∂ H > ρ > 0 for some ρ > 0 and two open sets U and U(cid:48)
pi 0
˜ ˜ ˜
in this region. Then for a generic (autonomous) perturbation H (p˜)+εH (θ,p˜)
0 1
there is an orbit (θ˜ ,p˜ )(t) whose action component connects U with U(cid:48), namely,
ε ε
7
Figure 1: Description of generic perturbations
Figure 2: Resonant net
p˜ (0) ∈ U and p˜ (t) ∈ U(cid:48) for some t = t . This can be proved using energy
ε ε ε
reduction to time periodic system of two and a half degrees of freedom (see e.g.
[6, Section 45]).
2. Generic instability of resonant totally elliptic points
In [44] stability of resonant totally elliptic fixed points of symplectic maps in
dimension 4 is studied. It is shown that generically a convex, resonant, totally
elliptic point of a symplectic map is Lyapunov unstable.
3. Relation with Mather’s approach
Theorem 1 was announced by Mather [56]. Some parts are the proof of written
in [57]. Our approach is quite different from the one initiated in these two
8
manuscripts. Here we summarize the differences:
• We start by constructing a net of normally hyperbolic invariant cylinders
“over” resonant segments Γ∗.
• We prove that certain variationally defined invariant (Aubry) sets belongs
these cylinders.
• We show that these sets have the same structure as Aubry-Mather sets for
twists maps.
• In a single resonance, mainly done in [13], the fact that these invariant sets
belong to invariant cylinders allows us to adapt techniques from [9, 23, 24].
Namely, we apply variational techniques for proving existence of Arnold
diffusion for a priori unstable systems (see section 9.8 for more details).
• One important obstacle is the problem of regularity of barrier functions
(see section 9), which outside of the realm of twist maps is difficult to
overcome. It is our understanding that Mather [62] handles this problem
without proving existence of invariant cylinders.
• In a double resonance we also construct normally hyperbolic invariant
cylinders. This leads to a fairly simple and explicit structure of mini-
mal orbits near a double resonance. In particular, in order to switch from
one resonance to another we need only one jump (see section 3.5 for an
heuristic explanation and Key Theorem 10 for the precise claim).
• It is our understanding that Mather’s approach [62] requires an implicitly
defined number of jumps. His approach resembles his proof of existence of
diffusing orbits for twist maps inside a Birkhoff region of instability [55].
4. Convexity of H on an open set
0
Suppose H (p) is strictly convex on an open set
0
Conv = {p ∈ B2 : Hessian ∂2 H (p) is strictly positive definite}.
pipj 0
Then for any connected component Conv(cid:48) ⊂ Conv Theorem 1 applies. More
exactly, for any pair of open sets U,U(cid:48) ⊂ Conv(cid:48) fix a smooth segment Γ∗ ⊂
Conv(cid:48) and r ≥ 4. Then there is an open and dense set U = U ⊂ Sr and a
Γ∗
nonnegative function ε = ε (H ) with ε | > 0. Let V = {(cid:15)H : H ∈ U, 0 <
0 0 1 0 U 1 1
ε < ε }, then for an open and dense set of (cid:15)H ∈ V the Hamiltonian system
0 1
H = H + (cid:15)H has an orbit {(θ ,p )(t)} whose action component satisfies
(cid:15) 0 1 (cid:15) (cid:15) t
p (0) ∈ U, p (t) ∈ U(cid:48) for some t = t > 0.
(cid:15) (cid:15) (cid:15)
9
Notice that Theorem 1 guarantees that this orbit {(θ ,p )(t)} has action com-
√ (cid:15) (cid:15) t
ponent O( ε) close to Γ∗. This is a convexity region for H . Therefore, one can
0
extend H to another Hamiltonian H(cid:101) , which is convex outside of Conv. Then
0 0
apply Theorem 1 and notice that under the condition that action component
p ∈ Conv orbits of H +(cid:15)H and H(cid:101) +(cid:15)H coincide.
0 1 0 1
5. Non-convex Hamiltonians
IntheHamiltonianH isnon-convexforallp ∈ B2,forexample,H (p) = p2−p2,
0 0 1 2
the problem of Arnold diffusion is wide open. With our approach it is closely
related to another deep open problem of extending Mather theory and weak
KAM theory beyond convex Hamiltonians.
6. The Main Result
One interesting application of Theorem 1 is the following Theorem, which is
our main result.
Theorem 2. For any ρ > 0 and any r ≥ 4 there is an open and dense set
U = U ⊂ Sr and a nonnegative function ε = ε (H ,ρ) with ε | > 0. Let V =
ρ 0 0 1 0 U
{(cid:15)H : H ∈ U, 0 < ε < ε }, then for an open and dense set of (cid:15)H ∈ W (cid:40) V
1 1 0 1
the Hamiltonian system H = H + (cid:15)H has a ρ-dense orbit {(θ ,p ,t)(t)} in
(cid:15) 0 1 (cid:15) (cid:15) t
T2 ×B2 ×T, namely, its ρ-neighborhood contains T2 ×B2 ×T.
Remark 1.2. To prove this theorem one needs to take a finite set of resonant
lines {Γ = Γ }N so that Γ∗ = ∪N Γ is ρ/3-dense in the action domain B2.
j (cid:126)kj j=1 j=1 j
Moreover, unperturbed resonant orbits for each p ∈ Γ∗ have angular components
θ(t) = θ(0) + tp ( mod 1) that are ρ/3-dense in T2. Since p˙ = O(ε) and by
√
Theorem 1 for diffusing orbits action p (t) is O( ε)-close to Γ∗, we have ρ-
ε
density of (θ ,p ,t)(t) in T2 ×B2 ×T.
ε ε
As the number of resonant lines N increases, the size of admissible perturbation
ε (H ,ρ) goes to zero. Examples of Hamiltonians having orbits accumulating to
0 1
“large” sets and shadow infinitely many resonant lines are proposed in [45, 46].
The proof of the main Theorem naturally divides into two major parts:
• I Geometric: construct a connected net of normally hyperbolic invariant man-
ifolds (NHIMs) along the selected resonant lines Γ∗.
• II Variational: construct orbits diffusing along this net.
The idea to construct a connected net of NHIMs appeared in Kaloshin-Zhang-
Zheng [46]. [46] describes the construction of an example of a C∞-Hamiltonian H
in a small Ck–neighborhood k ≥ 2 of H (p) = 1(cid:107)p(cid:107)2 such that, on H−1(1/2), the
0 2
Hamiltonian flow of H has an orbit dense in a set of positive measure.
10