Orbits of nearly integrable systems accumulating to KAM tori

103 Pages·2014·0.85 MB·English

Orbits of nearly integrable systems accumulating to
KAM tori
M. Guardia, V. Kaloshin
July 29, 2014
Contents
1 Introduction 3
1.1 The main result: time-periodic setting . . . . . . . . . . . . . . . . . . . . 3
1.2 Whitney KAM topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The quasi-ergodic hypotheses and Arnold diffusion . . . . . . . . . . . . . . 5
2 Outline of the proof of Theorem 2 6
2.1 Net of Dirichlet resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Deformation of the Hamiltonian and a tree of invariant cylinders . . . . . . 13
2.2.1 Po¨schel normal form and the molification of the Hamiltonian . . . . 14
2.2.2 Different regimes along resonances . . . . . . . . . . . . . . . . . . . 15
2.2.3 Normal form and invariant cylinders along single resonances . . . . 17
2.2.4 Normal form and invariant cylinders in the core of double resonances 19
2.3 Localization of the Aubry sets . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Aubry sets in the transition zones . . . . . . . . . . . . . . . . . . . 26
2.3.2 Aubry sets in the core of the double resonances . . . . . . . . . . . 26
2.4 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 First step of the proof: Selection of resonances 27
3.1 Approximation of Diophantine vectors . . . . . . . . . . . . . . . . . . . . 28
3.2 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Selection of resonances for all Diophantine frequencies: proof of Key Theorem 1 31
4 The deformation procedure 34
4.1 The molification of the Hamiltonian: proof of Lemma 2 . . . . . . . . . . . 34
4.2 Perturbation along resonant segments . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 The perturbation along the resonance of S . . . . . . . . . . . . . 36
0
4.2.2 The perturbation along resonances of S for n 1 . . . . . . . . . . 37
n
≥
1
5 Different regimes and normal forms while approaching Diophantine frequencies 38
5.1 Covering the Dirichlet segments by neighborhoods of strong double resonances 39
5.2 Normal form in the transition zones . . . . . . . . . . . . . . . . . . . . . . 40
5.2.1 Proof of Theorem 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Bounemoura Normal Form near in the core of the double resonances . . . . 46
5.3.1 Application of Corollary 5 . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.2 Proof of Theorem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 End of the proof of Key Theorem 4 . . . . . . . . . . . . . . . . . . . . . . 53
6 The normally hyperbolic cylinders in the transition zones 54
6.1 Change to slow-fast variables . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Existence of the cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.3 Proof of Theorem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Aubry sets in transition zones: proof of Key Theorem 7 64
7.1 Vertical estimates: proof of Theorem 14 . . . . . . . . . . . . . . . . . . . . 65
7.2 Horizontal estimates: proof of Theorem 15 . . . . . . . . . . . . . . . . . . 66
7.2.1 Localization for the slow angle: proof of Proposition 2 . . . . . . . . 68
7.2.2 Localization for the slow action: proof of Proposition 3 . . . . . . . 73
8 Equivalent forcing classes and shadowing 74
9 Perturbation of single averaged potentials 74
9.1 Perturbation of families of functions on the circle . . . . . . . . . . . . . . 77
9.2 Condition on local minimum . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.3 Bifurcation of global minima . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.4 Deformation of single averaged potential along all Dirichlet resonant segments 83
9.5 H¨older norms and approximations . . . . . . . . . . . . . . . . . . . . . . . 84
9.6 Smoothness of compositions and the inverse . . . . . . . . . . . . . . . . . 87
A A time-periodic KAM 91
A.1 Regularity of straightening of the union of KAM tori: a simplified version . 93
A.2 Anisotropic norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.3 Regularity of straightening of the union of KAM tori . . . . . . . . . . . . 95
A.4 Application of Po¨schel theorem to our setting . . . . . . . . . . . . . . . . 96
B Generic Tonelli Hamiltonians and a Generalized Mapertuis Principle 96
B.1 A generalized Mapertuis principle . . . . . . . . . . . . . . . . . . . . . . . 99
B.2 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2
1 Introduction
1.1 The main result: time-periodic setting
Let U be a convex bounded open subset of Rn. Consider a 2 smooth strictly convex
C
Hamiltonian H (I), I U, i.e. for some D > 1 we have
0
∈
D 1 v ∂2H (I)v,v D v for any I U and v Rn. (1)
− k k ≤ h I 0 i ≤ k k ∈ ∈
Fix r 2 and consider the space of r–perturbations: r(U Tn T) H (I,ϕ,t).
1
≥ C C × × ∋
Denote the unit sphere with respect to the standard r norm, given by maximum of all
C
partial derivative of order up to r, by
r = H r : H = 1 .
1 1 r
S { ∈ C k kC }
For a non-integer r be use the standard H¨older norm (see Section 9.5).
In this paper we study dynamics of nearly integrable systems
H (I,ϕ,t) = H (I)+εH (I,ϕ,t). (2)
ε 0 1
Assume that for some r 2,
≥
H 1, H 1. (3)
0 r+3 1 r
k kC ≤ k kC ≤
Consider small η > 0, τ > 0. A vector ω Rn is called (η,τ)-Diophantine if
∈
ω k +k η (k,k ) n τ for each (k,k ) (Zn 0 ) Z.
0 0 − − 0
| · | ≥ | | ∈ \{ } ×
Denote by
= ω : ω k +k η (k,k ) n τ (4)
η,τ 0 0 − −
D | · | ≥ | |
this set of frequencies. Denote U = H (U) the set of values of the gradient of H . It
(cid:8) ′ 0 (cid:9) 0
∇
is a bounded open set in Rn. Let U U be the set of points whose η-neighborhoods
η′ ⊂ ′
belong to U . Denote by Leb the Lebesgue measure on Rn Tn T and U := U .
′ × × Dη,τ Dη,τ∩ η′
Theorem 1. (time-periodic KAM) Let η,τ > 0, r > 2n+2τ +2. Assume (1) and (3).
Then there exist ε = ε (H ,η,τ) > 0 and c = c (H ,r,n) > 0 such that for any ε with
0 0 0 0 0 0
0 < ε < ε and any ω U the Hamiltonian H +εH has a (n+1)-dimensional (KAM)
0 ∈ Dη,τ 0 1
invariant torus and dynamics restricted to is smoothly conjugate to the constant
ω ω
T T
flow (ϕ˙,t˙) = (ω,1) on Tn+1. Moreover,
Leb(∪ω∈Dη,τ∩Uη′ Tω) > (1−c0η)Leb(U′ ×Tn ×T).
This theorem was essentially proven by Po¨schel [Po¨s82]. We add the actual derivation
in Appendix A. One can enlarge the set of KAM tori by lowering η and improve the lower
bound to 1 c √ε, but we do not rely on this improvement. In this paper we study the
0
−
Arnold diffusion phenomenon for these systems in the case n = 2 and consider only the
tori with frequencies in . Denote the set of all such KAM tori by
η,τ
D
KAMU := . (5)
η,τ ∪ω∈DηU,τ Tω
3
Theorem 2. (The First Main Result) Let η,τ > 0. Then there is r > 0 such that for any
0
r r and any r+3 smooth strictly convex Hamiltonian H there exists a r dense set
0 0
≥ C C
of perturbations D r such that for any H D there is ε = ε(η,τ,r,H ,H ) (0,ε )
1 0 1 0
⊂ S ∈ ∈
(where ε is the constant defined in Theorem 1) with the property that the Hamiltonian
0
H +εH has an orbit (I ,ϕ )(t) accumulating to all KAM tori from KAMU , i.e.
0 1 ε ε η,τ
KAMU (I ,ϕ )(t).
η,τ ⊂ ∪t∈R ε ε
An autonomous version of this result can be obtained using the standard energy re-
duction (see e.g. [Arn89, Sect 45]).
This result can be considered a weak form of the so-called quasi-ergodic hypothesis.
The quasi-ergodic hypothesis was posed by Ehrenfest and Birkhoff and asserts that a
typical Hamiltonian has a dense orbit in a typical energy surface. The result presented
in this paper does not obtain full dense orbits but orbits dense in a set of large measure,
and it only deals with nearly integrable systems. Note that the quasiergodic hypothesis
is not always true. Herman showed a counterexample in T2n [ δ,δ]2. He obtained open
× −
sets of Hamiltonian systems with a KAM persistent tori of codimension 1 in the energy
surface (see [Yoc92]).
Theorem 2 is a considerable improvement of the result from [KZZ09], where we give a
construction of a Hamiltonian of the form 1 3 I2+εH (I,ϕ) having an orbit accumu-
2 j=1 j 1
lating to a positive set of KAM tori. An example of this form with an orbit accumulating
P
to a fractal set of tori of maximal Hausdorff dimension is constructed in [KS12].
For the union of KAM tori KAMU one of the basic questions, studied in this paper,
η,τ
is the question of Lyapunov stability of these tori. In particular, we show that
For a class of nearly integrable Hamiltonian systems all KAM tori in fixed diophatine
class KAMU are Lyapunov unstable.
η,τ
LyapunovstabilityofaKAMtorusiscloselyrelatedtoaquestionofLyapunovstability
of a totally elliptic fixed point. An example of a 4-dimensional map with a Lyapunov
unstable totally elliptic fixed point was constructed by P. Le Calvez and R. Douady
[LCD83]. For resonant elliptic points of 4-dimensional maps instability was established
in [KMV04]. A class of examples of nearly integrable Hamiltonians having an unstable
KAM torus was recently obtained by J. Zhang and C. Q. Cheng [CZ13].
Douady [Dou88] proved that the stability or instability property of a totally elliptic
point is a flat phenomenon for mappings. Namely, if a symplectic mapping f
∞ ∞ 0
C C
satisfies certain nondegeneracy hypotheses, then there are two mapping f and g such that
f f and f g are flat mappings at the origin and
0 0
• − −
the origin is Lyapunov unstable for f and Lyapunov stable for g.
•
This shows that Lyapunov stability is not an open property. Thus, the only chance
to have robustness in the Main Theorem is to support perturbations of H away from
1
KAM tori. Moreover, the closer we approach to KAM tori the smaller is size of those
perturbations. This naturally leads us to a Whitney topology relative to the union of
KAM tori.
4
1.2 Whitney KAM topology
Denote s(B2 T3,KAMU ) — the space of s functions with the natural s-topology
C × η,τ C C
such that they tend to 0 as (I,ϕ) approaches KAMU inside the complement.
η,τ
Let s be a positive integer. Let M be one of U U R2, (U U ) T3, or
\ Dη,τ ⊂ \ Dη,τ ×
U T3 KAMU . If f is a s real valued function on M, the s-norm of f
× \ η,τ C C
f = sup ∂αf(x) ,
s
k kC x M, α sk k
∈ | |≤
where the supremum is taken over the absolute values of all partial derivatives ∂α of order
s. Definition of s-norm for non-integer s is in section 9.5.
≤ C
Introduce a strong s-topology. We endow it with the strong s-topology on the space
C C
of functions on a non-compact manifold or the s Whitney topology. A base for this
C
topology consists of sets of the following type. Let Ξ = ϕ ,U be a locally finite set
i i i Λ
{ }∈
of charts on M, where M is as above. Let K = K be a family of compact subsets
i i Λ
{ }∈
of M, K U . Let also ε = ε be a family of positive numbers. A strong basic
i i i i Λ
neighborho⊂od s(f,Ξ,K,ε) is g{ive}n∈by
N
i Λ (fϕ )(x) (gϕ )(x) ε x K .
i i s i i
∀ ∈ k − k ≤ ∀ ∈
The strong topology has all possible sets of this form. Let ε be small positive. Endow
0
s(U T3,KAMU ) with the strong topology.
C × η,τ
Theorem 3. (The Second Main result) Let η,τ > 0. Then there is r > 0 such that for
0
any r r and any r+3 smooth strictly convex Hamiltonian H there exists a r dense
0 0
≥ C C
set of perturbations D r such that for any H D there is ε = ε(η,τ,H ,H ) > 0
1 0 1
⊂ S ∈
with the property that there is a set W open in r-Whitney KAM topology such that for
C
any ∆H W the Hamiltonian H +ε(H +∆H ) has an orbit (I ,ϕ )(t) such that
1 0 1 1 ε ε
∈
KAMU (I ,ϕ )(t).
η,τ ⊂ ∪t∈R ε ε
Theorem 3 certainly implies Theorem 2.
1.3 The quasi-ergodic hypotheses and Arnold diffusion
The quasi-ergodic hypothesis asks for the existence of a dense orbit in a typical energy
surface of a typical Hamiltonian system. Instead Arnol’d diffusion asks whether a typical
nearly integrable Hamiltonian has orbits whose actions make a drift with size independent
of the perturbative parameter.
The study of Arnol’d diffusion was initiated by Arnol’d in his seminal paper [Arn64],
where he obtained a concrete Hamiltonian system with an orbit undergoing a small drift
in action. In the last decades there has been a huge progress in the area. The works of
Arnold diffusion can beclassified intwo different groups, the ones which deal with a priori
unstable systems and the ones dealing with a priori stable systems (as defined in [CG94].
5
Thefirstonesarethosewhosefirstorderpresent somehyperbolicity andhavebeenstudied
both by gemometric methods [DdlLS06, DdlLS08, DH09, Tre04, Tre12, DdlLS13] and
variational methods [Ber08, CY04, CY09]. A priori stable systems are those who are
close to an integrable Hamiltonian system, whose phase space is foliated by quasiperiodic
tori. The existence of Arnold diffusion in a priori stable systems was only known in
concrete examples [?, Dou88] until the recent works [Mat03, BKZ11, KZ12, Che12].
Acknowledgement: We would like to thank Ke Zhang, Georgi Popov, Jacques F´ejoz,
Kostya Khanin, Jinxin Xue, Yong Zheng, Jianlu Zhang, Hakan Elliasson, John Mather
for useful conversations. The authors thank the Institute for Advanced Study for its hos-
pitality, where a part of the work was done. The first author is partially supported by the
SpanishMCyT/FEDERgrantMTM2012-31714andtheCatalanSGRgrant2009SGR859.
The second author acknowledges support of a NSF grant DMS-1157830.
2 Outline of the proof of Theorem 2
The proof of Theorem 2 has several parts which we describe in this section. The proofs
of each part are given in Sections 3–8. We start by listing these parts. Recall that
U H (U) is the set of points whose η-neighborhoods belong to H (U) and U :=
η′ ⊂ ∇ 0 ∇ 0 Dη,τ
U . Fix R 1 (to be determined) and a sequence of radii R , where
DRη,τ ∩= Rη′1+2τ for 0ea≫ch n Z . Consider another “reciprocal” sequence{ ρn}n∈Z+ , ρ =
Rn+3+15τ forneach n Z . F∈or e+ach k (Z2 0) Z denote (Γ ) = ω U{:nk}n(∈ωZ,+1) =n0
n− ∈ + ∈ \ × S k { ∈ ′ · }
the corresponding resonant segment.
1. A tree of Dirichlet resonant segments:
Weconstruct asequence ofgridsofDiophantinefrequencies in n U such
• Dη,τ ⊂ Dη,τ
that 3ρ -neighborhood of n contains U and ρ -neighborhoods of points
n Dη,τ Dη,τ n
of n are pairwise disjoint.
Dη,τ
To each grid n we associate a collection of pairwise disjoint Voronoi cells
• Dη,τ
(see Figure 1): open neighborhoods
Vor (ω ) such that B (ω ) Vor (ω ) B (ω ).
{ n n }ωn∈Dηn,τ ρn n ⊂ n n ⊂ 3ρn n
We construct a collection of generations of resonant segments
S = S , S = ωn ,
∪n∈Z+ n n ∪kn∈Jn{Skn}
where J (Z2 0) Z is a collection of resonances, ωn (Γ ). We
n ⊂ \ × Skn ⊂ S kn
choosethiscollectionsuchthatmostintersectionbetweenthedifferentresonant
segments are empty and if nonempty we have quantitative estimates.
By construction each segment of generation n is contained in a Voronoi cell
•
Vor (ω ) for some ω n , i.e. ωn Vor (ω ). We say that segments ωn
n n n ∈ Dη,τ Skn ⊂ n n Skn
from S belong to the generation n.
n
6
We call a resonance segment from S a Dirichlet resonant segment.
By construction Dirichlet segments accumulates to all the Diophantine frequencies
in U and satisfy several properties (see Key Theorem 1 for details).
Dη,τ
ω15
ωj14 j ωj7 ωj8
ω6
j
ωj5 ω1
j
ω ω9
j j
ω2
ω4 j
j
ω3 ω10
j j
ω11
j
ω13 ω12
j j
Figure 1: Voronoi cells
These Dirichlet segments have a tree structure except that segments of the same
generation can have multiple intersections (junctions)! See Section 3. Notice that
this part is number theoretic and independent of the Hamiltonian H !
0
2. Global (P¨oschel) normal form and molification: Inordertoconstructdiffusionorbits
consider the Hamiltonian H in the Po¨schel normal form
ε
N = H Φ = H (I)+εR(I,ϕ,t)
ε ε ◦ P¨os 0′
(see Theorem 5). Our main result does not have loss of derivatives. To compensate
the loss of derivatives in the Po¨schel normal form we mollify our Hamiltonian H in
ε
the complement to KAM tori KAMU to H so that it is away from KAMU 1
η,τ ε′ C∞ η,τ
(see Section 4.1). Let N := H Φ .
ε′ ε′ ◦ P¨os
It is convenient toidentify resonances inthe actionspace. To each Dirichlet segment
ωn S with k J we associated a Diophantine frequency ω n . For each
Skn ⊂ n n ∈ n n ∈ Dη,τ
ω n denote I := (∂ H ) 1(ω ) and
n ∈ Dη,τ n I 0′ − n
ωn := I U : ∂ H (I) ωn .
Ikn { ∈ η I 0′ ∈ Skn}
Similarly, we can define Voronoi cells in action space:
Vor (I ) = I U : ∂ H (I) Vor (ω ) .
n n { ∈ η I 0′ ∈ n n }
1Certainly high norms of H′ blow up to infinity as we approachKAMU
ε η,τ
7
By construction we have ωn Vor (ω ). In the proof we always perturb away
Skn ⊂ n n
from KAM tori KAMU and this does not affect H . It is convenient to describe
η,τ 0′
perurbations as well as diffusing conditions in Po¨schel coordinates. Notice that in
Po¨schel coordinates all KAM tori are flat, i.e.
Φ (KAMU ) = (I,ϕ,t) : ∂ H (I) U .
P¨os η,τ { I 0′ ∈ Dη,τ}
3. Deformation of the Hamiltonian: We modify the Hamiltonian N to N by a small
ε′ ε′′
r perturbationsupportedawayfromKAMtoriKAMU sothattheresultingHamil-
C η,τ
tonian N is non-degenerate (in a way that we specify later). This perturbation is
ε′′
done in Section 4. Notice that this modification procedure is also done by induction:
N (I,ϕ,t) = N (I,ϕ,t)+ ∆Nn(I,ϕ,t), where
ε′′ ε′ ε
n N 0
∈X∪{ }
the 0-th generation perturbation ∆N0 is supported in a (√ε)-neighborhood
• ε O
of all horizontal and vertical resonant lines with k R .
k 0
S | | ≤
the n-th generation perturbation ∆Nn is supported in n-th order Voronoi cells,
• ε
i.e. ∆Nn(I,ϕ,t) = 0 for any I Vor (I ) for any I with ∂ H (I ) n .
ε 6∈ n n n I 0′ n ∈ Dη,τ
The zero step is done as in [KZ12] and it allows us to construct a net of normally
hyperbolic invariant cylinders along the resonances in S .
0
Then-thstepwemodifyourHamiltonianinVoronoicellsofordernclosetoDirichlet
resonant segments of order n so that it satisfies some non-degeneracy conditions,
while non-degeneracy conditions of orders k < n from previous steps hold true.
4. Resonant Normal Forms: Fix a generation n and a Dirichlet segment ωn of this
Ikn
generation. InSection5, wederiveResonantNormalFormsforN forpoints(I,ϕ,t)
ε′′
such that I is close to ωn. We obtain normal forms of two different types:
Ikn
A single resonant one
•
N Φkn(ψ,J,t) = kn(J)+ kn(ψs,J)+ kn(ψs,ψf,J,t)
ε′′,kn ◦ H0 Z R
where kn only depends on the slow angle ψs = k (ψ,t), (ψs,ψf,t) form a
n
Z ·
basis and kn is small relatively to non-degeneracy of kn. This normal form
R Z
is similar to [BKZ11].
A double resonant one
•
N Φkn,k′(ψ,J,t) = kn,k′(J)+ kn,k′(ψ,J)
ε′′ ◦ H0 Z
where kn,k′ onlydepends onthetwoslowanglesψs = k (ψ,t), ψs = k (ψ,t).
Z 1 n· 2 ′·
We follow the normal form procedure explained in [Bou10].
8
– (core of the double resonance) Thanks to the deformation procedure we
can completely remove the time dependence in the normal form in the core
of double resonances.
– (transition zones to a single resonance) In [KZ12] the double resonant
normal form Hamiltonian is a small pertubation of a two degrees of free-
dom mechanical system. Here we cannot make this reduction as a“non-
mechanical” remainder becomes large. To resolve the issue we describe
the leading order by a more general two degrees of freedom system. To
study such systems we apply a generalized Mapertuis principle (see Ap-
pendix B) and use the technique developped in Appendix A, [KZ12].
5. A tree of normally hyperbolic invariant cylinders: In Section 6, using the non-
degeneracy obtained in the Deformation step and Resonant Normal Forms from the
previous step, we build a tree of normally hyperbolic invariant cylinders along all
Dirichlet resonant segments.
In the single resonant regime, we obtain cylinders along the resonant segments ωn
Ikn
(see Key Theorem 3). We denote them by ωn. In the double resonance regime we
Ckn
obtain several cylinders by analogy with [KZ12]. This is explained in Key Theorems
5 and 6 and Corollary 2.
6. Localization of Aubry sets: We prove the existence of certain Aubry sets local-
ized inside the cylinders obtained in the previous step. We also establish a graph
property, which essentially says that these Aubry sets have the same structure as
Aubry-Mather sets of twist maps. This is done in Section 7.
7. Shadowing: In order to construct diffusing orbits along a chain of Aubry sets we
use the notion of c-equivalence proposed by Bernard. It essentially consists of three
parts: apriori unstable (diffusion along one cylinder), bifurcations (jump from one
cylinder to another in the same homology), a turn jump from a cylinder with one
homology to a cylinder with a different one. The first two regimes are essentially
done in [BKZ11] using a lemma from [CY04, CY09]. The last one is similar to
[KZ12].
Now we describe each step and split the proof of Theorem 2 in several Key Theorems.
2.1 Net of Dirichlet resonances
The first step is to construct a tree of generations of special resonant segments
S = S , S = ωn
{ n}n∈Z n {Skn}kn∈Fn
in the frequency space U R2 such that the following properties hold
′
⊂
9
each segment ωn belongs to the intersection of the resonant segment and the corre-
• Skn
sponding Voronoi cell ωn (Γ ) Vor (ω ) i.e. there are k (Z2 0) Z
and ω n 1 suchStkhnat⊂thSis iknnclu∩sion hn−o1lds.n−1 n ∈ \ ×
n−1 ∈ Dη,−τ
Any segment ωn of generation n intersects only segments of the generations n 2
• Skn −
and n 1 and the segments of generation n+1 and n+2 associated to frequencies
−
n+1 Vor (ω ) and n+2 Vor (ω ) respectively. However, it does not
Dη,τ ∩ n−1 n−1 Dη,τ ∩ n−1 n−1
intersect any segment of any prevous generation generation n+k or n k, k 3.
− ≥
We have quantitative information about properties of intersections of segments of
•
generation n inside of the same Voronoi cell.
The union ωn is connected.
• ∪n ∪kn∈Fn Skn
The closure ωn contains all Diophantine frequencies in U (see (4)).
• ∪n ∪kn∈Fn Skn Dη,τ
We shall call these segments Dirichlet resonant segments. We define them to satisfy
quantitative estimates on speed of approximation. Recall that an elementary pigeon hole
principle show that for any bounded set U R2 and any any ω = (ω ,ω ) U there is a
1 2
⊂ ∈
sequence k (ω) (Z2 0) Z such that
n
∈ \ ×
k (ω) (ω,1) k (ω) 2 and k (ω) as n .
n n − n
| · | ≤ | | | | → ∞ → ∞
We choose our segments so that
for each ωn (Γ ) there is a Diophantine number ω n such that
• Skn ⊂ S kn n ∈ Dη,τ
k (ω,1) k 2+3τ.
n n −
| · | ≤ | |
if ωn ωn′ = , then we have anupper bound onratio k / k as well as a uniform
• Skn ∩Skn′ 6 ∅ | n| | n′|
lower angle of intersection between Γkn and Γkn′ .
The tree of Dirichlet resonances is obtained in two steps.
in Section 3.1 we fix a Diophantine ω U and construct a connected “zigzag”
• ∗ ∈ Dη,τ
approaching ω .
∗
in Section 3.3 we define Dirichlet resonant segments and show how to deal with all
•
frequencies in U simultaneously.
Dη,τ
It turns out that we cannot just construct the “zigzag” for each frequency and then
consider the union over all frequencies in U , because we need estimates on the complex-
Dη,τ
ity of the intersection among resonant segments2. Now we define the sequence of discrete
sets n .
{Dη,τ}n∈Z+
2Too many segments, too many intersections to control
10