Table of Contents
Preface xi
Acknowledgments xiii
I Introduction and the general scheme 1
1 Introduction 3
1.1 Statement of the result 3
1.2 Scheme of diffusion 7
1.3 Three regimes of diffusion 11
1.4 The outline of the proof 12
1.5 Discussion 14
2 Forcing relation 17
2.1 Sufficient condition for Arnold diffusion 17
2.2 Diffusion mechanisms via forcing equivalence 18
2.3 Invariance under the symplectic coordinate changes 20
2.4 Normal hyperbolicity and Aubry-Mather type 22
3 Normal forms and cohomology classes at single resonances 24
3.1 Resonant component and non-degeneracy conditions 24
3.2 Normal form 26
3.3 The resonant component 29
4 Double resonance: geometric description 31
4.1 The slow system 31
4.2 Non-degeneracy conditions for the slow system 32
4.3 Normally hyperbolic cylinders 34
4.4 Local maps and global maps 36
5 Double resonance: forcing equivalence 39
5.1 Choice of cohomologies for the slow system 39
5.2 Aubry-Mather type at a double resonance 42
5.3 Connecting to Γk1,k2 and ΓSRk1 44
5.3.1 Connecting to the double resonance point 44
5.3.2 Connecting single and double resonance 45
5.4 Jump from non-simple homology to simple homology 49
5.5 Forcing equivalence at the double resonance 49
II Forcing relation and Aubry-Mather type 53
6 Weak KAM theory and forcing equivalence 55
6.1 Periodic Tonelli Hamiltonians 55
6.2 Weak KAM solution 57
6.3 Pseudographs, Aubry, Mañé, and Mather sets 59
6.4 The dual setting, forward solutions 60
6.5 Peierls barrier, static classes, elementary solutions 62
6.6 The forcing relation 64
6.7 The Green bundles 65
7 Perturbative weak KAM theory 66
7.1 Semi-continuity 66
7.2 Continuity of the barrier function 68
7.3 Lipschitz estimates for nearly integrable systems 70
7.4 Estimates for nearly autonomous systems 71
8 Cohomology of Aubry-Mather type 77
8.1 Aubry-Mather type and diffusion mechanisms 77
8.2 Weak KAM solutions are unstable manifolds 83
8.3 Regularity of the barrier functions 86
8.4 Bifurcation type 88
III Proving forcing equivalence 91
9 Aubry-Mather type at the single resonance 93
9.1 The single maximum case 93
9.2 Aubry-Mather type at single resonance 94
9.3 Bifurcations in the double maxima case 96
9.4 Hyperbolic coordinates 97
9.5 Normally hyperbolic invariant cylinder 100
9.6 Localization of the Aubry and Mane sets 102
9.7 Genericity of the single-resonance conditions 103
10 Normally hyperbolic cylinders at double resonance 106
10.1 Normal form near the hyperbolic fixed point 107
10.2 Shil' nikov's boundary value problem 108
10.3 Properties of the local maps 110
10.4 Periodic orbits for the local and global maps 114
10.5 Normally hyperbolic invariant manifolds 118
10.6 Cyclic concatenations of simple geodesies 119
11 Aubry-Mather type at the double resonance 121
11.1 High-energy case 121
11.2 Simple non-critical case 125
11.3 Simple critical case 126
11.3.1 Proof of Aubry-Mather type using local coordinates 126
11.3.2 Construction of the local coordinates 129
12 Forcing equivalence between kissing cylinders 133
12.1 Variational problem for the slow mechanical system 133
12.2 Variational problem for original coordinates 136
12.3 Scaling limit of the barrier function 139
12.4 The jump mechanism 140
IV Supplementary topics 145
13 Generic properties of mechanical systems on the two-torus 147
13.1 Generic properties of periodic orbits 147
13.2 Generic properties of minimal orbits 153
13.3 Non-degeneracy at high-energy 156
13.4 Unique hyperbolic minimizer at very high energy 158
13.5 Generic properties at the critical energy 160
14 Derivation of the slow mechanical system 162
14.1 Normal forms near maximal resonances 162
14.2 Affine coordinate change, rescaling, and energy reduction 172
14.3 Variational properties of the coordinate changes 177
15 Variational aspects of the slow mechanical system 182
15.1 Relation between the minimal geodesies and the Aubry sets 182
15.2 Characterization of the channel and the Aubry sets 185
15.3 The width of the channel 188
15.4 The case E = 0 190
Appendix: Notations 195
References 199
