Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Name: Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom | (AMS-208)
Brand: Princeton University Press
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(AMS-208)

Vadim Kaloshin Ke Zhang(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 3. November 2020

224 Seiten

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978-0-691-20493-2 (ISBN)

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Inhalt

Cover
Title
Copyright
Dedication
Contents
Preface
Acknowledgments
I Introduction and the general scheme
1 Introduction
1.1 Statement of the result
1.2 Scheme of diffusion
1.3 Three regimes of diffusion
1.4 The outline of the proof
1.5 Discussion
2 Forcing relation
2.1 Sufficient condition for Arnold diffusion
2.2 Diffusion mechanisms via forcing equivalence
2.3 Invariance under the symplectic coordinate changes
2.4 Normal hyperbolicity and Aubry-Mather type
3 Normal forms and cohomology classes at single resonances
3.1 Resonant component and non-degeneracy conditions
3.2 Normal form
3.3 The resonant component
4 Double resonance: geometric description
4.1 The slow system
4.2 Non-degeneracy conditions for the slow system
4.3 Normally hyperbolic cylinders
4.4 Local maps and global maps
5 Double resonance: forcing equivalence
5.1 Choice of cohomologies for the slow system
5.2 Aubry-Mather type at a double resonance
5.3 Connecting to GK1,K2 and G^SR K1
5.3.1 Connecting to the double resonance point
5.3.2 Connecting single and double resonance
5.4 Jump from non-simple homology to simple homology
5.5 Forcing equivalence at the double resonance
II Forcing relation and Aubry-Mather type
6 Weak KAM theory and forcing equivalence
6.1 Periodic Tonelli Hamiltonians
6.2 Weak KAM solution
6.3 Pseudographs, Aubry, Mañé, and Mather sets
6.4 The dual setting, forward solutions
6.5 Peierls barrier, static classes, elementary solutions
6.6 The forcing relation
6.7 The Green bundles
7 Perturbative weak KAM theory
7.1 Semi-continuity
7.2 Continuity of the barrier function
7.3 Lipschitz estimates for nearly integrable systems
7.4 Estimates for nearly autonomous systems
8 Cohomology of Aubry-Mather type
8.1 Aubry-Mather type and diffusion mechanisms
8.2 Weak KAM solutions are unstable manifolds
8.3 Regularity of the barrier functions
8.4 Bifurcation type
III Proving forcing equivalence
9 Aubry-Mather type at the single resonance
9.1 The single maximum case
9.2 Aubry-Mather type at single resonance
9.3 Bifurcations in the double maxima case
9.4 Hyperbolic coordinates
9.5 Normally hyperbolic invariant cylinder
9.6 Localization of the Aubry and Mañé sets
9.7 Genericity of the single-resonance conditions
10 Normally hyperbolic cylinders at double resonance
10.1 Normal form near the hyperbolic fixed point
10.2 Shil'nikov's boundary value problem
10.3 Properties of the local maps
10.4 Periodic orbits for the local and global maps
10.5 Normally hyperbolic invariant manifolds
10.6 Cyclic concatenations of simple geodesics
11 Aubry-Mather type at the double resonance
11.1 High-energy case
11.2 Simple non-critical case
11.3 Simple critical case
11.3.1 Proof of Aubry-Mather type using local coordinates
11.3.2 Construction of the local coordinates
12 Forcing equivalence between kissing cylinders
12.1 Variational problem for the slow mechanical system
12.2 Variational problem for original coordinates
12.3 Scaling limit of the barrier function
12.4 The jump mechanism
IV Supplementary topics
13 Generic properties of mechanical systems on the two-torus
13.1 Generic properties of periodic orbits
13.2 Generic properties of minimal orbits
13.3 Non-degeneracy at high-energy
13.4 Unique hyperbolic minimizer at very high energy
13.5 Generic properties at the critical energy
14 Derivation of the slow mechanical system
14.1 Normal forms near maximal resonances
14.2 Affine coordinate change, rescaling, and energy reduction
14.3 Variational properties of the coordinate changes
15 Variational aspects of the slow mechanical system
15.1 Relation between the minimal geodesics and the Aubry sets
15.2 Characterization of the channel and the Aubry sets
15.3 The width of the channel
15.4 The case E = 0
Appendix: Notations
References

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