16. Twist maps and Arnold diffusion for diffeomorphisms
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Jean-Pierre Marco
Abstract
We investigate a diffusion problem in the group Dκ of Cκ symplectic diffeomorphisms of the annulus A2,with compact support contained in T2×I2,where I ⊂ ℝ is a fixed interval containing [0, 1] in its interior. We endow Dκ with the uniform Cκ topology.
Let (θ, r) be the usual angle-action coordinates on A2. We say that a diffeomorphism inDκ satisfies the diffusion propertywhen it admits orbitswhose action r1 starts close to 0 and later gets close to 1. Our problem is to describe the occurrence of the diffusion property in Dκ.
We introduce the set Fκ ⊂ Dκ of diffeomorphisms of the form f(x1, x2) = (f1(x1), f2(x2)). The restriction of f1 to T × [0, 1] is a symplectic twist map, which leaves the circles T×{0} and T×{1} invariantwith Diophantine rotation, while f2 is a symplectic diffeomorphism that admits a hyperbolic fixed point O2 (whose expansion and contraction dominate those of f1) together with a transverse homoclinic point P2. For κ large enough, a diffeomorphism in Fκ does not satisfy the diffusion property and so, by analogy with the Hamiltonian setting for Arnold diffusion, it can be legitimately considered as an “unperturbed” system.
Our main result, in the spirit of Arnold’s questions, is the following: when κ is large enough, given f ∈ Fκ, there is a ball Bκ(f, ε) in Dκ such that for any g ∈ Bκ(f, ε), there is a ̃g ∈ Dκ−1, arbitrarily close to g in the Cκ−1 topology, which satisfies the diffusion property.
The diffeomorphisms we deal with here can be seen as models for Poincare sections of near-integrable convex Hamiltonian systems in the neighborhood of double resonances, following the approach of [27].
Abstract
We investigate a diffusion problem in the group Dκ of Cκ symplectic diffeomorphisms of the annulus A2,with compact support contained in T2×I2,where I ⊂ ℝ is a fixed interval containing [0, 1] in its interior. We endow Dκ with the uniform Cκ topology.
Let (θ, r) be the usual angle-action coordinates on A2. We say that a diffeomorphism inDκ satisfies the diffusion propertywhen it admits orbitswhose action r1 starts close to 0 and later gets close to 1. Our problem is to describe the occurrence of the diffusion property in Dκ.
We introduce the set Fκ ⊂ Dκ of diffeomorphisms of the form f(x1, x2) = (f1(x1), f2(x2)). The restriction of f1 to T × [0, 1] is a symplectic twist map, which leaves the circles T×{0} and T×{1} invariantwith Diophantine rotation, while f2 is a symplectic diffeomorphism that admits a hyperbolic fixed point O2 (whose expansion and contraction dominate those of f1) together with a transverse homoclinic point P2. For κ large enough, a diffeomorphism in Fκ does not satisfy the diffusion property and so, by analogy with the Hamiltonian setting for Arnold diffusion, it can be legitimately considered as an “unperturbed” system.
Our main result, in the spirit of Arnold’s questions, is the following: when κ is large enough, given f ∈ Fκ, there is a ball Bκ(f, ε) in Dκ such that for any g ∈ Bκ(f, ε), there is a ̃g ∈ Dκ−1, arbitrarily close to g in the Cκ−1 topology, which satisfies the diffusion property.
The diffeomorphisms we deal with here can be seen as models for Poincare sections of near-integrable convex Hamiltonian systems in the neighborhood of double resonances, following the approach of [27].
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
-
Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
-
Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
