Symplectic twist maps : global variational techniques / Christophe Golé.

This book concentrates mainly on the theorem of existence of periodic orbits for higher dimensional analogs of Twist maps. The setting is that of a discrete variational calculus and the techniques involve Conley-Zehnder-Morse Theory. They give rise to the concept of ghost tori which are of interest...

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Bibliographic Details
Main Author: Golé, Christophe
Format: eBook
Language:English
Published: Singapore ; River Edge, NJ : World Scientific, ©2001.
Series:Advanced series in nonlinear dynamics ; v. 18.
Subjects:
Online Access:Click for online access
Table of Contents:
  • 0. Introduction. 1. Fall from paradise. 2. Billiards and broken geodesies. 3. An ancestor of symplectic topology
  • 1. Twist maps of the annulus. 4. Monotone twist maps of the annulus. 5. Generating functions and variational setting. 6. Examples. 7. The Poincare-Birkhoff theorem
  • 2. The Aubry-Mather theorem. 8. Introduction. 9. Cyclically ordered sequences and orbits. 10. Minimizing orbits. 11. CO orbits of all rotation numbers. 12. Aubry-Mather sets
  • 3. Ghost circles. 14. Gradient flow of the action. 15. The gradient flow and the Aubry-Mather theorem. 16. Ghost circles. 17. Construction of ghost circles. 18. Construction of disjoint ghost circles. 19. Proof of lemma 18.5. 20. Proof of theorem 18.1. 21. Remarks and applications. 22. Proofs of monotonicity and of the Sturmian lemma
  • 4. Symplectic twist maps. 23. Symplectic twist maps of T[symbol] x IR[symbol]. 24. Examples. 25. More on generating functions. 2.6. Symplectic twist maps on general cotangent bundles of compact manifolds
  • 5. Periodic orbits for symplectic twist maps of T[symbol] x IR[symbol]. 27. Presentation of the results. 28. Finite dimensional variational setting. 29. Second variation and nondegenerate periodic orbits. 30. The coercive case. 31. Asymptotically linear systems. 32. Ghost tori. 33. Hyperbolicity Vs. action minimizers
  • 6. Invariant manifolds. 34. The theory of Kolmogorov-Arnold-Moser. 35. Properties of invariant tori. 36. (Un)stable manifolds and heteroclinic orbits. 37. Instability, transport and diffusion
  • 7. Hamiltonian systems vs. twist maps. 38. Case study: The geodesic flow. 39. Decomposition of Hamiltonian maps into twist maps. 40. Return maps in Hamiltonian systems. 41. Suspension of symplectic twist maps by Hamiltonian flows
  • 8. Periodic orbits for Hamiltonian systems. 42. Periodic orbits in the cotangent of the n-torus. 43. Periodic orbits in general cotangent spaces. 44. Linking of spheres
  • 9. Generalizations of the Aubry-Mather theorem. 45. Theory for functions on lattices and PDE's. 46. Monotone recurrence relationst. 47. Anti-integrable limit. 48. Mather's theory of minimal measures. 49. The case of hyperbolic manifolds. 50. Concluding remarks
  • 10. Generating phases and symplectic topology. 51. Chaperon's method and the theorem Of Conley-Zehnder. 52. Generating phases and symplectic geometry.