Symplectic Twist Maps

By: Christophe Gole

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Hardcover | 1 December 1999

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Hardcover
324 Pages

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21.84 x 15.49 x 2.29

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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 in the dimension 2 case (ghost circles). The debate is oriented somewhat toward the open problem of finding orbits of all (in particular, irrational) rotation vectors.

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"... this book constitutes a fundamental monograph on the subject and gives a beautiful treatment of many results in the area that previously have only been accessible in original research papers." Mathematical Reviews

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Non-Fiction Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Mathematics Geometry

Applied Mathematics Chaos Theory Topology Booktopia Publisher Services World Scientific Publishing

Introduction	p. 1
Fall from Paradise	p. 1
Billiards and Broken Geodesics	p. 4
An Ancestor of Symplectic Topology	p. 8
Twist Maps of the Annulus	p. 11
Monotone Twist Maps of the Annulus	p. 11
Generating Functions and Variational Setting	p. 17
Examples	p. 21
The Poincare-Birkhoff Theorem	p. 27
The Aubry-Mather Theorem	p. 31
Introduction	p. 31
Cyclically Ordered Sequences And Orbits	p. 34
Minimizing Orbits	p. 36
CO Orbits Of All Rotation Numbers	p. 40
Aubry-Mather Sets	p. 41
Appendix: Cyclically Ordered Sequences and Circle Maps	p. 46
Ghost Circles	p. 53
Gradient Flow of the Action	p. 54
The Gradient Flow and the Aubry-Mather Theorem	p. 57
Ghost Circles	p. 59
Construction of Ghost Circles	p. 63
Construction of Disjoint Ghost Circles	p. 67
Proof of Lemma 18.5	p. 70
Proof of Theorem 18.1	p. 73
Remarks and Applications	p. 78
Proofs of Monotonicity and of the Sturmian Lemma	p. 82
Symplectic Twist Maps	p. 87
Symplectic Twist Maps of T[superscript n] x [actual symbol not reproducible][superscript n]	p. 88
Examples	p. 91
More on Generating Functions	p. 95
Symplectic Twist Maps on General Cotangent Bundles of Compact Manifolds	p. 99
Periodic Orbits for Symplectic Twist Maps of T[superscript n] x [actual symbol not reproducible][superscript n]	p. 103
Presentation Of The Results	p. 103
Finite Dimensional Variational Setting	p. 107
Second Variation and Nondegenerate Periodic Orbits	p. 110
The Coercive Case	p. 112
Asymptotically Linear Systems	p. 114
Ghost Tori	p. 116
Hyperbolicity Vs. Action Minimizers	p. 118
Invariant Manifolds	p. 123
The Theory of Kolmogorov-Arnold-Moser	p. 123
Properties of Invariant Tori	p. 127
(Un)Stable Manifolds and Heteroclinic orbits	p. 135
Instability, Transport and Diffusion	p. 141
Hamiltonian Systems vs. Twist Maps	p. 145
Case Study: The Geodesic Flow	p. 146
Decomposition of Hamiltonian Maps into Twist Maps	p. 154
Return Maps in Hamiltonian Systems	p. 164
Suspension of Symplectic Twist Maps by Hamiltonian Flows	p. 165
Periodic Orbits for Hamiltonian Systems	p. 173
Periodic Orbits in the Cotangent of the n-Torus	p. 174
Periodic Orbits in General Cotangent Spaces	p. 177
Linking of Spheres	p. 186
Generalizations of the Aubry-Mather Theorem	p. 191
Theory for Functions on Lattices and PDE's	p. 192
Monotone Recurrence Relationst	p. 196
Anti-Integrable Limit	p. 197
Mather's Theory of Minimal Measures	p. 199
The Case of Hyperbolic Manifolds	p. 212
Concluding Remarks	p. 216
Generating Phases and Symplectic Topology	p. 217
Chaperon's Method and the Theorem Of Conley-Zehnder	p. 218
Generating Phases and Symplectic Geometry	p. 224
Overview of Symplectic Geometry	p. 233
Symplectic Vector Spaces	p. 233
Subspaces of a Symplectic Vector Space	p. 236
Symplectic Linear Maps	p. 238
Symplectic Manifolds	p. 241
Cotangent Bundles	p. 243
Hamiltonian Systems	p. 247
Some Topological Tools	p. 259
Hands on Introduction to Homology Theory	p. 260
Morse Theory	p. 267
Controlling the Topology of Invariant Sets	p. 273
Topological Proofs	p. 276
Generating Phases Quadratic at Infinity	p. 284
Covering Spaces, Lifts and Fundamental Group	p. 288
Bibliography	p. 293
Index	p. 303
Table of Contents provided by Blackwell. All Rights Reserved.

Symplectic Twist Maps

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