The Geometry of the Group of Symplectic Diffeomorphisms

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Format: Paperback
Pub. Date: 2001-03-01
Publisher(s): Birkhauser
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Summary

The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the minimal amount of energy required in order to generate a given mechanical motion? This variational problem admits an interpretation in terms of a remarkable geometry on the group discovered by Hofer in 1990. Hofer's geometry serves as a source of interesting problems and gives rise to new methods and notions which extend significantly our vision of the symplectic world. In the past decade this new geometry has been intensively studied in the framework of symplectic topology with the use of modern techniques such as Gromov's theory of pseudo-holomorphic curves, Floer homology and Guillemin-Sternberg-Lerman theory of symplectic connections. Furthermore, it opens up the intriguing prospect of using an alternative geometric intuition in dynamics. The book provides an essentially self-contained introduction into these developments and includes recent results on diameter, geodesics and growth of one-parameter subgroups in Hofer's geometry, as well as applications to dynamics and ergodic theory. It is addressed to researchers and students from the graduate level onwards.

Table of Contents

Preface ix
Introducing the Group
The origins of Hamiltonian diffeomorphisms
1(2)
Flows and paths of diffeomorphisms
3(1)
Classical mechanics
4(2)
The group of Hamiltonian diffeomorphisms
6(4)
Algebraic properties of Ham (M, Ω)
10(3)
Introducing the Geometry
A variational problem
13(1)
Biinvariant geometries on Ham (M, Ω)
14(1)
The choice of the norm: Lp vs. L∞
15(1)
The concept of displacement energy
16(5)
Lagrangian Submanifolds
Definitions and examples
21(2)
The Liouville class
23(3)
Estimating the displacement energy
26(3)
The ∂-Equation
Introducing the ∂-operator
29(2)
The boundary value problem
31(1)
An application to the Liouville class
32(1)
An example
33(4)
Linearization
The space of periodic Hamiltonians
37(2)
Regularization
39(2)
Paths in a given homotopy class
41(2)
Lagrangian Intersections
Exact Lagrangian isotopies
43(3)
Lagrangian intersections
46(2)
An application to Hamiltonian loops
48(3)
Diameter
The starting estimate
51(1)
The fundamental group
52(2)
The length spectrum
54(1)
Refining the estimate
55(2)
Growth and Dynamics
Invariant tori of classical mechanics
57(2)
Growth of one-parameter subgroups
59(4)
Curve shortening in Hofer's geometry
63(1)
What happens when the asymptotic growth vanishes?
64(1)
Length Spectrum
The positive and negative parts of Hofer's norm
65(1)
Symplectic fibrations over S2
66(3)
Symplectic connections
69(5)
An application to length spectrum
74(1)
Deformations of Symplectic Forms
The deformation problem
75(1)
The ∂-equation revisited
76(2)
An application to coupling
78(1)
Pseudo-holomorphic curves
79(2)
Persistence of exceptional spheres
81(2)
Ergodic Theory
Hamiltonian loops as dynamical objects
83(2)
The asymptotic length spectrum
85(2)
Geometry via algebra
87(2)
Geodesics
What are geodesics?
89(3)
Description of geodesics
92(1)
Stability and conjugate points
93(1)
The second variation formula
94(5)
Analysis of the second variation formula
99(2)
Length minimizing geodesics
101(4)
Floer Homology
Near the entrance
105(2)
Morse homology in finite dimensions
107(2)
Floer homology
109(4)
An application to geodesics
113(2)
Towards the exit
115(2)
Non-Hamiltonian Diffeomorphisms
The flux homomorphism
117(2)
The flux conjecture
119(2)
Links to ``hard'' symplectic topology
121(1)
Isometries in Hofer's geometry
122(3)
Bibliography 125(6)
Index 131(2)
List of Symbols 133

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