Overview

The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the niminal 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 extensively 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.

Full Product Details

Author:   Leonid Polterovich
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2001 ed.
Dimensions:   Width: 17.00cm , Height: 0.80cm , Length: 24.00cm
Weight:   0.560kg
ISBN:  

9783764364328

ISBN 10:   3764364327
Pages:   136
Publication Date:   01 March 2001
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

Preface.- 1 Introducing the Group.- 1.1 The origins of Hamiltonian diffeomorphisms.- 1.2 Flows and paths of diffeomorphisms.- 1.3 Classical mechanics.- 1.4 The group of Hamiltonian diffeomorphisms.- 1.5 Algebraic properties of Ham(M, Q).- 2 Introducing the Geometry.- 2.1 A variational problem.- 2.2 Biinvariant geometries on Ham(M, Q).- 2.3 The choice of the norm: Lp vs. Loa.- 2.4 The concept of displacement energy.- 3 Lagrangian Submanifolds.- 3.1 Definitions and examples.- 3.2 The Liouville class.- 3.3 Estimating the displacement energy.- 4 The $$ \bar \partial $$-Equation.- 4.1 Introducing the $$ \bar \partial $$-operator.- 4.2 The boundary value problem.- 4.3 An application to the Liouville class.- 4.4 An example.- 5 Linearization.- 5.1 The space of periodic Hamiltonians.- 5.2 Regularization.- 5.3 Paths in a given homotopy class.- 6 Lagrangian Intersections.- 6.1 Exact Lagrangian isotopies.- 6.2 Lagrangian intersections.- 6.3 An application to Hamiltonian loops.- 7 Diameter.- 7.1 The starting estimate.- 7.2 The fundamental group.- 7.3 The length spectrum.- 7.4 Refining the estimate.- 8 Growth and Dynamics.- 8.1 Invariant tori of classical mechanics.- 8.2 Growth of one-parameter subgroups.- 8.3 Curve shortening in Hofer’s geometry.- 8.4 What happens when the asymptotic growth vanishes?.- 9 Length Spectrum.- 9.1 The positive and negative parts of Hofer’s norm.- 9.2 Symplectic fibrations over S2.- 9.3 Symplectic connections.- 9.4 An application to length spectrum.- 10 Deformations of Symplectic Forms.- 10.1 The deformation problem.- 10.2 The $$ \bar \partial $$-equation revisited.- 10.3 An application to coupling.- 10.4 Pseudo-holomorphic curves.- 10.5 Persistence of exceptional spheres.- 11 Ergodic Theory.- 11.1 Hamiltonian loops as dynamical objects.- 11.2 Theasymptotic length spectrum.- 11.3 Geometry via algebra.- 12 Geodesics.- 12.1 What are geodesics?.- 12.2 Description of geodesics.- 12.3 Stability and conjugate points.- 12.4 The second variation formula.- 12.5 Analysis of the second variation formula.- 12.6 Length minimizing geodesics.- 13 Floer Homology.- 13.1 Near the entrance.- 13.2 Morse homology in finite dimensions.- 13.3 Floer homology.- 13.4 An application to geodesics.- 13.5 Towards the exit.- 14 Non-Hamiltonian Diffeomorphisms.- 14.1 The flux homomorphism.- 14.2 The flux conjecture.- 14.3 Links to “hard” symplectic topology.- 14.4 Isometries in Hofer’s geometry.- List of Symbols.

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