Symplectic Invariants and Hamiltonian Dynamics pp 143-192 | Cite as

# Geometry of compactly supported symplectic mappings in ℝ^{2n}

## Abstract

In this chapter we shall study the group *D* of those symplectic diffeomorphisms of ℝ^{2n} which are generated by time-dependent Hamiltonian vector fields having compactly supported Hamiltonians. We shall construct, in particular, an astonishing bi-invariant metric *d* on *D*, following H. Hofer [116]. Defining the energy *E(ψ)* of an element *ψ* ∈ *D* by means of the oscillations of generating Hamiltonians, the metric *d* will be defined by *d(ϕ, ψ)* = *E(ϕ*^{−1}*ψ*). It is of *C*^{0}-nature. The verification of the property that *d(ϕ, ψ)* = 0 if and only if *ϕ* = *ψ*, is not easy. It is based on the action principle. The metric *d* is intimately related, on the one hand, to the capacity function *c*_{0} introduced in Chapter 3 and hence to periodic orbits and, on the other hand, to a special capacity *e* defined on subsets of ℝ^{2n} and satisfying *e(U)* ≥ *c*_{0}*(U)*. The capacity *e* is called the displacement energy: *e*(*U*) measures the distance between the identity map and the set of ψ ∈ *D* which displaces *U* from itself, in the sense that *ψ(U)* ∩ *U* = ∅. A crucial role in our considerations will be played by the action spectrum, *σ (ψ)*, of the fixed points of an element *ψ* ∈ *D*. It turns out to be a compact nowhere dense subset of ℝ. In contrast to the simple variational technique used for a fixed Hamiltonian in Chapter 3, a minimax principle will be designed which is applicable simultaneously to all Hamiltonians generating the elements of *D*. It singles out a distinguished action *γ(ϕ)* ∈ σ*(ϕ)*. The map γ : (*D, d*) → ℝ is a continuous section of the action spectrum bundle over *D*; it is the main technical tool in this section. Its properties will also be used in order to establish infintely many periodic orbits for certain elements of *D*, and to describe the geodesics of (*D*, *d*). We mention that the completion of the group *V* with respect to the metric *d* is not understood.

## Keywords

Periodic Solution Periodic Orbit Special Capacity Bifurcation Diagram Group Versus## Preview

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