Geometry of compactly supported symplectic mappings in ℝ2n
In this chapter we shall study the group D of those symplectic diffeomorphisms of ℝ2n 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 . 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−1ψ). It is of C0-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 c0 introduced in Chapter 3 and hence to periodic orbits and, on the other hand, to a special capacity e defined on subsets of ℝ2n and satisfying e(U) ≥ c0(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.
KeywordsPeriodic Solution Periodic Orbit Special Capacity Bifurcation Diagram Group Versus
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