We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M . This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses LiuTian's construction of S 1 -invariant virtual moduli cycles. As a corollary, we find that any semifree action of S 1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of M . We also establish a version of the area-capacity inequality for quasicylinders.

In this paper, we examine the homotopy classes of positive loops in Sp (2) and Sp(4). We show that two positive loops are homotopic if and only if they are homotopic through positive loops. The Behavior of Positive Paths and Lifting LemmasA useful tool for describing the movement of eigenvalues along a positive path is the splitting number. The notion of splitting number arises from Krein theory, described in [2] and [3], and is explained further in Lalonde and McDuff [6]. They define the non-degenerate Hermitian symmetric form β on C 2n by β(v, w) = −iw T Jv where J is the standard 2n × 2n block matrix with the identity in the lower left box and minus the identity in the upper right box. They prove theHence, for any simple eigenvalue λ ∈ S 1 we may define σ(λ) = ±1 where β(v, v) ∈ σ(λ)R + . Using properties of β, we can check that σ(λ) = −σ(λ). As an illustration, when n = 1, the matrix J = 0 −1 1 0

No abstract

No abstract