Infinitely Many Leaf-wise Intersection Points on Cotangent Bundles

In the article [AF08] we showed that for a special class of perturbations of the Rabinowitz action functional critical points give rise to leaf-wise intersection points. In this article we prove existence of infinitely many leaf-wise intersections points for generic Hamiltonian functions on simply connected cotangent bundles. Along the way we prove that the perturbed Rabinowitz action functional is generically Morse as we announced in [AF08]. 1. Main Result We consider a closed hypersurface Σ ⊂ (M,ω = dλ) in an exact symplectic manifold (M,ω) such that (Σ, α := λ|Σ) is a contact manifold. Moreover, we assume that Σ bounds a compact region in M and that M is convex at infinity, that is, M is isomorphic to the symplectization of a compact contact manifold at infinity. Σ is foliated by the leaves of the characteristic line bundle which is spanned by the Reeb vector field R of α. For x ∈ Σ we denote by Lx the leaf through x. Furthermore, we denote by Hamc(M,ω) the group of compactly supported Hamiltonian diffeomorphism φH generated by smooth, time-dependent Hamiltonian functions H. Given φH ∈ Hamc(M,ω), a leaf-wise intersection point of φH consists of x ∈ Σ with φH(x) ∈ Lx. These were introduced and studied first by Moser [Mos78]. For the history of the problem we refer the reader to [AF08]. Let (M,Σ) = (T ∗L,S∗ gL) be the (unit) cotangent bundle of a closed manifold L with respect to the Riemannian metric g. We denote by Hg the set of smooth, time-dependent Hamiltonian functions for which there exist infinitely many leaf-wise intersection points of φH . Theorem 1.1. Let L be simply connected. Then for a generic metric g the set Hg is generic in C∞(S1 ×M). 2. The Rabinowitz action functional and leaf-wise intersection points We consider here a more general set-up, where we drop the assumption of Σ being of contact type. Definition 2.1. A pair (F,H) of Hamiltonian functions F,H : S1×M −→ R is called Moser pair if they satisfy F (t, ·) = 0 ∀t ∈ [12 , 1] and H(t, ·) = 0 ∀t ∈ [0, 1 2 ] , (2.1) 2000 Mathematics Subject Classification. 53D40, 37J10, 58J05.