Abstract
Let α be a contact form on S 3 , let ξ be its Reeb vector-field and let v be a non-singular vector-field in ker α. Let C β be the space of curves x on S 3 such ẋ = aξ + bv, ȧ = 0, a ≩ 0. Let L + , respectively L − , be the set of curves in C β such that b ≥ 0, respectively b ≤ 0. Let, for x ∈ C β , J(x) = ∫ 0 1 α x (ẋ)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1 - equivariant homology of C β , relative to L + ∪ L − and to some specially designed ”bottom set”, see section 4, are achieved in the Morse complex of (J,C β ) by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of ξ; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k−1) = 1+(2k−2), 1 for the linking, (2k−2) for the S 1 -equivariance, we find that the equivariant contributions of a critical point at infinity to L + and to L − are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for ξ turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the ”bottom sets” of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that π n+1 (S n ) = Z 2 , n ≥ 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of ξ excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L + and L − . It therefore follows from the present work that either a periodic orbit of index 1 connects L + and L − (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of ξ, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.