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Let $\alpha$ be a contact form on $S^3$, let $\xi$ be its Reeb vector-fieldand let $v$ be a non-singular vector-field in $ker\alpha$. Let $C_\beta$ be thespace of curves $x$ on $S^3$ such $\dot x=a\xi+bv, \dot a=0, a \gneq 0$. Let$L^+$, respectively $L^-$, be the set of curves in $C_\beta$ such that $b\geq0$, respectively $b \leq 0$. Let, for $x \in C_\beta$,$J(x)=\int_0^1\alpha_x(\dot x)dt$. We establish in this paper that an infinitenumber of cycles in the $S^1$-equivariant homology of $C_\beta$,{\bf relative}to $L^+ \cup L^-$ and to some specially designed "bottom set", see section 4,are achieved in the Morse complex of $(J, C_\beta)$ by unions of unstablemanifolds of critical points (at infinity)which must include periodic orbits of$\xi$; ie unions of unstable manifolds of critical points at infinity alonecannot achieve these cycles. The topological argument of existence of aperiodic orbit for $\xi$ turns out to be surprisingly close, in spirit, to thelinking/equivariant argument of P.H. Rabinowitz in [12]. The objects and theframeworks are strikingly different, but the original proof of [12] can berecognized in our proof, which uses degree theory, the Fadell-Rabinowitz index[8] and the fact that $\pi_{n+1}(S^n)=\mathbb{Z}_2, n\geq 3$. The argumentshold under the basic assumption that no periodic orbit of index $1$ connects$L^+$ and $L^-$. To a certain extent, the present result runs, especially inthe case of three-dimensional overtwisted [8] contact forms, against theexistence of non-trivial algebraic invariants defined by the periodic orbits of$\xi$ and independent of what $ker \alpha$ and/or $\alpha$ are.

A Linking/$S^1$ Equivariant Variational Argument in the space of Dual  Legendrian curves and the proof of the Weinstein Conjecture on $S^3$ "in the  large"