This is an essay rather than a paper, but very suggestive for the study of periodic orbits variational problems.
\(S^1\)-invariant homology involves cycles having a boundary as singular chains, so not cycles of singular homology. Critical points at infinity investigated in this paper are critical points of periodic orbits variational problem for contact vector-fields realized by such cycles. Existence of critical points at infinity is shown taking the standard geodesic problem on \(S^2\) as an example (§2, Th.1). Then, the usefulness of contact form homology [A. Bahri, Classical and quantic periodic motions of multiply polarized spin-particles, Pitman Research Notes in Mathematics Series. 378. Harlow: Longman (1998; Zbl 0891.58004) and Flow lines and algebraic invariants in contact form geometry. Progress in Nonlinear Differential Equations and their Applications 53. Boston, MA: Birkhäuser (2003; Zbl 1079.58007)] to the topological study of such phenomena is asserted (§3). Suggestive studies of variational spaces on \(S^3\) with the standard contact structure \(\alpha_0\) to tie contact form homology and \(S^1\)-equivariant cohomology of the loop space are also presented (§5).
Critical points at infinity are studied via the index of critical points [cf. W. Kilingberg, Closed geodesics on surfaces of genus 0. Annali della Scuola Norm. Sup. di Pisa, 6 (1979)]. Let \(x_\mu\) be a critical point of index \(\mu\). Then the unstable manifold of \(x_{2\nu+1}\) is shown to contain the unstable manifold of \(x_{2\nu-1,S^1}\) (point to circle Morse relation) and the unstable manifold of \(x^\infty_{2\nu}= x^\infty_{2\nu-1,S^1}\) (point circle Morse relation, maybe for another \(S^1\)-action), in its boundary (Th.1). Studies in §2, show that the variational space of the variational problem \(J\) has an intersection operator \(\partial\) which mixes critical points and critical points at infinity. But it does not count the contribution of the critical circles. Usefulness of the contact form homology is asserted by this observation (§3. The definition of the contact form homology is sketched in this section). In the absence of periodic orbits, the index of E. R. Fadell and P. H. Rabinowitz [Generalized cohomological index theories for Lie group action with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978; Zbl 0403.57001)] of the critical set at infinity is lower than 1. This is proved in §4 (Th.2).
These results show the importance to recognize completely and “geometrically” the homology and homotopy classes involved in the existence of periodic orbits. To response to this quest, variational spaces on \(S^3\) with the standard contact structure \(\alpha_0\) are studied in §5. \(\alpha_0\) is the lift of the Liouville form on \({\mathbb P}{\mathbb R}^3\) and the periodic orbits of the Reeb vector-field of this Liouville form on the unit sphere cotangent bundle project onto the closed geodesics of \(S^2\). There are the one action functional and three variational spaces \(\Lambda(S^3)\), \(\Lambda({\mathbb P}{\mathbb R}^3)\) and \(\Lambda(S^2)\). The functional of geodesic variational problem on \(\Lambda(S^2)\) is derived from the action functional on \(\Lambda({\mathbb P}{\mathbb R}^3)\). Same variational problem is defined on the spaces \(\text{Imm}_i(S^1, S^2)\), \(i= 0,1\), the spaces of immmersed curves of Maslov index \(i\). They rebuilds the \(\Lambda({\mathbb R}{\mathbb P}^3)\), topologically and \(\text{Imm}_0(S^1,S-2)\) rebuild variational space \(C_\beta\) for \((S^3,\alpha_0)\). Let \(L_0\) be the subspace of \(C_\beta\) defined by the constraint \(\int^1_0 b=0\). Then the author conjectures existence of a decreasing pseudo-gradient for the action functional which leaves \(L_0\) invariant (§5. Conjectural Claim. ii)).
Some additional observations on critical points at infinity are stated in §6, the last section.