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DCDS

Unstable manifolds of critical points at infinity
in the variational problems relating to periodic orbits of Reeb
vector-fields in Contact Form Geometry are viewed in this paper
as part of the attaching maps along which these
variational problems attach themselves to natural
generalizations that they have. The specific periodic orbit problem
for the Reeb vector-field $\xi_0$ of the standard contact
structure/form of $S^3$ is studied;
the extended variational problem is
the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells.
Some circle and ''loop" actions on the loop space of $S^3$, that
are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space
of $S^2$, are also defined. ''Conjugacy" relations relating these actions are
established.

DCDS

After briefly recalling the pseudo-holomorphic approach in contact
form geometry and after sketching the ways with which this approach
defines invariants, we introduce another approach, of more technical
type, which starts with a variational problem on Legendrian curves.
We show how this approach leads also to the definition of a homology.

Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].

Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.

This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].

Ideally, this homology would be generated by a part of the Morse complex of the variational problem which would involve only periodic orbits. Because of the lack of compactness, it has some additional part which we had characterized in an earlier work [5].

Taking a variant of this approach, we give here a much more restrictive characterization of this additional part which should allow to compute it precisely.

This should indicate that the lack of compactness, seen as creation of additional punctures in the pseudo-holomorphic approach, is much more limited than what would be theoretically allowed and leaves hope that it can be completely computed. The proof of all our claims will be published in [6].

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