The Omega limit set of a family of chords

Abstract

In this paper we study the limit behavior of a family of chords on compact energy hypersurfaces of a family of Hamiltonians. Under the assumption that the energy hypersurfaces are all of contact type, we give results on the Omega limit set of this family of chords. Roughly speaking, such a family must either end in a degeneracy, in which case it joins another family, or can be continued.This gives a Floer theoretic explanation of the behavior of certain families of symmetric periodic orbits in many well-known problems, including the restricted three-body problem.

Bifurcations of symmetric periodic orbits via Floer homology

“…) is a family of non-degenerate Reeb chords such that the family does not extend over 0 and such that the ω-limit set Ω ⊂ Σ 0 is isolated in the set of Reeb chords on Σ 0 , then we have the same conclusion as the theorem, see [7,Theorem B]. Note that Belbruno, Frauenfelder, and van Koert do not use the local eLFRH, but use an elementary method based on Floer's stretching method for time-dependent gradient flow lines, which does not require gluing.…”

The Limit Set of a Family of Periodic Orbits

“…Thus there must exist a second smooth family of parametrised periodic orbits in U with the same limit set K as in Figure 3. See also [BFK19,Theorem 6.1]. It should also be interesting to study bifurcations via SFT-Euler characteristic, see [FM21].…”