> Suppose I have a symplectic manifold $(M, \omega)$ and a (real codimension 1) hypersurface $\Sigma$ with chosen normal vector-field $\nu$.  This induces a vector field on $\Sigma$, which I call $R = R(\Sigma, \nu)$ (note that changing $\nu$ only scales my $R$). By deforming $\Sigma$ to $\Sigma'$ through embedded hypersurfaces, can I arrange so that the only invariant sets of the flow of $R'$ on $\Sigma'$ are non-degenerate periodic orbits?

**Why I am asking this:** I am thinking about this question because I am trying to understand what some $h$-principle arguments say in a concrete example.  Giroux recently announced a proof that any almost contact manifold admits a contact structure.  A key step was a lemma he called the "multiple bypass surgery lemma".  I don't understand the proof sketch of this lemma, so I thought I would try to work through the details in a toy example... I hit this hurdle in trying to get it to work, though I may be taking the wrong path entirely.