New version of the question:

Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form whose restriction to $\ker \alpha$ is non-degenerate. In particular, I don't ask that $\omega$ be closed. From this pair, I define a vector field $R$ with the properties that $\alpha(R) = 1$, and $\omega(R, \cdot) = 0$.

A class of examples come from hypersurfaces in a symplectic manifold. In those examples, the 2-form is the restriction of the ambient symplectic form (and is thus closed), and the 1-form is obtained by contracting the symplectic form with a normal vector field.

Can I deform the pair $(\alpha, \omega)$ so that the only invariant sets of the resulting vector field are non-degenerate periodic orbits?

More generally, does this class of vector fields have any rigidity to it?

Explanation of the change: The first version of this question restricted attention to the class of examples of hypersurfaces in a symplectic manifold. From what Alvarez Paiva commented, it seems unlikely that this is doable. I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.)

Why I am asking this: I started thinking about this question in trying to understand some examples related to the question of putting a contact structure on an almost contact manifold. I don't think that my line of thinking is related to the original question anymore, but I am still curious to understand how soft almost contact is (compared to contact).

New version of the question:

Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form whose restriction to $\ker \alpha$ is non-degenerate. In particular, I don't ask that $\omega$ be closed. From this pair, I define a vector field $R$ with the properties that $\alpha(R) = 1$, and $\omega(R, \cdot) = 0$.

A class of examples come from hypersurfaces in a symplectic manifold. In those examples, the 2-form is the restriction of the ambient symplectic form (and is thus closed), and the 1-form is obtained by contracting the symplectic form with a normal vector field.

Suppose $V$ is a closed manifold. Can I deform the pair $(\alpha, \omega)$ so that the only minimal invariant sets of the resulting vector field are non-degenerate periodic orbits?

More generally, does this class of vector fields have any rigidity to it?

Explanation of the change: The first version of this question restricted attention to the class of examples of hypersurfaces in a symplectic manifold. From what Alvarez Paiva commented, it seems unlikely that this is doable. I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.)

Why I am asking this: I started thinking about this question in trying to understand some examples related to the question of putting a contact structure on an almost contact manifold. I don't think that my line of thinking is related to the original question anymore, but I am still curious to understand how soft almost contact is (compared to contact).

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.

New version of the question:

Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form whose restriction to $\ker \alpha$ is non-degenerate. In particular, I don't ask that $\omega$ be closed. From this pair, I define a vector field $R$ with the properties that $\alpha(R) = 1$, and $\omega(R, \cdot) = 0$.

A class of examples come from hypersurfaces in a symplectic manifold. In those examples, the 2-form is the restriction of the ambient symplectic form (and is thus closed), and the 1-form is obtained by contracting the symplectic form with a normal vector field.

Can I deform the pair $(\alpha, \omega)$ so that the only invariant sets of the resulting vector field are non-degenerate periodic orbits?

More generally, does this class of vector fields have any rigidity to it?

Explanation of the change: The first version of this question restricted attention to the class of examples of hypersurfaces in a symplectic manifold. From what Alvarez Paiva commented, it seems unlikely that this is doable. I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.)

Why I am asking this: I started thinking about this question in trying to understand some examples related to the question of putting a contact structure on an almost contact manifold. I don't think that my line of thinking is related to the original question anymore, but I am still curious to understand how soft almost contact is (compared to contact).