Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid  \alpha_{| D}=d\alpha_{|D}=0\}.$$
When $D$ is an integrable distribution, the last condition $d\alpha_{|D}=0$ is redundant.
The quotient complex $\Omega^*(M)/\Omega^*(M,D)$ produces a cohomology. In this 
 complex we can replace the space of coboundaries(exact forms along distribution) with its smooth topology closure. The reason for closure is explained in [this paper which is linked in the MO post in the next paragraph](https://arxiv.org/pdf/math/0204110.pdf).

So we get a cohomology $H^*(M,D)$, the cohomology of $M$ along distribution $D$.

Now assume that the flow of a vector field $\tilde{X}$ on $M$ preserves $D$.

>Is the dynamical Lefschetz trace formula discussed in ["Hilbert 16th problem and  dynamical Lefschetz trace formula"](https://mathoverflow.net/questions/333425/hilbert-16th-problem-and-dynamical-lefschetz-trace-formula) still valid?



**Motivation:**  Theorem 9.3 page 29 of [the following paper](http://www.mi-ras.ru/~snovikov/23.pdf) says that if a vector field on $S^3$ is transversal to a codimension one foliation, then the vector field must have a parameter family of periodic orbits. On the other hand one of the main concerns of the above MO post is finiteness of closed orbits for a vector field on $S^3$ which is a result of lifting of a (generic) vector field on 2-sphere. So we should almost give up finding a foliation transversal to our vector field. **Instead of a foliation, we search a (non integrable) transversal distribution which is preserved by the flow of our vector field.** So we need an appropriate analogy of leafwise de Rham cohomology when there is not any leaf, that is a non-integrable distribution. The above cohomology we introduce is a candidate for such appropriate cohomology.
 


So we hope that the lifting $\tilde{X}$ (to $S^3$ or $S^2\times S^1$) of a Poincare compactification of a quadratic plannar vector field would be a geodesible vector field. Then the orthogonal direction $D$(with respect to an appropriate metric adapted to $\tilde{X}$) is obviously preserved by flow of $\tilde{X}$. 


(For more explaination on the later paragraph please  see  [this post](
https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273648#273648)

So we try to investigate the dynamics of $\tilde{X}$ via a possible distribution analogy of Dynamical trace formula).

>Precise and detailed motivations are mentioned in the above linked MO post entitled "Hilbert 16th problem and dynamical Lefschetz trace formula".