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.
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" still valid?
Motivation: Theorem 9.3 page 29 of the following paper 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]( Finding a 1-form adapted to a smooth flow)
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".
