Added, June 19, 2019:The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting dynamical interpretation. For example we hope that the index can help us to find an upper bound for the number of attractors of a dynamical system. According to comment conversations in this post we realize that ellipticity or hypoelipticity is a very relevant or perhaps a necessary conditions for existence of "Index". Now the subject and materials of this recently hold conference, "Fredholm theory of Non elliptic operatores seems to be related to this post.

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately this operator is not an elliptic operator.

From the dynamical view point,what type of **elliptic** operators, or at least Fredholm diff. operators, can be associated with $X$?

I mean, for a given non vanishing vector field $X$, what interesting elliptic operator $D$ can be constructed such that its fredholm index contains some information about the dynamical behavior of $X$. For example: the number of attractores, or the number of isolated compact invariant sets, etc..

**EditL:** For a possible related post see the following:

Topics in pseudo-differential operators. 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) pp. 167–305 Edizioni Cremonese, Rome $\endgroup$ – Liviu Nicolaescu Oct 6 '14 at 16:47