Elliptic operators corresponds to non vanishing vector fields

Question

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:

How to compute the index of such operator?

Answer 1

Perhaps you would be interested in Witten's proof of the Poincare-Hopf theorem. Given a smooth nondegenerate vector field $V$ on a smooth closed manifold $M$, the theorem asserts that the Euler characteristic of $M$ is equal to the sum of the signs of the critical points of $V$. Perhaps this isn't as interesting as the dynamical behavior that you mentioned in your question, but it's a start.

Witten's approach is to use $V$ to perturb the de Rham complex by replacing the de Rham differential $d$ with the operator

$$d_t = d + t i_v \colon\: \Omega^*(M) \to \Omega^*(M)$$

where $t$ is a real number and $i_V$ is the interior product with $V$. He looked at the corresponding perturbed de Rham operator $D_t = d_t + d_t^*$ (where the adjoint is defined using a choice of Riemannian metric) and as usual viewed it as a graded Dirac-type operator on the graded Clifford module $\Omega^*(M)$. $D_t$ is elliptic and hence Fredholm, and since the index of an operator is determined by its symbol class the index of $D_t$ is just the index of the usual de Rham operator $D$ which is the Euler characteristic of $M$.

On the other hand, one can calculate that $$D_t^2 = D^2 + t^2 ||V||^2 + t T$$ where $T$ is some bundle map. For large values of $t$ the potential term $t^2 ||V||^2$ becomes very large except in a tiny neighborhood of the critical set of $V$, so one can show that the eigenvectors of $D_t$ concentrate near the critical set. Combining this observation with the McKean-Singer formula for the index of $D_t$ and some asymptotic analysis proves the Poincare-Hopf theorem.

There are a variety of generalizations of this result in the literature - perturbing other operators, relaxing the nondegeneracy assumption, etc. I don't know this literature too well and so I don't quite know how much dynamics to expect, but it's worth a look.

Answer 2

The following does not answer your question directly, but I could not resist writing it down.

Some interesting properties of $X$ will arise if you consider the operator $F_\epsilon(g)=D(g)+\epsilon\Delta g$ and let $\epsilon\to0$. (Here, $D$ is your $D$ and $\Delta$ is the Laplace--Beltrami or you can replace it with any other uniformly elliptic 2-nd order operator)

This has a probabilistic interpretation of adding a small noisy perturbation to the dynamical system and then letting the noise amplitude go to zero. Of course, over finite time intervals the perturbed dynamics converge to the unperturbed deterministic motion, but over infinite time horizon there are often interesting residual effects after "zeroing" the noise.

One keyword is "Freidlin--Wentzell theory".