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?

The Laplacian plus the operator you mention is an elliptic operator associated with the vector field. Do you want the operator to be of first order or to depend linearly on the vector field? Any additional assumptions would help. — Joonas Ilmavirta, Oct 02 '14 at 22:13
@JoonasIlmavirta Ah! I realy thank you very much for your comment. As you said $\Delta +D_{X}$ is a an elliptic operator.Now my next question is "what is the dynamical interpretation for the fredholm index of this ellitic operator". My main motivation: The codimension of the range of derivational operator is an upper bound for the number of limit cycles of $X$. But two difficulities: D is not elliptic the second $D$ is not fredholm in the case of existence of a limit cycle surrounding a non resonance singularities. Thanks again for your interesting comment. I wrote a related motivation in — Ali Taghavi, Oct 02 '14 at 22:25
...in http://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators — Ali Taghavi, Oct 02 '14 at 22:25
@JoonasIlmavirta do you think there are some dynamical interpretation for the fredholm index of the elliptic operator which you proposed? What about if the laplacian correspond to a metric which has some compatibility with $X$? By compatibility I mean some situation like thihttp://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsgeodesiable-flows: — Ali Taghavi, Oct 02 '14 at 22:30
I'm not exactly sure what you mean by a dynamical interpretation, but my first intuition is to consider the Hamiltonian flow associated with the operator. The flow associated with the Hamiltonian is the geodesic flow (a free particle), and the extra term adds a force described by the vector field to the equations of motion. I am not familiar enough with the Fredholm index in this context to be able to say anything meaningful about it. — Joonas Ilmavirta, Oct 03 '14 at 04:50
@JoonasIlmavirta By dynamical interpretation, I mean , for example"The number of attractors of $X$" In the second part of this note I explained some thing related to this concept http://arxiv.org/abs/1302.0001. In this note I tried to make a remedy for non elliptic ness of the derivational operator. But what do you mean by "Hamiltonian flow associated with the operator"? — Ali Taghavi, Oct 03 '14 at 12:06
@JoonasIlmavirta As another example of dynamical interpretation for certain diff operator see the Veku theorem in .(Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic
physics, by B. Booss and D. D. Bleecker) — Ali Taghavi, Oct 03 '14 at 12:41
The $C^\infty(M)$-algebra of scalar differential operators generated by $L_X$, the Lie derivative along $X$, contains no elliptic operator if $\dim M>1$. — Liviu Nicolaescu, Oct 03 '14 at 12:59
Consider the Hamiltonian function $H:T^M\to\mathbb R$ defined by $H(x,p)=\frac12g_{ij}(x)p^ip^j+X_i(x)p^i$. The corresponding Hamiltonian flow on the cotangent space is a dynamical system corresponding to $X$. If $X=0$, you get geodesics. A simpler dynamical system corresponding to $X$ is the system $\dot x=X(x)$ on $M$, but this does not look "as elliptic" as the Hamiltonian one. Do you want the dynamical system to live on $M$ itself rather than $T^M$? The dynamical viewpoint would be clearer if you gave the dynamical system (if you have one). — Joonas Ilmavirta, Oct 03 '14 at 14:28
why you assume $X$ is non vanishing? If not, for example $X=-\nabla f$, where $f$ is a Morse function. There is a Laplacian-like operator, called Witten Laplatian, related the stable/unstale cells of the dynamical and the cohomology of maniflold. — shu, Oct 04 '14 at 09:31
@LiviuNicolaescu Thanks for the comment.where is a proof of this statement?is it elementary?What about the algebra of diff operatores on $\Gamma TM$ generated by $\nabla_{X}$ or $[X,.]$?I seach for an operator associated with $X$ such that some interesting quantity of this operator can count the number of attractors. — Ali Taghavi, Oct 04 '14 at 20:08
@shu thanks for your information on Witten Laplacian. My reason that I assumeed the nonvanishing condition is the following; Some times ago a researcher suggested me to consider the operator $D(U)=PU_{x}+QU_{y}+i(QU_{x}-PU_{y})$ as complex diff operator associated with vector field $P\partial_{x}+Q\partial_{y}$. This operator is elliptic at non singular points of $X$. This situation was my main motivation to consider the following question..... — Ali Taghavi, Oct 04 '14 at 20:18
... the following question http://mathoverflow.net/questions/182139/lifting-a-quadratic-system-to-a-non-vanishing-vector-field-on-s3 — Ali Taghavi, Oct 04 '14 at 20:19
@ Ali Compute the principal symbol and notice it is not invertible. It vanishes for any covector $\xi$ such that $\xi(X)=0$. — Liviu Nicolaescu, Oct 04 '14 at 20:50
@LiviuNicolaescu another question: Am i right to think that the index of every operator $D_{X}+\Delta$, the same operator which proposed byJoonas, is zero where $D_{X}$ is the derivation operator — Ali Taghavi, Oct 04 '14 at 20:53
@LiviuNicolaescu what type of nonelliptic operatores are known to have finite fredholm index? — Ali Taghavi, Oct 06 '14 at 15:44
On a compact manifold a partial differential operator is Fredholm if and only if it is elliptic. — Liviu Nicolaescu, Oct 06 '14 at 16:30
@LiviuNicolaescu and what type of results on non elliptic on non compact without boundary? — Ali Taghavi, Oct 06 '14 at 16:35
You can find this in Theorem 5, Chapter IV of R. Seeley's memoir Topics in pseudo-differential operators. 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) pp. 167–305 Edizioni Cremonese, Rome — Liviu Nicolaescu, Oct 06 '14 at 16:47
@ Ali You first need to understand well the meaning of ellipticity before you attack more complicated issues. — Liviu Nicolaescu, Oct 06 '14 at 16:50
@LiviuNicolaescu I think it is a linear map $D$ on $\Gamma E$ for which principle symbole is an invertible bundle morphism on $q^{} E$ where $q:T^{}M\to M$ is the natural projection. The definition of principle symbol is written clearly in Nakahara. Are you meaning some other complicated definition of ellipticity? — Ali Taghavi, Oct 06 '14 at 16:55
@LiviuNicolaescu for a covector $\Xi$ and a section $s$ $P(D)(\Xi,s)=D(1/n!fs)$ where $f$ is a function with $df=\Xi$. are you meaning some more complicated definition of ellipticity? — Ali Taghavi, Oct 06 '14 at 17:01
@LiviuNicolaescu I enter the chat but I am not sure that it works. Did you received my message? — Ali Taghavi, Oct 06 '14 at 17:13
@LiviuNicolaescu According to your comment "On a compact manifold a partial differential operator is Fredholm if and only if it is elliptic" what is the error of the following statement:Let $f:S^{2}\to \mathbb{R}$ be a morse function with exactly two critical points a minimum at S and a maximum at N(for example $f(x,y,z)=z$). Now consider the gradient vector field $X=\nabla f$. Now I think the derivational operator $D_{X}$ is a non elliptic operator on $C^{\infty}(M)$ which is a fredholm operator(of codimension 2 and the kernel is one dimensional). Am I mistaken? — Ali Taghavi, Oct 07 '14 at 08:49
@LiviuNicolaescu Now I reviewed your definition of symbol in your lecture on Atiyah singer index theorem.I have a question: you spend at least one or two pages to define the symbol. it seems that you take it very difficult. why you do not simply define the symbol as in the Nakahara book or in a paper of of Atiyah?(As I wrote in the above comments). What is the advantage of your complicated definition? Is there a real application and a motivation for this complication? — Ali Taghavi, Oct 07 '14 at 19:25
@ChrisGerig Are you considering the smooth function? That is : do you believe that the codimension is infinite if we consider the operator on smooth functions? — Ali Taghavi, Mar 04 '15 at 08:52
Yes; I think the spherical harmonics won't be in the image, because they can't be integrated (when solving for the corresponding function in the domain). For example, in order for $D_X(f)=\cos\phi$ you need $f=\sin^2\theta\ln|\sin\phi|$ which blows up at $\phi=0,\pi$. — Chris Gerig, Mar 04 '15 at 18:08
@ChrisGerig But what is the error of the following argument which shows the codimension is "2"?: Let $g$ be an smooth function with $g(N)=g(S)=0$ then $D_{X}(f)=g$ for $f(x)=\int_{-\infty}^{+\infty} g(\phi_{t}(x))dt$ where $\phi$ is the flow of the vector field.$ Since the singularities are hyperbolic, then this integral is well defined(converges). In this note I tried to extend this idea to count the number of limit cycles as a fredholm index. However there is a gap in this not but a true (and weaker) version... — Ali Taghavi, May 12 '15 at 16:40
I don't know enough to turn this into an answer, but you might have a look at some of the stuff that symplectic geometers and low dimensional topologists are doing. Some of their most powerful tools - like symplectic homology and Heegaard-Floer homology - are based on index theory for nonlinear Fredholm operators, and there are deep connections with dynamical systems. — Paul Siegel, Jul 06 '15 at 12:48
I just want to reiterate what Liviu Nicolaescu has already said: Any PDO defined using only the vector field $X$ and no other PDEO is never elliptic. The best way to understand this is to look up the most general definition of an elliptic partial differential operator and test it against examples such as $\nabla_X$, $[X,\cdot]$, and any other example you can think of. Any PDO defined using only $X$ is essentially an ODE along the integral curves of $X$. If such an operator is Fredholm, it is due to the global dynamics of the operator and not a local property of the PDO such as ellipticity. — Deane Yang, Jul 07 '15 at 22:27
@PaulSiegel Thanks for your very interesting comment. I do not know why I did not realized your comment. I did not received any announce for your comment. — Ali Taghavi, Aug 05 '15 at 05:19
@DeaneYang Thanks for your very interesting comment. I had the same problem with your comment as I explained above.Can I ask you to more explain on your last statement. — Ali Taghavi, Aug 05 '15 at 05:20
@JoonasIlmavirta What about $\Delta +\partial^{2}/\partial X^{2}$? Is its index independent of choosing a vector field?Is it always elliptic? — Ali Taghavi, Aug 05 '15 at 05:27
@AliTaghavi, that is always elliptic but I don't know about the index off the top of my head. — Joonas Ilmavirta, Aug 05 '15 at 06:53
@LiviuNicolaescu Is the index of the operator in my previous comment independent of $X$? — Ali Taghavi, Aug 05 '15 at 07:04
@JoonasIlmavirta any way your first comment on this question was very interesting for me and that comment is a motiviation to this home genus second order operator. — Ali Taghavi, Aug 05 '15 at 07:07
Ali, I assumed that you wanted the PDO to be first order and the top order term to be $X$. — Deane Yang, Aug 05 '15 at 15:33
@DeaneYang According to my previous comment is the index of $\Delta +\epsilon \partial^{2}/\partial X^{2}$ independent of $X$ and $\epsilon$, hence $0$? So this operator is useless from dynamical view point? — Ali Taghavi, Aug 06 '15 at 08:31
@PaulSiegel can I ask you to introduce me a precise reference which need the minimum background. — Ali Taghavi, Aug 06 '15 at 08:50
A couple of quick little comments: 1) The operator $\Delta + \epsilon X^2$ depends not only on $X$ but on the Riemannian metric used to define $\Delta$. 2) Perhaps a better thing to look at is $X^2 + \epsilon^2 \Delta$ and ask what happens as $\epsilon \rightarrow 0$? — Deane Yang, Aug 16 '15 at 18:25
@DeaneYang Do you mean the index of $\Delta+\epsilon X^{2}$ depends on $X$ and $\epsilon$? If yes, what is the mistake of the following argument. $\Delta=\epsilon X^{2}$ is a path of fredholm operator hence the index is fix. — Ali Taghavi, Aug 17 '15 at 22:01
@AliTaghavi For the last ten years, Bismut has been working on so-called hypoelliptic operators related to the geodesic spray on $T^*M$, see [http://www.math.u-psud.fr/~bismut/]. This is a manifold version of the Fokker-Planck equation, and it gives Fredholm operators. Maybe, you can set up similar operators on $M$ itself under nice assumptions on $X$? — Sebastian Goette, Nov 13 '15 at 07:19
@SebastianGoette I sincerely thank you very much. I just realize that there are non elliptic operator which are Fredholm. I confess that I did not pay good attention to your valuable comments. Thanks a Lot. i hope that it works for dynamical interpretations for a vector field. — Ali Taghavi, May 15 '17 at 13:51
@DeaneYang Thank you for interesting suggestion $X^2+ \epsilon \Delta$ It seems that the index is unbounded when epsilon goes to zero. — Ali Taghavi, Jun 03 '17 at 06:14
@DeaneYang because the codimension of the range $X^2$ is infinite in case of existence of at least two attractors(either limit cycle or singularity — Ali Taghavi, Jun 03 '17 at 06:38
Please see the very interesting comment by Lukas Geyer https://math.stackexchange.com/questions/1163800/elliptic-and-fredholm-partial-differential-operators — Ali Taghavi, Jun 03 '17 at 06:40
and the last version of the following https://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators — Ali Taghavi, Jun 03 '17 at 06:41
after about 13 years i am still wonder to find an appropriate operator associated to a vector field whose index or other quantities can be used to count the number of limit cycles — Ali Taghavi, Jun 03 '17 at 06:43
@JoonasIlmavirta I found the Sibreg _Witten equation very similar to your idea. Can one find some relations in this contex? Thanks again for your interesting comment https://www.physicsoverflow.org/41645/the-seiberg-witten-equations-for-vector-fields — Ali Taghavi, Oct 26 '18 at 10:09
@LiviuNicolaescu may I ask you to read the update version of this question? — Ali Taghavi, Jun 29 '19 at 20:24

score 10 · Answer 1 · answered Jul 06 '15 at 12:42

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.

I suppose this is actually quite similar to Yuri Bakhtin's answer, but perhaps the added focus on index theory will be helpful. — Paul Siegel, Jul 06 '15 at 12:50
thank you very much for your interesting answer. I try to learn it. — Ali Taghavi, Jul 07 '15 at 13:56

score 7 · Answer 2 · answered Oct 03 '14 at 20:20

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".

Thank you very much for your answer. I try to understand this interesting probabilistic interpretation you mentioned. — Ali Taghavi, Oct 04 '14 at 19:49

Elliptic operators corresponds to non vanishing vector fields

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