Let X=P∂∂x+Q∂∂y be a polynomial vector field on R2. Consider the following (Moyal) operator on C[x,y]:
˜DX(f)=fx∗P+fy∗Q where ∗ is the Moyal product. This is a noncommutativized version of the standard derivation DX(f)=fxP+fyQ:
I have two questions:
1) Motivated by the Leibniz formula for DX, I ask that: what is a natural formula for ˜DX(f∗g)=?
2)Can the number of limit cycles of X be bounded by the codimension of the range of ˜DX?
In particular is the following statement, true?
- For a limit cycle γ of a vector field X on the plane and for an smooth function f on R2, there is a point p on γ such that ˜DX(f)(p)=0.
A (commutative) motivation for the second question is the question in the following post
Note The * statement is a Moyal version of a key lemma to prove the commutative version of the second question. By commutative version I mean we replace the Moyal product by the usual product.
This post imported from StackExchange MathOverflow at 2015-01-10 19:26 (UTC), posted by SE-user Ali Taghavi