# zbMATH — the first resource for mathematics

On real analytic planar vector fields near the characteristic set. (English) Zbl 0960.35016
Grinberg, E. L. (ed.) et al., Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 251, 429-438 (2000).
Let $$L= A(x,y){\partial\over\partial x}+ B(x,y){\partial\over\partial y}$$ be a planar vector field, where $$A$$, $$B$$ are $$\mathbb{C}$$-valued and real analytic functions, defined in the annulus $$\{(x,y)\in \mathbb{R}^2$$, $$1- a<\sqrt{x^2+ y^2}< 1+ a$$, $$a\in (0,1)\}$$. Let $$S^1$$ be the characteristic set of $$L$$, i.e. the $$L_p$$ and $$\overline L_p$$ are dependent on $$S^1$$. The author has proved that the vector field $$L$$ of finite type along $$S^1$$ is equivalent near $$S^1$$ to the rotationally invariant vector field $$R_k= {\partial\over\partial r}- ik(r- 1)^{k-1}{\partial\over \partial\vartheta}$$, where $$(r,\vartheta)$$ are the polar coordinates and $$L$$ is of finite type $$k-1$$. Further, the vector field $$L$$ of infinite type along $$S^1$$ is equivalent to the rotationally invariant vector field $$X_q= q(r- 1){\partial\over\partial r}+ i{\partial\over\partial\vartheta}$$, where $$q$$ is a rotation number.
For the entire collection see [Zbl 0941.00009].

##### MSC:
 35F05 Linear first-order PDEs 37C10 Dynamics induced by flows and semiflows 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
##### Keywords:
differentiable equivalence; first integral; uniformization