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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].

35F05 Linear first-order PDEs
37C10 Dynamics induced by flows and semiflows
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems