Structurally stable perturbations of polynomials in the Riemann sphere. (English) Zbl 1153.37329
Summary: The perturbations of complex polynomials of one variable are considered in a wider class than the holomorphic one. It is proved that under certain conditions on a polynomial \(p\) of the plane, the \(C^r\) conjugacy class of a map \(f\) in a \(C^{1}\) neighborhood of \(p\) depends only on the geometric structure of the critical set of \(f\). This provides the first class of examples of structurally stable maps with critical points and nontrivial nonwandering set in dimension greater than one.
MSC:
| 37C20 | Generic properties, structural stability of dynamical systems |
| 37C75 | Stability theory for smooth dynamical systems |
| 58K05 | Critical points of functions and mappings on manifolds |
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