Structurally stable perturbations of polynomials in the Riemann sphere

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 Cr$C^r$ conjugacy class of a map f   in a C¹$C^1$ 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.