Holomorphic motions in the parameter space for the relaxed Newton's method.

Abstract

It is a well known fact, that for certain polynomials $f$ the relaxed Newton's method $N_{f,h}(z) = z - h\frac{f(z)}{f'(z)}$ associated with $f$ has some extraneous attracting cycles. In the case of cubic polynomials the set of these bad conditioned polynomials has been intensively studied and described by means of quasi--holomorphic surgery and holomorphic motions, cf.~\cite{haeseler:1988}. In the present paper we will generalize this description to polynomials of higher degree.