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(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. [12]. In the present paper we will generalize this description to polynomials of higher degree.