Extraneous fixed points,basin boundaries and chaotic dynamics for Schröder and König rational iteration functions

Summary The Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newton's method. In both schemes, iteration functionsf_m(z) are constructed so that sequencesz_n+1=f_m(z_n) converge locally to a rootz^* ofg(z) asO(|z_n–z^*|^m). It is well known that attractive cycles, other than the zerosz^*, may exist for Newton's method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsK_m(z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z²–1, Cayley's classical result for the basins of attraction of Newton's method is extended for allK_m(z). The existence of chaotic {z_n} sequences is also demonstrated for these iteration methods.